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Embedding planar 5-graphs in three pages
Discrete Applied Mathematics xxx (xxxx) xxx
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Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Embedding planar 5-graphs in three pages Xiaxia Guan, Weihua Yang
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Department of Mathematics, Taiyuan University of Technology, Taiyuan Shanxi 030024, China
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Article history: Received 7 August 2018 Received in revised form 5 November 2019 Accepted 25 November 2019 Available online xxxx Keywords: Book embedding Planar graph Pagenumber
a b s t r a c t A book embedding of a graph G is an embedding of its vertices along the spine of a book, and an embedding of its edges to the pages so that no two edges on the same page cross. A planar graph of maximum degree k is called a planar k-graph. Bekos et al. described an O(n2 ) time algorithm for computing two-page book embeddings for planar 4-graphs. In this paper, we embed planar 5-graphs into books of three pages by an O(n2 ) time algorithm. © 2019 Elsevier B.V. All rights reserved.
1. Introduction The concept of a book embedding of a graph was introduced by Atneosen [1]. A book consists of a line called the spine and some half-planes called pages, sharing the spine as a common boundary. A book embedding of a graph G = (V , E) consists of a (linear) layout L of its nodes along the spine ℓ (for simplicity we use ℓ to denote the spine sometimes) of a book (i.e., L : V → {1, 2, . . . , n}) and the assignment of each edge to the pages so that two edges embedded on the same page do not cross. We say that two edges (a, b) and (c , d) such that L(a) < L(b) and L(c) < L(d) on the same page cross in the layout L if and only if L(a) < L(c) < L(b) < L(d) or L(c) < L(a) < L(d) < L(b). We also say an edge (a, b) (L(a) < L(b)) nests a vertex v if and only if L(a) < L(v ) < L(b) or L(b) < L(v ) < L(a), and an edge (a, b) nests an edge (c , d) (L(c) < L(d)) if and only if L(a) < L(c), L(d) < L(b) or L(b) < L(c), L(d) < L(a). A central goal in the study of book embedding is to find the minimum number of pages in which a graph can be embedded, and find an algorithm to embed the graph into a book with the minimum number of pages. The minimum number of pages in which a graph can be embedded is called the pagenumber or book thickness or stack number of the graph. Determining the pagenumber of a graph G is a hard problem. It remains a difficult problem even when the layout L is fixed, since determining if a given layout admits a k-page book embedding is NP-complete [23]. Book embedding of graphs plays an important role in several technical applications, for examples, direct interconnection networks [17], VLSI design [8], fault-tolerant processor arrays [21], sorting with parallel stacks [11], single-row routing [22], and ordered sets [19]. The book embedding of graphs has been discussed for many graph families, see [4,5,10,18]. The most famous ones are the planar graphs. Bernhart and Kainen [4] firstly characterized the graphs with pagenumber one as the outerplanar graphs and the graphs with pagenumber two as the sub-Hamiltonian planar graphs (the subgraphs of planar Hamiltonian graphs). Deciding whether the pagenumber of general planar graphs is two is NP-hard [8]. Moreover, Bernhart and Kainen [4] conjectured that planar graphs have unbounded pagenumber, but this was disproved in [7] and [13]. Buss and Shor [7] proposed a nine-page algorithm. Heath [13] reduced the number to seven. Istrail [16] found an algorithm that embeds planar graphs in six pages. Later, Yannakakis [25] showed that planar graphs admit a four-page book embedding, which ∗ Corresponding author. E-mail address: [email protected] (W. Yang). https://doi.org/10.1016/j.dam.2019.11.020 0166-218X/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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can be constructed in linear-time. Yannakakis [24] claimed that four pages are necessary without giving a formal proof for this claim. Later, Dujmovic and Wood [9] conjectured that the pagenumber of planar graphs is four. Bekos et al. [3] also posed the same conjecture. So far, the pagenumber of planar graphs is still open. One natural direction is to consider the pagenumber of specific planar graphs. A planar graph of maximum degree k is called a planar k-graph. The pagenumber of planar k-graphs is clearly at most four. Ewald [12] proved that maximal planar graphs of maximum degree at most 6 are Hamiltonian, hence, their pagenumber is two. However, there are planar k-graphs (k ≤ 6) that are not subgraphs of maximal planar graphs of maximum degree at most 6, i.e., the pagenumber of planar k-graphs (k ≤ 6) may not be two. Heath [14] found a linear algorithm that embeds planar 3-graphs in two pages. Later, Bekos et al. [2] described an O(n2 ) algorithm to embed planar 4-graphs into books of two pages. That is, the pagenumber of planar 3-graphs or planar 4-graphs is two. Bekos et al. also posed a question whether the result can be extended to planar 5-graphs. In this paper, we embed planar 5-graphs into three pages by an O(n2 ) time algorithm, and thus the pagenumber of planar 5-graphs is three at most. The rest of the paper is organized as follows. Section 2 gives an algorithm that every planar 5-graph admits a three-page book embedding and its correctness is proved in Section 3. 2. The algorithm In this section, we describe an algorithm to embed planar 5-graphs in three-page books. The algorithm is similar to that of Bekos et al. [2], which is given by a recursive combinatorial construction. Our algorithm is different from the idea in [25] by Yannakakis that embeds planar graphs into four pages. Several basic terms and further definitions are necessary. A graph is said to be planar if it can be drawn in the plane so that its edges do not cross. Such a drawing of a planar graph G is called a planar embedding of G (we sometimes refer to a planar embedding of a planar graph G as a plane graph), see [6]. The bridgeless subgraphs of a connected graph G are the connected components formed by deleting all bridges (cut edges) of G. Then the bridgeless subgraphs and the bridges of G have a natural tree structure. Note that a bridgeless subgraph is a maximal 2-edge-connected subgraph or an isolated vertex. If we contract all bridgeless subgraphs into vertices, we obtain a tree; each vertex of this tree is a block-vertex. Let p1 , p2 and p3 be the three pages in clockwise order around ℓ in the algorithm. It is well known that the pagenumber of a graph equals the maximum pagenumber of its 2-connected subgraphs [4], we therefore assume that our input graph G is 2-connected. From now on, we discuss a fixed planar embedding of a planar graph (i.e., a plane graph). The general idea of our algorithm is as follows (see Fig. 1). Firstly, we remove from G the cycle Cout being the outer boundary of G and contract each bridgeless subgraph of the remaining graph into a single vertex (block-vertex). We denote the implied graph by F . Note that F is a forest (since F is not necessarily connected). Secondly, we embed the cycle Cout such that (i) the order of the vertices of Cout along the spine ℓ is fixed, (ii) all edges of Cout are on the same page, except for the edge that connects the leftmost vertex and the rightmost vertex for Cout along the spine ℓ. Thirdly, we embed the following elements: (i) the chords of Cout , (ii) the edges between Cout and F , (iii) the forest F . Finally, we take each block-vertex with a cycle C being the outer boundary of the bridgeless subgraph it corresponds to in G − Cout in turn and lay out its interior in a similar way. To formalize the idea mentioned above, we consider an arbitrary bridgeless subgraph with a simple cycle C as its outer boundary. If its outer boundary is a non-simple cycle, then we may embed each simple subcycle in turn for a non-simple cycle, and the order of embedding for these subcycles will be discussed later. We denote the subgraph of G contained in C by Gin (C ) and the subgraph of G outside C by Gout (C ), and let Gout (C ) = G − Gin (C ) and Gin (C ) = G − Gout (C ). Let v1 , v2 , . . . , vk be a counterclockwise order of the vertices of C from some vertex v1 . At the next level (embedding the edges in interior of bridgeless subgraphs in Gin (C )) we use cycles in clockwise order. For the recursive step, we assume the following invariant properties: IP-1: Gout (C ) has a three-page book embedding. IP-2: The book embedding of Gout (C ) is consistent with the given planar embedding of G. In other words, the clockwise order of the edges around vertex v in the book embedding of Gout (C ) is consistent with the counterclockwise order of the edges in the given planar embedding of G. IP-3: The vertices of C are placed in the order v1 , . . . , vk along ℓ, i.e., L(v1 ) < L(v2 ) < · · · < L(vk ), and all edges of C are on the same page, except for the edge (v1 , vk ). Without loss of generality, we place edges (vi , vi+1 ) (1 ≤ i < k) of C to page p1 and edge (v1 , vk ) to page p2 (at the next level we place edges (vi , vi+1 ) (1 ≤ i < k) of C to page p2 and edge (v1 , vk ) to page p3 ). IP-4: If C is not the cycle Cout (the outer boundary of G), the degree of either v1 or vk is at most 4 in Gin (C ). Without loss of generality, we assume degGin (C ) (vk ) ≤ 4. This is to assure that v is placed at the left of vk , where v ∈ Gin (C ). IP-5: If deg(v1 ) = 4 in Gin (C ), then it is incident to zero or two chords of C . If deg(v1 ) = 5 in Gin (C ), then it is incident to zero or three chords of C . This is to assure that v is placed at the right of v1 , where v ∈ Gin (C ). Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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Fig. 1. The general idea of the algorithm (the dotted edges denote the removed edges in (2)).
Note that the embedding specified in IP-2 is maintained throughout the whole embedding process. Furthermore, combined with IP-1 it is sufficient to ensure planarity. Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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Fig. 2. The marked edges for Fig. 1(1) (the yellow edges). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Now, we assume the edges of C are embedded. We describe how to assign the chords of C and the edges between C and F to pages. Let vi be a vertex of C , i = 1, . . . , k. Since G is a plane graph with maximum degree five, vi is incident to at most three non-embedded edges which are not assigned to pages up to now (the edges belong to Gin (C ) − E(C )). We denote the edges incident to vi in Gin (C ) − E(C ) that follow (vi , v(i+1) mod k ) in clockwise order (as defined by given planar embedding) by e1 , e2 and e3 . Following the naming scheme of Bekos et al. block-vertices that are adjacent to vertices of cycle C are referred to as anchors, and block-vertices that are adjacent to other block-vertices only are referred to as ancillaries. Consider an anchor a, let vl,a be the leftmost vertex of C that is adjacent to a along ℓ. If there is exactly one edge between a and vl,a (i.e., (a, vl,a ) is simple), we mark this edge. Otherwise, we mark the edge with the largest subscript among those incident to vl,a (i.e., the rightmost edge incident to the leftmost vertex vl,a of C adjacent to an anchor a) (see Fig. 2). Hence, each anchor is incident to exactly one marked edge and each vertex of C is incident to at most three marked edges. Let v be a vertex of C . We distinguish four cases based on the number of marked edges to assign the chords and the edges between C and F . Case 1. v is adjacent to three anchors a1 , a2 and a3 through three marked edges e1 , e2 and e3 , respectively (for example v1 in Fig. 1(3)). Note that v cannot be the rightmost vertex of C due to IP-4. Then, a1 , a2 and a3 are placed from left to right and directly to the right of v along ℓ. Moreover, e1 , e2 and e3 are placed to page p1 (see Fig. 3(1)). Case 2. v is adjacent to two anchors a1 and a2 through two marked edges ei and ej , respectively. If deg(v ) = 4 in Gin (C ), i.e., the two marked edges ei and ej are edges e1 and e2 , respectively, then e1 and e2 are placed to page p1 . Furthermore, if v ̸ = vk , then a1 and a2 are placed consecutively directly to the right of v (see Fig. 3(2)). Otherwise, a1 and a2 are placed consecutively directly to the left of v (see Fig. 3(3)). If deg(v ) = 5 in Gin (C ), v ̸ = vk due to degree restriction of vk by IP-4. In this case, the two marked edges are placed to page p1 and the non-marked edge is placed to page p2 . We distinguish three sub-cases for the exact placements of a1 and a2 : (i) If the two marked edges ei and ej are edges e1 and e2 , a1 and a2 are placed from left to right along ℓ and directly to the left of v (see Fig. 3(4) or v5 in Fig. 1(3)). (ii) If the two marked edges ei and ej are edges e1 and e3 , we place a1 directly to the left of v and a2 directly to the right of v (see Fig. 3(5) or v4 in Fig. 1(3)). (iii) If the two marked edges ei and ej are edges e2 and e3 , a1 and a2 are placed directly to the right of v and L(a1 ) < L(a2 ) (see Fig. 3(6) or v2 in Fig. 1(3)). Case 3. v is adjacent to one anchor a by the marked edge e. Suppose that deg(v ) = 3 in Gin (C ), then e is placed to page p1 . If v = vk , then a is placed directly to the left of v (see Fig. 3(7)). Otherwise, a is placed directly to the right of v (see Fig. 3(8)). Assume deg(v ) = 4 in Gin (C ). If v = vk , then a is placed directly to the left of v . We distinguish two sub-cases for the placements of edges. If e2 is the marked edge, then e1 is placed to page p3 and e2 is placed to page p2 (see Fig. 3(9) or v11 in Fig. 1(3)). Otherwise, e1 is the marked edge, then e1 is placed to page p1 and e2 is placed to page p2 (see Fig. 3(10)). If v ̸= vk , the marked edge is placed to page p1 and the non-marked edge is placed to page p2 . If e2 is the marked edge, a is placed directly to the right of v (see Fig. 3(11)). Otherwise, a is placed directly to the left of v (see Fig. 3(12)). Edges e1 and e2 are placed to page p3 (see v3 in Fig. 1(3)). Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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Fig. 3. The edges colored black are in p1 , the edges colored red are in p2 , and the edges colored blue are in p3 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. The edges colored black are in p1 , the edges colored red are in p2 , the edges colored blue are in p3 , and the edges colored gray are the non-embedded edges. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Now we assume deg(v ) = 5 in Gin (C ). It follows that v is not the rightmost vertex of C by IP-4. In this case, we distinguish three sub-cases. (i) If e1 is the marked edge, then a is placed directly to the left of v . Edge e1 is placed to page p1 , and e2 and e3 are placed to page p2 (see Fig. 3(13)). (ii) If e2 is the marked edge, then a is placed directly to the right of v . Moreover, e1 is placed to page p2 and e2 , and e3 are placed to page p3 (see Fig. 3(14) or v8 in Fig. 1(3)). (iii) If e3 is the marked edge, then a is placed directly to the right of v . e3 is to page p1 , and e2 and e3 are placed to page p2 (see Fig. 3(15) or v6 and v10 in Fig. 1(3)). Case 4. If v is not incident to any marked edge and deg(v ) ̸ = 2 in Gin (C ), then the edges incident to v are placed to page p2 (see Fig. 4). Up to now, we have laid out anchors and drawn the chords of C and the edges between C and F in pages. We next describe how to embed ancillaries and the edges of F . The edges of connecting two anchors are placed to p2 . Let F be a new forest formed by ancillaries, T be a connected component of F (i.e., a tree). It is obvious that F is a subgraph of F . The anchored tree T for T is obtained by adding all anchors adjacent to each ancillary of T . Suppose that T is rooted at the anchor that is leftmost along ℓ, we assign the vertices of T in order along ℓ implied by the specific Death First Search (DFS) traversal of T . If the children a′ and a′′ of a are in the counterclockwise order of edges around a, when starting from edge (a, p(a)) (where p(a) is the parent of a), then L(a′ ) < L(a′′ ). Moreover, the edges of F¯ are placed to p2 (see Fig. 5). To define the order of the trees in F , we create an auxiliary digraph GTaux whose vertices correspond to trees and there is a directed edge (vT 1 , vT 2 ) in GTaux if and only if T 1 has an anchor that is between two consecutive anchors of T 2 . Lemma 2.1 (Bekos et al. [2]). Auxiliary digraph GTaux is a directed acyclic graph. Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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Fig. 5. The vertices colored gray are ancillaries, and the vertices colored black are anchors.
Fig. 6. The vertices colored gray are ancillaries, the edges colored black are in p1 , the edges colored red are in p2 , and the edges colored blue are in p3 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 7. The edges colored black are in p1 , the edges colored red are in p2 , and the edges colored blue are in p3 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Lemma 2.1 implies an embedding of the trees in the order defined by a topological sorting of GTaux . In other words, if T 1 has an anchor that is between two consecutive anchors of T 2 along ℓ, then the tree T 1 will be embedded before T 2 . Up to now, Gin (C ) has been embedded, such that every bridgeless subgraph of Gin (C ) is contracted into a single vertex along ℓ and each edge is assigned to one of pages p1 , p2 and p3 . Next, we describe how to recursively proceed. Let a be a block-vertex of Gin (C ) with a simple outer boundary 𭟋a : w0 → w1 → . . . → wm → w0 being the outer boundary of the bridgeless subgraph it corresponds to in Gin (C ), we discuss the non-simple case later. We consider two cases (see Fig. 6): Case 1. a is an anchor and w0 is incident to the marked edge e. We place w0 as the rightmost vertex of 𭟋a on ℓ, w1 as the leftmost vertex of 𭟋a on ℓ, and wi on the left of wi+1 for i = 1, . . . , m − 1 and there are no vertices in between (see Fig. 7(1)). Assign (w0 , w1 ) to page p3 and the remaining edges of 𭟋a to page p2 (at the next level the edges embedded in counterclockwise order are similar to Gin (C ) in Gin (𭟋a )). This placement is infeasible only if there is an edge or two edges incident to w0 between (w0 , w1 ) and the marked edge e in the counterclockwise order of the edges around w0 when starting from (w0 , w1 ). In this case, w0 is to the left of w1 , . . . , wm , i.e., w0 is the leftmost vertex of 𭟋a on ℓ (this may violate IP-4, however, in this case, anchors a in Gin (𭟋a ) placed to left (right) of vr ,a ̸ = v1 (vr ,a = v1 ), where vr ,a is the rightmost vertex of 𭟋a adjacent to a along ℓ). This is impossible if there is also an edge (w0 , w′) incident to w0 between (w0 , w1 ) and the marked edge e in clockwise order of the edges around w0 when starting from (w0 , w1 ). We shall address the problem if (w0 , w′) is placed to page p2 (see Fig. 8). Case 2. a is an ancillary and w is its parent in the labeled anchored tree T to which a belongs to. We place w0 to be the leftmost vertex of 𭟋a on ℓ, w1 to be the rightmost vertex of 𭟋a on ℓ, and wi to the left of wi+1 for i = 1, . . . , m − 1 and there are no vertices in between. Moreover, (w0 , w ) is to page p3 and the remaining edges are to page p2 . This placement is infeasible in two cases. Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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Fig. 8. The placement of ancillary is determined by Lemma 2.2.
Fig. 9. Fa is not a simple cycle.
Fig. 10. The placement of a non-simple cycle being the outer boundary of the bridgeless subgraph b4 in Fig. 1 (The edges colored black are in p1 , the edges colored red are in p2 , the edges colored blue are in p3 ). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
One case is that the edges (except the edge (w0 , w1 )) incident to w0 are placed between (w0 , wm ) and (w0 , w ) in the counterclockwise order of the edges around w0 when starting from (w0 , wm ), say (w0 , w ′ ) (and (w0 , w ′′ )). Lemma 2.2 (Bekos et al. [2]). Ancillary a can be repositioned on ℓ such that: (i) a is placed between two consecutive anchors of T . (ii) The embedding specified by IP-2 is preserved and the edges (w0 , w ), (w0 , w ′ ) and (a, w ′′ ) are on page p2 and crossing-free. (iii) w0 is the leftmost vertex of 𭟋a and wi is to the left of wi+1 for i = 1, . . . , m − 1; All edges of 𭟋a are placed to page p2 , except for (w0 , wm ). By Lemma 2.2, the vertices of subtrees of T rooted at w ′ and w ′′ are placed to the left of w0 and right of w . The other case is that there is an edge e′ (and edge e′′ ) incident to w0 between (w0 , wm ) and (w0 , w ) in the counterclockwise (clockwise, respectively) order of the edges around w0 when starting from (w0 , wm ). In this case, we place the vertices of subtrees of T rooted at w ′ to right of w , and place w ′′ to the left of w0 and to the right of w (see Fig. 9). We assume 𭟋a being a non-simple cycle, i.e., 𭟋a consists of simple cycles (each of them is called a subcycle). Note that any two subcycles share at most one vertex of 𭟋a and each vertex of 𭟋a is incident to at most two subcycles. For 𭟋a , we define a tangency graph Gtan whose vertices correspond to subcycles of 𭟋a and there is an edge between every pair of subcycles that share a vertex, then Gtan is a tree. Since the degree of w0 is at most 5, w0 lies in at most two subcycles. We firstly suppose that w0 lies in exactly one subcycle. Let Gtan be rooted at the subcycle containing w0 . Then we place the subcycles of 𭟋a in the order implied by the Breadth First Search (BFS) traversal of Gtan . Assume w0 lies in two subcycles, say C1 and C2 that correspond to vertices c1 and c2 of Gtan , respectively. We place the subcycles of 𭟋a to the left (right) of w0 in the order by the BFS-traversal of the subtree rooted at c1 (c2 respectively) (see Fig. 10). We summarize the embedding of edges in the graph obtained by contracting bridgeless subgraphs of Gin (C ). The edges assigned to page p1 are the edges on C . The edges on page p2 consist of the edges in F , (v1 , vk ), the edges incident to a vertex of C in Gin (C ), and the edges between anchors. The edges embedded to page p3 are edges e2 and e3 in the case Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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that e2 is the marked edge and e1 and e3 are the non-marked edges, and e2 in the case that deg(vk ) = 4 in Gin (C ) and e2 is the marked edge, and e1 is the non-marked edge. 3. Proof of correctness In this section, we shall show that the algorithm is correct, i.e., no two edges intersect on the same page and IP-1 up to IP-5 are satisfied for an arbitrary simple cycle C . Lemma 3.1 (Bekos et al. [2]). For each ancillary a of a labeled tree T there is (i) at least an anchor of T with label smaller than that of a and (ii) at least an anchor of T with label greater than that of a. Lemma 3.2.
If v ∈ Gin (C ), then v lies to the left of vk and to the right of v1 , where C : ⟨v1 → . . . → vk → v1 ⟩.
Proof. (i). If v is a vertex on C , then v must be to the right of v1 and to the left of vk due to IP-3. (ii). If v is not a vertex on C , then we distinguish two subcases. (ii-1). We firstly assign the vertices of F of Gin (C ) on the spine ℓ (F is the forest obtained by contracting the bridgeless subgraphs of Gin (C )). Suppose that block-vertex a is an anchor, we have L(v1 ) < L(a) < L(vk ) for each anchor a in Gin (C ) according to the placement of anchors, i.e., all anchors are placed between v1 and vk . If a is an ancillary, then Lemma 3.1 implies that there are at least two anchors ai and aj , such that L(ai ) < L(a) < L(aj ). Then each ancillary is placed between v1 and vk . (ii-2). For the bridgeless subgraph represented by a block-vertex a, the vertices of Gin (𭟋a ) are placed to the position between the two vertices placed to the right and to the left of a along the spine ℓ, i.e., the vertices of Gin (𭟋a ) are placed between v1 and vk according to (a). Therefore, v lies to the left of vk and to the right of v1 for all v ∈ Gin (C ). □ Recall the placement of the anchored tree T , all edges of T are placed to the same page. Therefore, the lemma below is true here. Lemma 3.3 (Bekos et al. [2]). The placement of the anchored tree T is planar. Lemma 3.4.
There is no conflict between edges assigned to pages p1 , p2 , and p3 in Gin (C ).
Proof. We firstly consider the edges embedded on p1 in Gin (C ): all edges on C except for the one that connects its outermost vertices. Clearly, the edges assigned on p1 have no crossings. We now show that all edges assigned to p2 can be embedded without crossings. Lemma 3.3 implies that the edges of T have no crossings. Note that there is a path P (consisting of edges in p2 ) joining a pair of consecutive anchors (say u1 and ul+1 ) of T and the algorithm must place an ancillary a of T between them. Since c is nested by an edge of P and all edges of T belong to p2 , an edge connecting a to an ancillary of T placed between another pair of consecutive anchors of T would cross P. By an argument similar to that of Lemma 11 in [2], it can be seen that there is a path P(u0 → ul+1 ): u0 → uj1 → . . . → ujp → ul+1 consisting of vertices of {u0 , . . . , ul+1 }, whose edges belong to p2 and for each edge of P(u0 → ul+1 ) there are no edges in p2 with endpoints in {u0 , . . . , ul+1 } that nests it. Next, P(u0 → ul+1 ) contains at least one vertex v of C , and the vertex v is incident to at least one marked edge. According to the placement of edges, there is exactly one edge incident to C in Gin (C ), a contradiction. This gives a planar embedding of edges assigned in page p2 . Finally, each edge in p3 is incident to some vertex v on C . Note that G is a plane graph and the clockwise order of edges incident to v in the book embedding of Gin (C ) is consistent with counterclockwise order of edges around v on Gin (C ). Therefore, the edges assigned to page p3 do not cross each other due to IP-2 and IP-1. □ According to Lemma 3.2, the edges of Gin (C ) do not cross the edges of Gout (C ) on the same page. There is no conflict between edges in the same pages for Gout (C ) and Gin (C ) due to IP-1 and Lemma 3.4, respectively. Moreover, the edges incident to C do not cross on the same pages due to IP-2. Therefore, no edges intersect in the same pages. We are now ready to describe that IP-1 up to IP-5 hold when a simple cycle Cs is recursively drawn. Firstly, we prove that IP-1 up to IP-5 are satisfied for Cout , i.e., the first step of recursion holds. Lemma 3.5 (Bekos et al. [2]). Any planar graph G admits a planar drawing with a chordless outer boundary. IP-1, IP-2 and IP-3 are clearly satisfied, since Gout (Cout ) = Cout for Cout . Lemma 3.5 implies IP-5. If there is a vertex v ∈ V (Cout ) with degGin (Cout ) (v ) ≤ 4, then it is chosen as vk and all the invariant properties are satisfied. However, if such a vertex does not exist, i.e., each vertex v ∈ V (Cout ) has degGin (Cout ) (v ) = 5, this violates IP-4. This case will be addressed
by the following lemma. For the placement of the edges adjacent to C , we only consider the case that there is a vertex vk of the degree at most 4. Therefore, we have to prove the following lemma. Lemma 3.6.
If C = Cout , then IP-4 does not necessarily hold.
Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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Proof. If C = Cout and IP-4 is not satisfied, then there is no vertex v of Cout with degGin (Cout ) (v ) ≤ 4. Suppose vm is adjacent to vertices b1 , b2 and b3 through edges e1 , e2 and e3 , where b1 , b2 and b3 belong to the bridgeless subgraphs corresponding to anchors a1 , a2 and a3 (or simply, b1 , b2 and b3 belong to anchors a1 , a2 and a3 ), respectively (see Fig. 11(1)). The vertices of Gin (C ) should be placed to the left of vm . We therefore consider the following five cases. Case 1. All the edges incident to vm are marked edges. Case 2. e2 and e3 are marked edges and e1 is a non-marked edge. Case 3. e1 and e3 are marked edges and e2 is a non-marked edge. Case 4. e3 is a marked edge and e1 and e2 are non-marked edges. Case 5. e2 is a marked edge and e1 and e3 are non-marked edges. For Case 1, b1 , b2 and b3 belong to distinct anchors. For Case 2 and Case 5, a1 ̸ = a3 and a2 ̸ = a3 , but a1 = a2 is possible. For Case 3, a1 ̸ = a2 and a1 ̸ = a3 , but a3 = a2 is possible. For Case 4, a2 = a3 , and a1 = a2 = a3 are possible. (1) For Case 1, we augment G by introducing a vertex vm+1 , adding the edges (vm , vm+1 ), (vm+1 , b2 ), (vm+1 , b3 ) and removing the edges (vm , b2 ), (vm , b3 ) (see Fig. 11(2)). We denote the augmented graph by Gaug . Then IP-4 holds for Gaug . We claim that b1 , b2 and b3 are placed to the left of vm in the embedding of the augmented graph Gaug . Let Caug be the aug aug aug outer boundary of the augmented graph Gaug , and a1 , a2 and a3 be the block-vertices containing b1 , b2 and b3 in Gaug − Caug , respectively. In this case, we have (vm , b2 ) being a marked edge and vm+1 belonging to another block-vertex (containing only vm+1 ) aug aug and is adjacent to vm . Note that both a2 and a3 are ancillaries. Since vm+1 is adjacent to exactly one vertex of Cout aug aug (i.e., vm ). Then, vm+1 is to the right of a1 and the left of vm , and (vm , vm+1 ) and (vm , a1 ) are placed to page p1 according aug aug to Fig. 3(3). Furthermore, (a2 , vm+1 ) and (a3 , vm+1 ) to page p2 (see Fig. 11(3)). There are no anchors between vm+1 and vm due to degGaug (vm ) = 4. Therefore, the rightmost anchor of Gaug − Caug is vm+1 . Then, all vertices of Gaug − Caug are to the left of vm+1 . We obtain a valid embedding of G by contracting vm and vm+1 back into vm (see Fig. 11(4)). (2) For Sub-Case 2.1: a1 ̸ = a2 , we also augment G (Gaug ) by introducing a vertex vm+1 , adding the edges (vm , vm+1 ), (vm+1 , b2 ), (vm+1 , b3 ) and removing the edges (vm , b2 ), (vm , b3 ) similar to Case 1 (see Fig. 11(2)). Then IP-4 holds for Gaug . We also claim that b1 , b2 and b3 are placed to the left of vm in the embedding of the augmented graph Gaug . aug In this case, (vm , a2 ) is a non-marked edge. Then vm+1 is to the right of a1 and to the left of vm (see Fig. 2(9)). Furthermore, (vm , vm+1 ) is placed to page p2 and (vm , b1 ) is placed to page p3 , and (vm+1 , b2 ) and (vm+1 , b2 ) to page p2 (see Fig. 11(5)). We also obtain a valid embedding of G by contracting vm and vm+1 back into vm (see Fig. 11(6)). For Sub-Case 2.2: a1 = a2 = a, we augment G by introducing a vertex vm+1 , adding the edges (vm , vm+1 ), (vm+1 , b1 ), (vm+1 , b2 ) and (vm+1 , b3 ) and removing the edges (vm , b1 ), (vm , b2 ) and (vm , b3 ) (see Fig. 11(7)). Suppose a1 = a2 = a, then vm+1 must belong to a, and b1 , b1 , b2 and vm+1 appear in the clockwise traversal of the outer boundary Ca of a. Then (vm , vm+1 ) is a marked edge. Hence, a is to the right of a1 and the left of vm according to Fig. 3(7). Moreover, (vm , vm+1 ) is placed to page p1 , and (vm+1 , b1 ) is placed to page p3 and (vm+1 , b3 ), (vm+1 , b2 ) and (b1 , b2 ) are placed to page p2 (see Fig. 11(8)). By argument similar to Case 1, we obtain a valid embedding of G by contracting vm and vm+1 back into vm (see Fig. 11(9)). (3) For Sub-Case 3.1: a2 = a3 , we obtain the augmented graph Gaug similar to Case 1 (see Fig. 11(2)). Then IP-4 holds also for Gaug . And b1 , b2 and b3 are placed to the left of vm in the embedding of the augmented graph Gaug . Suppose a2 = a3 = a, then vm+1 must belong to a, and b2 , b3 and vm+1 appear in the clockwise traversal of the outer aug boundary Ca of a. Then we have (vm , a1 ) being a marked edge and vm+1 belonging to another block-vertex (containing only vm+1 ) and is adjacent to vm . Note that a is an anchor. Since vm+1 is adjacent to exactly one vertex of Cout (i.e., vm ). aug aug aug Then, vm+1 is to the right of a1 and the left of vm (see Fig. 3(3)). Furthermore, (a3 , vm+1 ) and (a2 , a3 ) to page p2 and aug (a2 , vm ) to page p3 (see Fig. 11(10)). There are no anchors between vm+1 and vm due to degGaug (vk ) = 4. Therefore, the rightmost anchor of Gaug − Caug is vm+1 . Then, all vertices of Gaug − Caug are to the left of vm+1 . We obtain a valid embedding of G by contracting vm and vm+1 back into vm (see Fig. 11(11)). For Sub-Case 3.2: a2 ̸ = a3 , we augment G by introducing three vertices (say vm+1 , vm+2 and vm+3 ) to the right of vm , adding the edges (vm , vm+1 ), (vm+1 , vm+2 ), (vm+2 , vm+3 ), (vm , vm+2 ), (vm+3 , b2 ), and (vm+3 , b3 ), and removing the edges (vm , v1 ), (vm , b1 ) (see Fig. 11(12)). Note that (vm+3 , b2 ) is a non-marked edge, (vm+3 , b3 ) and (vm+3 , b2 ) are the marked edges. We have b1 , b2 and b3 are to the left of vm due to Fig. 3(12) and Fig. 3(9), respectively. Furthermore (vm , b1 ) is placed to page p1 , (vm+3 , b2 ) is placed to page p2 , and (vm+1 , b3 ) to page p3 (see Fig. 11(13)). There are no vertices of Gaug between vm and vm+3 , except for vm+1 and vm+2 . Hence, all vertices of Gaug − Caug are to the left of vm . If we contract vm , vm+1 , vm+2 and vm+3 back into vm , then we obtain a valid embedding of G (see Fig. 11(14)). (4) For Sub-Case 4.1: a2 = a3 ̸ = a1 , we also obtain the augmented graph Gaug similar to Case 1 (see Fig. 11(2)). Then IP-4 holds for Gaug . We also claim that b1 , b2 and b3 are placed to the left of vm in the embedding of the augmented graph Gaug . In this case, (vm , b2 ) is a non-marked edge. Then vm+1 is to the right of a1 and to the left of vm (see Fig. 3(9)). Furthermore, (vm , vm+1 ), (vm+1 , b3 ) and (b2 , b3 ) to page p2 , (vm+1 , b2 ) and (vm , b1 ) is placed to page p3 (see Fig. 11(15)). We also obtain a valid embedding of G by contracting vm and vm+1 back into vm (see Fig. 11(16)). For Sub-Case 4.2: a1 = a2 = a3 = a, obtain the augmented graph Gaug similar to Case 2.2 (see Fig. 11(7)). Then vm+1 must belong to a, and b1 , b2 , b3 and vm+1 appear in the clockwise traversal of the outer boundary Ca of a. Then (vm , vm+1 ) Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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Fig. 11. Dotted line is removed, the edges colored black are in p1 , the edges colored red are in p2 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
is a marked edge. Hence, a is to the right of a1 and the left of vm according to Fig. 3(7). Moreover, (vm , vm+1 ) is placed to page p1 , and (vm+1 , b2 ), (b2 , b3 ) and (b1 , b3 ) are placed to page p2 , and (vm+1 , b1 ) is placed to page p3 (see Fig. 11(17)). By argument similar to Case 1, we obtain a valid embedding of G by contracting vm and vm+1 back into vm (see Fig. 11(18)). For Sub-Case 4.3: a2 ̸ = a3 , we augment G by introducing a vertex vm+1 , adding the edges (vm , vm+1 ), (vm+1 , b1 ), (vm+1 , b2 ) and removing the edges (vm , b1 ), (vm , b2 ) (see Fig. 11(19)). Note that (vm , vm+1 ) is a non-marked edge and (vm , b3 ) is a marked edge. We have b1 , b2 and b3 are to the left of vm due to Fig. 3(9). Furthermore, (vm , vm+1 ) is placed to page p3 , (vm , b3 ), (vm+1 , b1 ) and (vm+1 , b2 ) are placed to page p2 (see Fig. 11(20)). Similar to that of Case 1, we obtain a valid embedding of G by contracting vm and vm+1 back into vm (see Fig. 11(21)). Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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(5) For Sub-Case 5.1: a1 ̸ = a2 . By argument similar to Sub-Case 2.1, we can obtain the embedding of G (see Fig. 11(5) and (6)). For Sub-Case 5.2: a1 = a2 = a. By argument similar to Sub-Case 2.2, we can obtain the embedding of G (see Fig. 11(8) and (9)). □ Lemma 3.7 (Bekos et al. [2]). Assume that all trees that precede T in a topological sorting of GTaux have been drawn on page p2 without crossings by preserving the embedding specified by IP-2. When T is drown, the embedding specified by IP-2 is also preserved. Lemma 3.8 (Bekos et al. [2]). Let v be a vertex of a cycle C with degree 2 in Gin (C ) that is not the left/right-most vertex of C . Let also vr (vl , resp.) be its next neighbor on C to its right (left, resp.). Since, edge (v, vr ) belongs to C , it is placed to the page p1 . In fact, it can also be placed to page p2 without crossings and while the embedding specified by IP-2 is maintained (the page p1 is the bottom-page, and the page p2 is the top-page in [2]). Lemma 3.9. Let v be a vertex of a cycle C with degree 3 or 2 in Gin (C ) that is not the left/right-most vertex of C . Let also vr (vl , resp.) be its next neighbor on C to its right (left, resp.). The algorithm places (v, vr ) to page p2 . In fact, it can also be placed to page p3 without crossings and while the embedding specified by IP-2 is maintained. Proof. If deg(v ) = 2 in Gin (C ), then claim holds according to Lemma 3.8. Suppose deg(v ) = 3 in Gin (C ). We distinguish two cases to discuss as follows. (i) Assume that v is incident to a non-marked edge e. Then the anchor a is placed to the left of v according to the placement of anchor. Thus, IP-2 is maintained for v . We only discuss whether IP-2 is maintained for vr . If no block-vertex is placed between vertices v and vr , then obviously edge (v, vr ) can be placed to page p2 without introducing edge-crossings and without changing the combinatorial embedding specified by IP-2 (recall that vertices v and vr are consecutive vertices of cycle C ). Thus, we may assume without loss of generality that there are block-vertices placed between vertices v and vr of cycle C . We will prove that we can move the block-vertices in between to the left of vertex v , so that vertices v and vr become consecutive along spine ℓ. The aforementioned move is not possible, only if there is an anchor c between vertices v and vr , such that edge (v, vr ) is placed to page p2 . To overcome this problem, we can potentially place vertex v between vertices c and vr . However, in this case, if (c , vr ) is placed to page p1 , then edges (c , vr ) and (v, vl ) would cross according to the placement of the edges incident to vertices in C . If (c , vr ) is placed to page p2 , then edges (c , vr ) and (v, a) would cross. In either cases, we can cope with problematic situation if edge (c , vr ) is replaced to page p3 . (ii) If v is adjacent to an anchor a through one marked edge e, then we move a to the right of vr and (v, vr ) can be placed to page p2 without crossings, which can be shown similar to the case that there are block-vertices placed between vertices v and vr of cycle C in (i). □ We are ready to describe how the recursive step holds as following: IP-1 up to IP-5 hold for an arbitrary simple cycle Cs . Each edge is embedded to one of three pages: page p1 , page p2 , and page p3 , and no two edges intersect on the same page. Therefore, IP-1 is satisfied. Lemma 3.7 implies IP-2. If Cs is the outer boundary of a block-vertex or a leaf of the tangency tree, then IP-3 trivially holds. If Cs is the outer boundary of a non-leaf of the tangency tree, it contains at least one edge on page p3 . This violates IP-3. In this case, we re-embed it to page p1 using Lemma 3.8. IP-4 holds. In fact, if Cs is the outer boundary of a block-vertex or root of the tangency tree of a non-simple outer boundary, then at least one vertex of Cs is adjacent to Gout (Cs ). If Cs is the outer boundary of internal node of the tangency tree of a non-simple outer boundary, then its leftmost vertex has at least two neighbors in Gout (Cs ). The following lemma implies that IP-5 does not necessarily hold for simple cycle Cs . Lemma 3.10.
IP-5 does not necessarily hold for an arbitrary simple cycle C .
Proof. If IP-5 does not hold for some simple cycle Cs : w1 → w2 → . . . → wm , then we may distinguish three cases. Case 1. degGin (C ) (w1 ) = 4 and w1 is incident to one chord of C . Case 2. deg(w1 ) = 5 in Gin (C ) and w1 is incident to one chord of C . Case 3. deg(w1 ) = 5 in Gin (C ) and w1 is incident to exactly two chords of C (say (w1 , wi1 ) and (w1 , wi2 ), where i1 ∈ {3, . . . , m − 1} and i1 < i2 ). For Case 1 and Case 2, w1 is incident to exactly one chord (say (w1 , wi ), i ∈ {3, . . . , m − 1}) of C . In general, (w1 , wi ) ∈ P(w1 → wj ), where P(w1 → wj ) is a path of chords from w1 to wj on page p2 . Suppose wq ∈ P(w1 → wj ), (wq , wx ) and (wq , wy ) are chords, x < y, we chose (wq , wy ) ∈ P(w1 → wj ). The restriction implies that P(w1 → wj ) is uniquely defined (see Fig. 12(1)). We refer to it as the separating path of chords of Cs , since it splits Gin (Cs ) into two subgraphs: (i) Gin (Cl ) with outer boundary Cl consisting of the edges (w1 , w2 ), (w2 , w3 ), . . . , (wj−1 , wj ) and the edges of P(w1 → wj ) (see Fig. 12(2)), and (ii) Gin (Cr ) with outer boundary Cr consisting of the edges (wj , wj+1 ), . . . , (wm−1 , wm ), (wm , w1 ) and the edges of P(w1 → wj ) (see Fig. 12(3)). Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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Fig. 12. Dotted line is removed, the edges colored black are in p1 , the edges colored red are in p2 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
We now describe how the two sub-instances Gin (Cl ) and Gin (Cr ) can be recursively solved. Note that v must be to the right of v1 , where v is a vertex of Gin (C ). According to the assignments of anchors, we consider the following sub-cases (for Case 1 and Case 2). Sub-Case 1.1: degGin (C ) (w1 ) = 4 and w1 is incident to exactly one chord e1 of C . Sub-Case 2.1: deg(w1 ) = 5 in Gin (C ) and w1 is incident to exactly one chord e2 of C . Sub-Case 2.2: deg(w1 ) = 5 in Gin (C ) and w1 is incident to exactly one chord e1 of C . Observe that if i ̸ = j, then Cl is a non-simple cycle, i.e., Cl consists of some simple subcycles, for which IP-5 holds (hence they can be recursively drawn), except for the first one, that is leftmost embedded along ℓ. Note that j ̸ = m. In fact, if j = m and deg(w1 ) = 3 (deg(w1 ) = 4) in Gin (Cr ), then e3 is a bridge according to the placements of chords or e2 is not a chord for some vertex vs (vs belong to P(w1 → wj )). If e3 is a bridge, a contradiction is obtained since G is 2-connected. If e2 is not a chord for some vertex vs , a contradiction is obtained since vs belongs to P(w1 → wj ) ((i) if e2 is a marked edge, then e3 is placed to page p2 (see to Fig. 3(14)); (ii) if e2 is a non-marked edge, then the edges incident to vs are placed to page p2 due to Case 4 in the placements of edges incident to C ). In Sub-Case 1.1, deg(w1 ) = 2 in Gin (Cl ), i.e., deg(w1 ) = 3 in Gin (Cr ). We modify Gin (Cl ) as follows. Remove w1 and join the edges (w2 , wi ) and (w2 , wm ). Then Gin (Cl′ ) (see Fig. 12(4)) has fewer vertices than Gin (C ). We can benefit from this by proceeding recursively, as we initially did with Gin (C ). Eventually, IP-5 should hold for some vertex wp , otherwise a graph with at most 3 vertices on its outer boundary should have a chord, a contradiction. To complete the embedding of Gin (C ), we remove (w2 , wi ) and (w2 , wm ), and connect the neighbors of wj in Gin (Cr′ ) with its copy in Gin (Cl′ ) (no crossings are introduced, since the two copies of wj in Gin (Cl′ ) and Gin (Cr′ ) are consecutive along ℓ). It remains to replace the copy of wj in Gin (Cr′ ) with w1 , and add (w1 , w2 ) and (w1 , wi ) (see Fig. 12(6)). In Sub-Case 2.1, deg(w1 ) = 3 in Gin (Cl ) and deg(w1 ) = 3 in Gin (Cr ). We modify Gin (Cl ) similarly to that of Sub-Case 1.1, we obtain a valid placement by placing exactly w1 and a1 to the right of wj and a2 to the right of wp , where a1 and a2 are adjacent to w1 through two marked edges e1 and e2 , respectively (see Fig. 12(7)). In Sub-Case 2.2, deg(w1 ) = 2 in Gin (Cl ) and deg(w1 ) = 4 in Gin (Cr ). Similarly to that of Sub-Case1.1, w1 , a1 and a2 are placed to the right of wj and the left of wj+1 , where a1 and a2 are adjacent to w1 through two marked edges e1 and e2 , respectively. Hence, we obtain a valid embedding (see Fig. 12(8)). For Case 3, we only consider this case due to the placements of anchors. Sub-Case 3.1: deg(w1 ) = 5 in Gin (C ) and w1 is incident to exactly two chords e1 and e2 of C . Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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Fig. 13. The non sub-hamiltonian graph of maximum degree 7.
In this case, we choose (w1 , wi2 ) ∈ P(w1 → wj ). If wq ∈ P(w1 → wj ), (wq , wx ) and (wq , wy ) are chords, x < y, then (wq , wy ) ∈ P(w1 → wj ). Hence, P(w1 → wj ) is uniquely defined, and w1 and a are placed directly to the right of wj by the argument of Sub-Case 1.1 (see Fig. 12(9)). □ Theorem 3.11. There is a quadratic-time algorithm to construct book embedding for planar graphs of maximum degree 5 on 3 pages. Proof. Given a planar graph of maximum degree 5 of n vertices, the computation of the bridgeless-subgraphs, the topological sorting of GTaux , BFS-traversals on the tangency trees Gtan and DFS-traversals on the anchored tree T can be done in linear time. Hence the algorithm runs in O(n2 ) time. □ 4. Conclusions and open problems We have noted previously that if the pagenumber of a maximal planar graph is two, then it is hamiltonian. Since there is a planar 8-graph which is non-hamiltonian in [4]. There also is a planar 7-graph requiring at least three pages [20] (the proof is rather straightforward) (see Fig. 13). Then the pagenumber of planar 8-graphs and planar 7-graphs is at least 3. This paper has presented an O(n2 ) time algorithm for embedding planar 5-graphs into books of three pages. The natural open problem is whether two pages are sufficient for planar 5-graphs. Hoffmann and Klemz [15] show that a 3-connected planar 5-graph is subhamiltonian and then it can be embedded in two pages. Acknowledgments The research is supported by NSFC (No. 11671296), Program for the Innovative Talents of Higher Learning Institutions of Shanxi, Natural Science Foundation of Shanxi Provincial (No. 201801D221193), and Shanxi Scholarship Council of China. References [1] G.A. Atneosen, On the Embeddability of Compacta in N-Books: Intrinsic and Extrinsic Properties (Ph.D. thesis), Department of Mathematics, Michigan State University, 1968. [2] M.A. Bekos, M. Gronemann, C.N. Raftopoulou, Two-page book embeddings of 4-planar graphs, Algorithmica 75 (2016) 158–185. [3] M.A. Bekos, M. Kaufmann, C. Zielke, The Book Embedding Problem from a SAT Solver Perspective, GD 2015, 2015. [4] F. Bernhart, P.C. Kainen, The book thickness of a graph, J. Combin. Theory Ser. B 27 (3) (1979) 320–331. [5] T. Bilski, Optimum embedding of complete graphs in books, Discrete Math. 182 (1998) 21–28. [6] J.A. Bondy, U.S.R. Murty, Graph Theory with Application, North-Holland, New York, 1976. [7] J.F. Buss, P.W. Shor, On the pagenumber of planar graphs, in: Proceedings of the 16th ACM Symposium on Theory of Computing, STOC84, 1984, pp. 98–100. [8] F.R.K. Chung, F.T. Leighton, A.L. Rosenberg, Embedding graphs in books a layout problem with applications to VLSI design, SIAM J. Algebr. Discrete Methods 8 (1987) 33–58. [9] V. Dujmovic, D.R. Wood, Graph treewidth and geometric thickness parameters, Discrete Comput. Geom. 37 (2007) 641–670. [10] H. Enomoto, T. Nakamigawa, K. Ota, On the pagenumber of complete bipartite graphs, J. Combin. Theory Ser. B 71 (1997) 111–120. [11] S. Even, A. Itai, Queues, stacks, and graphs, in: Z. Kohavi, A. Paz (Eds.), Theory of Machines and Computations, Academic Press, New York, 1971, pp. 71–86. [12] G. Ewald, Hamiltonian Circuits in Simplicial Complexes, D. Reidel Publishing Company, Dordrecht-Holland, 1973. [13] L. Heath, Embedding planar graphs in seven pages, in: Proceedings of the 25th IEEE Annual Symposium on Foundations of Computer Science, 1984, pp. 74–89. [14] L. Heath, Algorithms for Embedding Graphs in Books (Ph.D. thesis), University of North Carolina, Chapel Hill, 1985. [15] M. Hoffmann, B. Klemz, Triconnected planar graphs of maximum degree five are subhamiltonian, ESA-2019. https://algo2019.ak.in.tum.de/index. php/algo-program, http://drops.dagstuhl.de/opus/volltexte/2019/11179/. [16] S. Istrail, An algorithm for embedding planar graphs in six pages, 1986, manuscript.
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[17] N. Kapoor, M. Russell, I. Stojmenovic, A.Y. Zomaya, A genetic algorithm for finding the pagenumber of interconnection networks, J. Parallel Distrib. Comput. 62 (2002) 267–283. [18] D.J. Muder, M.L. Weaver, D.B. West, Pagenumber of complete bipartite graphs, J. Graph Theory 12 (1988) 469–489. [19] R. Nowakowski, A. Parker, Ordered sets, pagenumbers, and planarity, Order 6 (1989) 209–218. [20] S. Pupyrev, Private communication. http://be.cs.arizona.edu/static/img/deg7 non hamiltonian.png. [21] A.L. Rosenberg, The Diogenes approach to testable fault-tolerant arrays of processors, IEEE Trans. Comput. 32 (1983) 902–910. [22] H.C. So, Some theoretical results on the routing of multilayer printed-wiring boards, in: IEEE Intl. Symp. On Circuits and Systems, 1974, pp. 296–303. [23] A. Wigderson, The Complexity of the Hamiltonian Circuit Problem for Maximal Planar Graphs, Technical Report 298, Department of EECS, Princeton University, 1982. [24] M. Yannakakis, Four pages are necessary and sufficient for planar graphs (extended abstract), in: ACM Symposium on Theory of Computing, STOC86, 1986, pp. 104–108. [25] M. Yannakakis, Embedding planar graphs in four pages, J. Comput. System Sci. 38 (1989) 36–67.
Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Embedding planar 5-graphs in three pages Xiaxia Guan, Weihua Yang
∗
Department of Mathematics, Taiyuan University of Technology, Taiyuan Shanxi 030024, China
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Article history: Received 7 August 2018 Received in revised form 5 November 2019 Accepted 25 November 2019 Available online xxxx Keywords: Book embedding Planar graph Pagenumber
a b s t r a c t A book embedding of a graph G is an embedding of its vertices along the spine of a book, and an embedding of its edges to the pages so that no two edges on the same page cross. A planar graph of maximum degree k is called a planar k-graph. Bekos et al. described an O(n2 ) time algorithm for computing two-page book embeddings for planar 4-graphs. In this paper, we embed planar 5-graphs into books of three pages by an O(n2 ) time algorithm. © 2019 Elsevier B.V. All rights reserved.
1. Introduction The concept of a book embedding of a graph was introduced by Atneosen [1]. A book consists of a line called the spine and some half-planes called pages, sharing the spine as a common boundary. A book embedding of a graph G = (V , E) consists of a (linear) layout L of its nodes along the spine ℓ (for simplicity we use ℓ to denote the spine sometimes) of a book (i.e., L : V → {1, 2, . . . , n}) and the assignment of each edge to the pages so that two edges embedded on the same page do not cross. We say that two edges (a, b) and (c , d) such that L(a) < L(b) and L(c) < L(d) on the same page cross in the layout L if and only if L(a) < L(c) < L(b) < L(d) or L(c) < L(a) < L(d) < L(b). We also say an edge (a, b) (L(a) < L(b)) nests a vertex v if and only if L(a) < L(v ) < L(b) or L(b) < L(v ) < L(a), and an edge (a, b) nests an edge (c , d) (L(c) < L(d)) if and only if L(a) < L(c), L(d) < L(b) or L(b) < L(c), L(d) < L(a). A central goal in the study of book embedding is to find the minimum number of pages in which a graph can be embedded, and find an algorithm to embed the graph into a book with the minimum number of pages. The minimum number of pages in which a graph can be embedded is called the pagenumber or book thickness or stack number of the graph. Determining the pagenumber of a graph G is a hard problem. It remains a difficult problem even when the layout L is fixed, since determining if a given layout admits a k-page book embedding is NP-complete [23]. Book embedding of graphs plays an important role in several technical applications, for examples, direct interconnection networks [17], VLSI design [8], fault-tolerant processor arrays [21], sorting with parallel stacks [11], single-row routing [22], and ordered sets [19]. The book embedding of graphs has been discussed for many graph families, see [4,5,10,18]. The most famous ones are the planar graphs. Bernhart and Kainen [4] firstly characterized the graphs with pagenumber one as the outerplanar graphs and the graphs with pagenumber two as the sub-Hamiltonian planar graphs (the subgraphs of planar Hamiltonian graphs). Deciding whether the pagenumber of general planar graphs is two is NP-hard [8]. Moreover, Bernhart and Kainen [4] conjectured that planar graphs have unbounded pagenumber, but this was disproved in [7] and [13]. Buss and Shor [7] proposed a nine-page algorithm. Heath [13] reduced the number to seven. Istrail [16] found an algorithm that embeds planar graphs in six pages. Later, Yannakakis [25] showed that planar graphs admit a four-page book embedding, which ∗ Corresponding author. E-mail address: [email protected] (W. Yang). https://doi.org/10.1016/j.dam.2019.11.020 0166-218X/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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can be constructed in linear-time. Yannakakis [24] claimed that four pages are necessary without giving a formal proof for this claim. Later, Dujmovic and Wood [9] conjectured that the pagenumber of planar graphs is four. Bekos et al. [3] also posed the same conjecture. So far, the pagenumber of planar graphs is still open. One natural direction is to consider the pagenumber of specific planar graphs. A planar graph of maximum degree k is called a planar k-graph. The pagenumber of planar k-graphs is clearly at most four. Ewald [12] proved that maximal planar graphs of maximum degree at most 6 are Hamiltonian, hence, their pagenumber is two. However, there are planar k-graphs (k ≤ 6) that are not subgraphs of maximal planar graphs of maximum degree at most 6, i.e., the pagenumber of planar k-graphs (k ≤ 6) may not be two. Heath [14] found a linear algorithm that embeds planar 3-graphs in two pages. Later, Bekos et al. [2] described an O(n2 ) algorithm to embed planar 4-graphs into books of two pages. That is, the pagenumber of planar 3-graphs or planar 4-graphs is two. Bekos et al. also posed a question whether the result can be extended to planar 5-graphs. In this paper, we embed planar 5-graphs into three pages by an O(n2 ) time algorithm, and thus the pagenumber of planar 5-graphs is three at most. The rest of the paper is organized as follows. Section 2 gives an algorithm that every planar 5-graph admits a three-page book embedding and its correctness is proved in Section 3. 2. The algorithm In this section, we describe an algorithm to embed planar 5-graphs in three-page books. The algorithm is similar to that of Bekos et al. [2], which is given by a recursive combinatorial construction. Our algorithm is different from the idea in [25] by Yannakakis that embeds planar graphs into four pages. Several basic terms and further definitions are necessary. A graph is said to be planar if it can be drawn in the plane so that its edges do not cross. Such a drawing of a planar graph G is called a planar embedding of G (we sometimes refer to a planar embedding of a planar graph G as a plane graph), see [6]. The bridgeless subgraphs of a connected graph G are the connected components formed by deleting all bridges (cut edges) of G. Then the bridgeless subgraphs and the bridges of G have a natural tree structure. Note that a bridgeless subgraph is a maximal 2-edge-connected subgraph or an isolated vertex. If we contract all bridgeless subgraphs into vertices, we obtain a tree; each vertex of this tree is a block-vertex. Let p1 , p2 and p3 be the three pages in clockwise order around ℓ in the algorithm. It is well known that the pagenumber of a graph equals the maximum pagenumber of its 2-connected subgraphs [4], we therefore assume that our input graph G is 2-connected. From now on, we discuss a fixed planar embedding of a planar graph (i.e., a plane graph). The general idea of our algorithm is as follows (see Fig. 1). Firstly, we remove from G the cycle Cout being the outer boundary of G and contract each bridgeless subgraph of the remaining graph into a single vertex (block-vertex). We denote the implied graph by F . Note that F is a forest (since F is not necessarily connected). Secondly, we embed the cycle Cout such that (i) the order of the vertices of Cout along the spine ℓ is fixed, (ii) all edges of Cout are on the same page, except for the edge that connects the leftmost vertex and the rightmost vertex for Cout along the spine ℓ. Thirdly, we embed the following elements: (i) the chords of Cout , (ii) the edges between Cout and F , (iii) the forest F . Finally, we take each block-vertex with a cycle C being the outer boundary of the bridgeless subgraph it corresponds to in G − Cout in turn and lay out its interior in a similar way. To formalize the idea mentioned above, we consider an arbitrary bridgeless subgraph with a simple cycle C as its outer boundary. If its outer boundary is a non-simple cycle, then we may embed each simple subcycle in turn for a non-simple cycle, and the order of embedding for these subcycles will be discussed later. We denote the subgraph of G contained in C by Gin (C ) and the subgraph of G outside C by Gout (C ), and let Gout (C ) = G − Gin (C ) and Gin (C ) = G − Gout (C ). Let v1 , v2 , . . . , vk be a counterclockwise order of the vertices of C from some vertex v1 . At the next level (embedding the edges in interior of bridgeless subgraphs in Gin (C )) we use cycles in clockwise order. For the recursive step, we assume the following invariant properties: IP-1: Gout (C ) has a three-page book embedding. IP-2: The book embedding of Gout (C ) is consistent with the given planar embedding of G. In other words, the clockwise order of the edges around vertex v in the book embedding of Gout (C ) is consistent with the counterclockwise order of the edges in the given planar embedding of G. IP-3: The vertices of C are placed in the order v1 , . . . , vk along ℓ, i.e., L(v1 ) < L(v2 ) < · · · < L(vk ), and all edges of C are on the same page, except for the edge (v1 , vk ). Without loss of generality, we place edges (vi , vi+1 ) (1 ≤ i < k) of C to page p1 and edge (v1 , vk ) to page p2 (at the next level we place edges (vi , vi+1 ) (1 ≤ i < k) of C to page p2 and edge (v1 , vk ) to page p3 ). IP-4: If C is not the cycle Cout (the outer boundary of G), the degree of either v1 or vk is at most 4 in Gin (C ). Without loss of generality, we assume degGin (C ) (vk ) ≤ 4. This is to assure that v is placed at the left of vk , where v ∈ Gin (C ). IP-5: If deg(v1 ) = 4 in Gin (C ), then it is incident to zero or two chords of C . If deg(v1 ) = 5 in Gin (C ), then it is incident to zero or three chords of C . This is to assure that v is placed at the right of v1 , where v ∈ Gin (C ). Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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Fig. 1. The general idea of the algorithm (the dotted edges denote the removed edges in (2)).
Note that the embedding specified in IP-2 is maintained throughout the whole embedding process. Furthermore, combined with IP-1 it is sufficient to ensure planarity. Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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Fig. 2. The marked edges for Fig. 1(1) (the yellow edges). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Now, we assume the edges of C are embedded. We describe how to assign the chords of C and the edges between C and F to pages. Let vi be a vertex of C , i = 1, . . . , k. Since G is a plane graph with maximum degree five, vi is incident to at most three non-embedded edges which are not assigned to pages up to now (the edges belong to Gin (C ) − E(C )). We denote the edges incident to vi in Gin (C ) − E(C ) that follow (vi , v(i+1) mod k ) in clockwise order (as defined by given planar embedding) by e1 , e2 and e3 . Following the naming scheme of Bekos et al. block-vertices that are adjacent to vertices of cycle C are referred to as anchors, and block-vertices that are adjacent to other block-vertices only are referred to as ancillaries. Consider an anchor a, let vl,a be the leftmost vertex of C that is adjacent to a along ℓ. If there is exactly one edge between a and vl,a (i.e., (a, vl,a ) is simple), we mark this edge. Otherwise, we mark the edge with the largest subscript among those incident to vl,a (i.e., the rightmost edge incident to the leftmost vertex vl,a of C adjacent to an anchor a) (see Fig. 2). Hence, each anchor is incident to exactly one marked edge and each vertex of C is incident to at most three marked edges. Let v be a vertex of C . We distinguish four cases based on the number of marked edges to assign the chords and the edges between C and F . Case 1. v is adjacent to three anchors a1 , a2 and a3 through three marked edges e1 , e2 and e3 , respectively (for example v1 in Fig. 1(3)). Note that v cannot be the rightmost vertex of C due to IP-4. Then, a1 , a2 and a3 are placed from left to right and directly to the right of v along ℓ. Moreover, e1 , e2 and e3 are placed to page p1 (see Fig. 3(1)). Case 2. v is adjacent to two anchors a1 and a2 through two marked edges ei and ej , respectively. If deg(v ) = 4 in Gin (C ), i.e., the two marked edges ei and ej are edges e1 and e2 , respectively, then e1 and e2 are placed to page p1 . Furthermore, if v ̸ = vk , then a1 and a2 are placed consecutively directly to the right of v (see Fig. 3(2)). Otherwise, a1 and a2 are placed consecutively directly to the left of v (see Fig. 3(3)). If deg(v ) = 5 in Gin (C ), v ̸ = vk due to degree restriction of vk by IP-4. In this case, the two marked edges are placed to page p1 and the non-marked edge is placed to page p2 . We distinguish three sub-cases for the exact placements of a1 and a2 : (i) If the two marked edges ei and ej are edges e1 and e2 , a1 and a2 are placed from left to right along ℓ and directly to the left of v (see Fig. 3(4) or v5 in Fig. 1(3)). (ii) If the two marked edges ei and ej are edges e1 and e3 , we place a1 directly to the left of v and a2 directly to the right of v (see Fig. 3(5) or v4 in Fig. 1(3)). (iii) If the two marked edges ei and ej are edges e2 and e3 , a1 and a2 are placed directly to the right of v and L(a1 ) < L(a2 ) (see Fig. 3(6) or v2 in Fig. 1(3)). Case 3. v is adjacent to one anchor a by the marked edge e. Suppose that deg(v ) = 3 in Gin (C ), then e is placed to page p1 . If v = vk , then a is placed directly to the left of v (see Fig. 3(7)). Otherwise, a is placed directly to the right of v (see Fig. 3(8)). Assume deg(v ) = 4 in Gin (C ). If v = vk , then a is placed directly to the left of v . We distinguish two sub-cases for the placements of edges. If e2 is the marked edge, then e1 is placed to page p3 and e2 is placed to page p2 (see Fig. 3(9) or v11 in Fig. 1(3)). Otherwise, e1 is the marked edge, then e1 is placed to page p1 and e2 is placed to page p2 (see Fig. 3(10)). If v ̸= vk , the marked edge is placed to page p1 and the non-marked edge is placed to page p2 . If e2 is the marked edge, a is placed directly to the right of v (see Fig. 3(11)). Otherwise, a is placed directly to the left of v (see Fig. 3(12)). Edges e1 and e2 are placed to page p3 (see v3 in Fig. 1(3)). Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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Fig. 3. The edges colored black are in p1 , the edges colored red are in p2 , and the edges colored blue are in p3 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. The edges colored black are in p1 , the edges colored red are in p2 , the edges colored blue are in p3 , and the edges colored gray are the non-embedded edges. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Now we assume deg(v ) = 5 in Gin (C ). It follows that v is not the rightmost vertex of C by IP-4. In this case, we distinguish three sub-cases. (i) If e1 is the marked edge, then a is placed directly to the left of v . Edge e1 is placed to page p1 , and e2 and e3 are placed to page p2 (see Fig. 3(13)). (ii) If e2 is the marked edge, then a is placed directly to the right of v . Moreover, e1 is placed to page p2 and e2 , and e3 are placed to page p3 (see Fig. 3(14) or v8 in Fig. 1(3)). (iii) If e3 is the marked edge, then a is placed directly to the right of v . e3 is to page p1 , and e2 and e3 are placed to page p2 (see Fig. 3(15) or v6 and v10 in Fig. 1(3)). Case 4. If v is not incident to any marked edge and deg(v ) ̸ = 2 in Gin (C ), then the edges incident to v are placed to page p2 (see Fig. 4). Up to now, we have laid out anchors and drawn the chords of C and the edges between C and F in pages. We next describe how to embed ancillaries and the edges of F . The edges of connecting two anchors are placed to p2 . Let F be a new forest formed by ancillaries, T be a connected component of F (i.e., a tree). It is obvious that F is a subgraph of F . The anchored tree T for T is obtained by adding all anchors adjacent to each ancillary of T . Suppose that T is rooted at the anchor that is leftmost along ℓ, we assign the vertices of T in order along ℓ implied by the specific Death First Search (DFS) traversal of T . If the children a′ and a′′ of a are in the counterclockwise order of edges around a, when starting from edge (a, p(a)) (where p(a) is the parent of a), then L(a′ ) < L(a′′ ). Moreover, the edges of F¯ are placed to p2 (see Fig. 5). To define the order of the trees in F , we create an auxiliary digraph GTaux whose vertices correspond to trees and there is a directed edge (vT 1 , vT 2 ) in GTaux if and only if T 1 has an anchor that is between two consecutive anchors of T 2 . Lemma 2.1 (Bekos et al. [2]). Auxiliary digraph GTaux is a directed acyclic graph. Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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Fig. 5. The vertices colored gray are ancillaries, and the vertices colored black are anchors.
Fig. 6. The vertices colored gray are ancillaries, the edges colored black are in p1 , the edges colored red are in p2 , and the edges colored blue are in p3 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 7. The edges colored black are in p1 , the edges colored red are in p2 , and the edges colored blue are in p3 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Lemma 2.1 implies an embedding of the trees in the order defined by a topological sorting of GTaux . In other words, if T 1 has an anchor that is between two consecutive anchors of T 2 along ℓ, then the tree T 1 will be embedded before T 2 . Up to now, Gin (C ) has been embedded, such that every bridgeless subgraph of Gin (C ) is contracted into a single vertex along ℓ and each edge is assigned to one of pages p1 , p2 and p3 . Next, we describe how to recursively proceed. Let a be a block-vertex of Gin (C ) with a simple outer boundary 𭟋a : w0 → w1 → . . . → wm → w0 being the outer boundary of the bridgeless subgraph it corresponds to in Gin (C ), we discuss the non-simple case later. We consider two cases (see Fig. 6): Case 1. a is an anchor and w0 is incident to the marked edge e. We place w0 as the rightmost vertex of 𭟋a on ℓ, w1 as the leftmost vertex of 𭟋a on ℓ, and wi on the left of wi+1 for i = 1, . . . , m − 1 and there are no vertices in between (see Fig. 7(1)). Assign (w0 , w1 ) to page p3 and the remaining edges of 𭟋a to page p2 (at the next level the edges embedded in counterclockwise order are similar to Gin (C ) in Gin (𭟋a )). This placement is infeasible only if there is an edge or two edges incident to w0 between (w0 , w1 ) and the marked edge e in the counterclockwise order of the edges around w0 when starting from (w0 , w1 ). In this case, w0 is to the left of w1 , . . . , wm , i.e., w0 is the leftmost vertex of 𭟋a on ℓ (this may violate IP-4, however, in this case, anchors a in Gin (𭟋a ) placed to left (right) of vr ,a ̸ = v1 (vr ,a = v1 ), where vr ,a is the rightmost vertex of 𭟋a adjacent to a along ℓ). This is impossible if there is also an edge (w0 , w′) incident to w0 between (w0 , w1 ) and the marked edge e in clockwise order of the edges around w0 when starting from (w0 , w1 ). We shall address the problem if (w0 , w′) is placed to page p2 (see Fig. 8). Case 2. a is an ancillary and w is its parent in the labeled anchored tree T to which a belongs to. We place w0 to be the leftmost vertex of 𭟋a on ℓ, w1 to be the rightmost vertex of 𭟋a on ℓ, and wi to the left of wi+1 for i = 1, . . . , m − 1 and there are no vertices in between. Moreover, (w0 , w ) is to page p3 and the remaining edges are to page p2 . This placement is infeasible in two cases. Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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Fig. 8. The placement of ancillary is determined by Lemma 2.2.
Fig. 9. Fa is not a simple cycle.
Fig. 10. The placement of a non-simple cycle being the outer boundary of the bridgeless subgraph b4 in Fig. 1 (The edges colored black are in p1 , the edges colored red are in p2 , the edges colored blue are in p3 ). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
One case is that the edges (except the edge (w0 , w1 )) incident to w0 are placed between (w0 , wm ) and (w0 , w ) in the counterclockwise order of the edges around w0 when starting from (w0 , wm ), say (w0 , w ′ ) (and (w0 , w ′′ )). Lemma 2.2 (Bekos et al. [2]). Ancillary a can be repositioned on ℓ such that: (i) a is placed between two consecutive anchors of T . (ii) The embedding specified by IP-2 is preserved and the edges (w0 , w ), (w0 , w ′ ) and (a, w ′′ ) are on page p2 and crossing-free. (iii) w0 is the leftmost vertex of 𭟋a and wi is to the left of wi+1 for i = 1, . . . , m − 1; All edges of 𭟋a are placed to page p2 , except for (w0 , wm ). By Lemma 2.2, the vertices of subtrees of T rooted at w ′ and w ′′ are placed to the left of w0 and right of w . The other case is that there is an edge e′ (and edge e′′ ) incident to w0 between (w0 , wm ) and (w0 , w ) in the counterclockwise (clockwise, respectively) order of the edges around w0 when starting from (w0 , wm ). In this case, we place the vertices of subtrees of T rooted at w ′ to right of w , and place w ′′ to the left of w0 and to the right of w (see Fig. 9). We assume 𭟋a being a non-simple cycle, i.e., 𭟋a consists of simple cycles (each of them is called a subcycle). Note that any two subcycles share at most one vertex of 𭟋a and each vertex of 𭟋a is incident to at most two subcycles. For 𭟋a , we define a tangency graph Gtan whose vertices correspond to subcycles of 𭟋a and there is an edge between every pair of subcycles that share a vertex, then Gtan is a tree. Since the degree of w0 is at most 5, w0 lies in at most two subcycles. We firstly suppose that w0 lies in exactly one subcycle. Let Gtan be rooted at the subcycle containing w0 . Then we place the subcycles of 𭟋a in the order implied by the Breadth First Search (BFS) traversal of Gtan . Assume w0 lies in two subcycles, say C1 and C2 that correspond to vertices c1 and c2 of Gtan , respectively. We place the subcycles of 𭟋a to the left (right) of w0 in the order by the BFS-traversal of the subtree rooted at c1 (c2 respectively) (see Fig. 10). We summarize the embedding of edges in the graph obtained by contracting bridgeless subgraphs of Gin (C ). The edges assigned to page p1 are the edges on C . The edges on page p2 consist of the edges in F , (v1 , vk ), the edges incident to a vertex of C in Gin (C ), and the edges between anchors. The edges embedded to page p3 are edges e2 and e3 in the case Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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that e2 is the marked edge and e1 and e3 are the non-marked edges, and e2 in the case that deg(vk ) = 4 in Gin (C ) and e2 is the marked edge, and e1 is the non-marked edge. 3. Proof of correctness In this section, we shall show that the algorithm is correct, i.e., no two edges intersect on the same page and IP-1 up to IP-5 are satisfied for an arbitrary simple cycle C . Lemma 3.1 (Bekos et al. [2]). For each ancillary a of a labeled tree T there is (i) at least an anchor of T with label smaller than that of a and (ii) at least an anchor of T with label greater than that of a. Lemma 3.2.
If v ∈ Gin (C ), then v lies to the left of vk and to the right of v1 , where C : ⟨v1 → . . . → vk → v1 ⟩.
Proof. (i). If v is a vertex on C , then v must be to the right of v1 and to the left of vk due to IP-3. (ii). If v is not a vertex on C , then we distinguish two subcases. (ii-1). We firstly assign the vertices of F of Gin (C ) on the spine ℓ (F is the forest obtained by contracting the bridgeless subgraphs of Gin (C )). Suppose that block-vertex a is an anchor, we have L(v1 ) < L(a) < L(vk ) for each anchor a in Gin (C ) according to the placement of anchors, i.e., all anchors are placed between v1 and vk . If a is an ancillary, then Lemma 3.1 implies that there are at least two anchors ai and aj , such that L(ai ) < L(a) < L(aj ). Then each ancillary is placed between v1 and vk . (ii-2). For the bridgeless subgraph represented by a block-vertex a, the vertices of Gin (𭟋a ) are placed to the position between the two vertices placed to the right and to the left of a along the spine ℓ, i.e., the vertices of Gin (𭟋a ) are placed between v1 and vk according to (a). Therefore, v lies to the left of vk and to the right of v1 for all v ∈ Gin (C ). □ Recall the placement of the anchored tree T , all edges of T are placed to the same page. Therefore, the lemma below is true here. Lemma 3.3 (Bekos et al. [2]). The placement of the anchored tree T is planar. Lemma 3.4.
There is no conflict between edges assigned to pages p1 , p2 , and p3 in Gin (C ).
Proof. We firstly consider the edges embedded on p1 in Gin (C ): all edges on C except for the one that connects its outermost vertices. Clearly, the edges assigned on p1 have no crossings. We now show that all edges assigned to p2 can be embedded without crossings. Lemma 3.3 implies that the edges of T have no crossings. Note that there is a path P (consisting of edges in p2 ) joining a pair of consecutive anchors (say u1 and ul+1 ) of T and the algorithm must place an ancillary a of T between them. Since c is nested by an edge of P and all edges of T belong to p2 , an edge connecting a to an ancillary of T placed between another pair of consecutive anchors of T would cross P. By an argument similar to that of Lemma 11 in [2], it can be seen that there is a path P(u0 → ul+1 ): u0 → uj1 → . . . → ujp → ul+1 consisting of vertices of {u0 , . . . , ul+1 }, whose edges belong to p2 and for each edge of P(u0 → ul+1 ) there are no edges in p2 with endpoints in {u0 , . . . , ul+1 } that nests it. Next, P(u0 → ul+1 ) contains at least one vertex v of C , and the vertex v is incident to at least one marked edge. According to the placement of edges, there is exactly one edge incident to C in Gin (C ), a contradiction. This gives a planar embedding of edges assigned in page p2 . Finally, each edge in p3 is incident to some vertex v on C . Note that G is a plane graph and the clockwise order of edges incident to v in the book embedding of Gin (C ) is consistent with counterclockwise order of edges around v on Gin (C ). Therefore, the edges assigned to page p3 do not cross each other due to IP-2 and IP-1. □ According to Lemma 3.2, the edges of Gin (C ) do not cross the edges of Gout (C ) on the same page. There is no conflict between edges in the same pages for Gout (C ) and Gin (C ) due to IP-1 and Lemma 3.4, respectively. Moreover, the edges incident to C do not cross on the same pages due to IP-2. Therefore, no edges intersect in the same pages. We are now ready to describe that IP-1 up to IP-5 hold when a simple cycle Cs is recursively drawn. Firstly, we prove that IP-1 up to IP-5 are satisfied for Cout , i.e., the first step of recursion holds. Lemma 3.5 (Bekos et al. [2]). Any planar graph G admits a planar drawing with a chordless outer boundary. IP-1, IP-2 and IP-3 are clearly satisfied, since Gout (Cout ) = Cout for Cout . Lemma 3.5 implies IP-5. If there is a vertex v ∈ V (Cout ) with degGin (Cout ) (v ) ≤ 4, then it is chosen as vk and all the invariant properties are satisfied. However, if such a vertex does not exist, i.e., each vertex v ∈ V (Cout ) has degGin (Cout ) (v ) = 5, this violates IP-4. This case will be addressed
by the following lemma. For the placement of the edges adjacent to C , we only consider the case that there is a vertex vk of the degree at most 4. Therefore, we have to prove the following lemma. Lemma 3.6.
If C = Cout , then IP-4 does not necessarily hold.
Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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Proof. If C = Cout and IP-4 is not satisfied, then there is no vertex v of Cout with degGin (Cout ) (v ) ≤ 4. Suppose vm is adjacent to vertices b1 , b2 and b3 through edges e1 , e2 and e3 , where b1 , b2 and b3 belong to the bridgeless subgraphs corresponding to anchors a1 , a2 and a3 (or simply, b1 , b2 and b3 belong to anchors a1 , a2 and a3 ), respectively (see Fig. 11(1)). The vertices of Gin (C ) should be placed to the left of vm . We therefore consider the following five cases. Case 1. All the edges incident to vm are marked edges. Case 2. e2 and e3 are marked edges and e1 is a non-marked edge. Case 3. e1 and e3 are marked edges and e2 is a non-marked edge. Case 4. e3 is a marked edge and e1 and e2 are non-marked edges. Case 5. e2 is a marked edge and e1 and e3 are non-marked edges. For Case 1, b1 , b2 and b3 belong to distinct anchors. For Case 2 and Case 5, a1 ̸ = a3 and a2 ̸ = a3 , but a1 = a2 is possible. For Case 3, a1 ̸ = a2 and a1 ̸ = a3 , but a3 = a2 is possible. For Case 4, a2 = a3 , and a1 = a2 = a3 are possible. (1) For Case 1, we augment G by introducing a vertex vm+1 , adding the edges (vm , vm+1 ), (vm+1 , b2 ), (vm+1 , b3 ) and removing the edges (vm , b2 ), (vm , b3 ) (see Fig. 11(2)). We denote the augmented graph by Gaug . Then IP-4 holds for Gaug . We claim that b1 , b2 and b3 are placed to the left of vm in the embedding of the augmented graph Gaug . Let Caug be the aug aug aug outer boundary of the augmented graph Gaug , and a1 , a2 and a3 be the block-vertices containing b1 , b2 and b3 in Gaug − Caug , respectively. In this case, we have (vm , b2 ) being a marked edge and vm+1 belonging to another block-vertex (containing only vm+1 ) aug aug and is adjacent to vm . Note that both a2 and a3 are ancillaries. Since vm+1 is adjacent to exactly one vertex of Cout aug aug (i.e., vm ). Then, vm+1 is to the right of a1 and the left of vm , and (vm , vm+1 ) and (vm , a1 ) are placed to page p1 according aug aug to Fig. 3(3). Furthermore, (a2 , vm+1 ) and (a3 , vm+1 ) to page p2 (see Fig. 11(3)). There are no anchors between vm+1 and vm due to degGaug (vm ) = 4. Therefore, the rightmost anchor of Gaug − Caug is vm+1 . Then, all vertices of Gaug − Caug are to the left of vm+1 . We obtain a valid embedding of G by contracting vm and vm+1 back into vm (see Fig. 11(4)). (2) For Sub-Case 2.1: a1 ̸ = a2 , we also augment G (Gaug ) by introducing a vertex vm+1 , adding the edges (vm , vm+1 ), (vm+1 , b2 ), (vm+1 , b3 ) and removing the edges (vm , b2 ), (vm , b3 ) similar to Case 1 (see Fig. 11(2)). Then IP-4 holds for Gaug . We also claim that b1 , b2 and b3 are placed to the left of vm in the embedding of the augmented graph Gaug . aug In this case, (vm , a2 ) is a non-marked edge. Then vm+1 is to the right of a1 and to the left of vm (see Fig. 2(9)). Furthermore, (vm , vm+1 ) is placed to page p2 and (vm , b1 ) is placed to page p3 , and (vm+1 , b2 ) and (vm+1 , b2 ) to page p2 (see Fig. 11(5)). We also obtain a valid embedding of G by contracting vm and vm+1 back into vm (see Fig. 11(6)). For Sub-Case 2.2: a1 = a2 = a, we augment G by introducing a vertex vm+1 , adding the edges (vm , vm+1 ), (vm+1 , b1 ), (vm+1 , b2 ) and (vm+1 , b3 ) and removing the edges (vm , b1 ), (vm , b2 ) and (vm , b3 ) (see Fig. 11(7)). Suppose a1 = a2 = a, then vm+1 must belong to a, and b1 , b1 , b2 and vm+1 appear in the clockwise traversal of the outer boundary Ca of a. Then (vm , vm+1 ) is a marked edge. Hence, a is to the right of a1 and the left of vm according to Fig. 3(7). Moreover, (vm , vm+1 ) is placed to page p1 , and (vm+1 , b1 ) is placed to page p3 and (vm+1 , b3 ), (vm+1 , b2 ) and (b1 , b2 ) are placed to page p2 (see Fig. 11(8)). By argument similar to Case 1, we obtain a valid embedding of G by contracting vm and vm+1 back into vm (see Fig. 11(9)). (3) For Sub-Case 3.1: a2 = a3 , we obtain the augmented graph Gaug similar to Case 1 (see Fig. 11(2)). Then IP-4 holds also for Gaug . And b1 , b2 and b3 are placed to the left of vm in the embedding of the augmented graph Gaug . Suppose a2 = a3 = a, then vm+1 must belong to a, and b2 , b3 and vm+1 appear in the clockwise traversal of the outer aug boundary Ca of a. Then we have (vm , a1 ) being a marked edge and vm+1 belonging to another block-vertex (containing only vm+1 ) and is adjacent to vm . Note that a is an anchor. Since vm+1 is adjacent to exactly one vertex of Cout (i.e., vm ). aug aug aug Then, vm+1 is to the right of a1 and the left of vm (see Fig. 3(3)). Furthermore, (a3 , vm+1 ) and (a2 , a3 ) to page p2 and aug (a2 , vm ) to page p3 (see Fig. 11(10)). There are no anchors between vm+1 and vm due to degGaug (vk ) = 4. Therefore, the rightmost anchor of Gaug − Caug is vm+1 . Then, all vertices of Gaug − Caug are to the left of vm+1 . We obtain a valid embedding of G by contracting vm and vm+1 back into vm (see Fig. 11(11)). For Sub-Case 3.2: a2 ̸ = a3 , we augment G by introducing three vertices (say vm+1 , vm+2 and vm+3 ) to the right of vm , adding the edges (vm , vm+1 ), (vm+1 , vm+2 ), (vm+2 , vm+3 ), (vm , vm+2 ), (vm+3 , b2 ), and (vm+3 , b3 ), and removing the edges (vm , v1 ), (vm , b1 ) (see Fig. 11(12)). Note that (vm+3 , b2 ) is a non-marked edge, (vm+3 , b3 ) and (vm+3 , b2 ) are the marked edges. We have b1 , b2 and b3 are to the left of vm due to Fig. 3(12) and Fig. 3(9), respectively. Furthermore (vm , b1 ) is placed to page p1 , (vm+3 , b2 ) is placed to page p2 , and (vm+1 , b3 ) to page p3 (see Fig. 11(13)). There are no vertices of Gaug between vm and vm+3 , except for vm+1 and vm+2 . Hence, all vertices of Gaug − Caug are to the left of vm . If we contract vm , vm+1 , vm+2 and vm+3 back into vm , then we obtain a valid embedding of G (see Fig. 11(14)). (4) For Sub-Case 4.1: a2 = a3 ̸ = a1 , we also obtain the augmented graph Gaug similar to Case 1 (see Fig. 11(2)). Then IP-4 holds for Gaug . We also claim that b1 , b2 and b3 are placed to the left of vm in the embedding of the augmented graph Gaug . In this case, (vm , b2 ) is a non-marked edge. Then vm+1 is to the right of a1 and to the left of vm (see Fig. 3(9)). Furthermore, (vm , vm+1 ), (vm+1 , b3 ) and (b2 , b3 ) to page p2 , (vm+1 , b2 ) and (vm , b1 ) is placed to page p3 (see Fig. 11(15)). We also obtain a valid embedding of G by contracting vm and vm+1 back into vm (see Fig. 11(16)). For Sub-Case 4.2: a1 = a2 = a3 = a, obtain the augmented graph Gaug similar to Case 2.2 (see Fig. 11(7)). Then vm+1 must belong to a, and b1 , b2 , b3 and vm+1 appear in the clockwise traversal of the outer boundary Ca of a. Then (vm , vm+1 ) Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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Fig. 11. Dotted line is removed, the edges colored black are in p1 , the edges colored red are in p2 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
is a marked edge. Hence, a is to the right of a1 and the left of vm according to Fig. 3(7). Moreover, (vm , vm+1 ) is placed to page p1 , and (vm+1 , b2 ), (b2 , b3 ) and (b1 , b3 ) are placed to page p2 , and (vm+1 , b1 ) is placed to page p3 (see Fig. 11(17)). By argument similar to Case 1, we obtain a valid embedding of G by contracting vm and vm+1 back into vm (see Fig. 11(18)). For Sub-Case 4.3: a2 ̸ = a3 , we augment G by introducing a vertex vm+1 , adding the edges (vm , vm+1 ), (vm+1 , b1 ), (vm+1 , b2 ) and removing the edges (vm , b1 ), (vm , b2 ) (see Fig. 11(19)). Note that (vm , vm+1 ) is a non-marked edge and (vm , b3 ) is a marked edge. We have b1 , b2 and b3 are to the left of vm due to Fig. 3(9). Furthermore, (vm , vm+1 ) is placed to page p3 , (vm , b3 ), (vm+1 , b1 ) and (vm+1 , b2 ) are placed to page p2 (see Fig. 11(20)). Similar to that of Case 1, we obtain a valid embedding of G by contracting vm and vm+1 back into vm (see Fig. 11(21)). Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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(5) For Sub-Case 5.1: a1 ̸ = a2 . By argument similar to Sub-Case 2.1, we can obtain the embedding of G (see Fig. 11(5) and (6)). For Sub-Case 5.2: a1 = a2 = a. By argument similar to Sub-Case 2.2, we can obtain the embedding of G (see Fig. 11(8) and (9)). □ Lemma 3.7 (Bekos et al. [2]). Assume that all trees that precede T in a topological sorting of GTaux have been drawn on page p2 without crossings by preserving the embedding specified by IP-2. When T is drown, the embedding specified by IP-2 is also preserved. Lemma 3.8 (Bekos et al. [2]). Let v be a vertex of a cycle C with degree 2 in Gin (C ) that is not the left/right-most vertex of C . Let also vr (vl , resp.) be its next neighbor on C to its right (left, resp.). Since, edge (v, vr ) belongs to C , it is placed to the page p1 . In fact, it can also be placed to page p2 without crossings and while the embedding specified by IP-2 is maintained (the page p1 is the bottom-page, and the page p2 is the top-page in [2]). Lemma 3.9. Let v be a vertex of a cycle C with degree 3 or 2 in Gin (C ) that is not the left/right-most vertex of C . Let also vr (vl , resp.) be its next neighbor on C to its right (left, resp.). The algorithm places (v, vr ) to page p2 . In fact, it can also be placed to page p3 without crossings and while the embedding specified by IP-2 is maintained. Proof. If deg(v ) = 2 in Gin (C ), then claim holds according to Lemma 3.8. Suppose deg(v ) = 3 in Gin (C ). We distinguish two cases to discuss as follows. (i) Assume that v is incident to a non-marked edge e. Then the anchor a is placed to the left of v according to the placement of anchor. Thus, IP-2 is maintained for v . We only discuss whether IP-2 is maintained for vr . If no block-vertex is placed between vertices v and vr , then obviously edge (v, vr ) can be placed to page p2 without introducing edge-crossings and without changing the combinatorial embedding specified by IP-2 (recall that vertices v and vr are consecutive vertices of cycle C ). Thus, we may assume without loss of generality that there are block-vertices placed between vertices v and vr of cycle C . We will prove that we can move the block-vertices in between to the left of vertex v , so that vertices v and vr become consecutive along spine ℓ. The aforementioned move is not possible, only if there is an anchor c between vertices v and vr , such that edge (v, vr ) is placed to page p2 . To overcome this problem, we can potentially place vertex v between vertices c and vr . However, in this case, if (c , vr ) is placed to page p1 , then edges (c , vr ) and (v, vl ) would cross according to the placement of the edges incident to vertices in C . If (c , vr ) is placed to page p2 , then edges (c , vr ) and (v, a) would cross. In either cases, we can cope with problematic situation if edge (c , vr ) is replaced to page p3 . (ii) If v is adjacent to an anchor a through one marked edge e, then we move a to the right of vr and (v, vr ) can be placed to page p2 without crossings, which can be shown similar to the case that there are block-vertices placed between vertices v and vr of cycle C in (i). □ We are ready to describe how the recursive step holds as following: IP-1 up to IP-5 hold for an arbitrary simple cycle Cs . Each edge is embedded to one of three pages: page p1 , page p2 , and page p3 , and no two edges intersect on the same page. Therefore, IP-1 is satisfied. Lemma 3.7 implies IP-2. If Cs is the outer boundary of a block-vertex or a leaf of the tangency tree, then IP-3 trivially holds. If Cs is the outer boundary of a non-leaf of the tangency tree, it contains at least one edge on page p3 . This violates IP-3. In this case, we re-embed it to page p1 using Lemma 3.8. IP-4 holds. In fact, if Cs is the outer boundary of a block-vertex or root of the tangency tree of a non-simple outer boundary, then at least one vertex of Cs is adjacent to Gout (Cs ). If Cs is the outer boundary of internal node of the tangency tree of a non-simple outer boundary, then its leftmost vertex has at least two neighbors in Gout (Cs ). The following lemma implies that IP-5 does not necessarily hold for simple cycle Cs . Lemma 3.10.
IP-5 does not necessarily hold for an arbitrary simple cycle C .
Proof. If IP-5 does not hold for some simple cycle Cs : w1 → w2 → . . . → wm , then we may distinguish three cases. Case 1. degGin (C ) (w1 ) = 4 and w1 is incident to one chord of C . Case 2. deg(w1 ) = 5 in Gin (C ) and w1 is incident to one chord of C . Case 3. deg(w1 ) = 5 in Gin (C ) and w1 is incident to exactly two chords of C (say (w1 , wi1 ) and (w1 , wi2 ), where i1 ∈ {3, . . . , m − 1} and i1 < i2 ). For Case 1 and Case 2, w1 is incident to exactly one chord (say (w1 , wi ), i ∈ {3, . . . , m − 1}) of C . In general, (w1 , wi ) ∈ P(w1 → wj ), where P(w1 → wj ) is a path of chords from w1 to wj on page p2 . Suppose wq ∈ P(w1 → wj ), (wq , wx ) and (wq , wy ) are chords, x < y, we chose (wq , wy ) ∈ P(w1 → wj ). The restriction implies that P(w1 → wj ) is uniquely defined (see Fig. 12(1)). We refer to it as the separating path of chords of Cs , since it splits Gin (Cs ) into two subgraphs: (i) Gin (Cl ) with outer boundary Cl consisting of the edges (w1 , w2 ), (w2 , w3 ), . . . , (wj−1 , wj ) and the edges of P(w1 → wj ) (see Fig. 12(2)), and (ii) Gin (Cr ) with outer boundary Cr consisting of the edges (wj , wj+1 ), . . . , (wm−1 , wm ), (wm , w1 ) and the edges of P(w1 → wj ) (see Fig. 12(3)). Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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Fig. 12. Dotted line is removed, the edges colored black are in p1 , the edges colored red are in p2 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
We now describe how the two sub-instances Gin (Cl ) and Gin (Cr ) can be recursively solved. Note that v must be to the right of v1 , where v is a vertex of Gin (C ). According to the assignments of anchors, we consider the following sub-cases (for Case 1 and Case 2). Sub-Case 1.1: degGin (C ) (w1 ) = 4 and w1 is incident to exactly one chord e1 of C . Sub-Case 2.1: deg(w1 ) = 5 in Gin (C ) and w1 is incident to exactly one chord e2 of C . Sub-Case 2.2: deg(w1 ) = 5 in Gin (C ) and w1 is incident to exactly one chord e1 of C . Observe that if i ̸ = j, then Cl is a non-simple cycle, i.e., Cl consists of some simple subcycles, for which IP-5 holds (hence they can be recursively drawn), except for the first one, that is leftmost embedded along ℓ. Note that j ̸ = m. In fact, if j = m and deg(w1 ) = 3 (deg(w1 ) = 4) in Gin (Cr ), then e3 is a bridge according to the placements of chords or e2 is not a chord for some vertex vs (vs belong to P(w1 → wj )). If e3 is a bridge, a contradiction is obtained since G is 2-connected. If e2 is not a chord for some vertex vs , a contradiction is obtained since vs belongs to P(w1 → wj ) ((i) if e2 is a marked edge, then e3 is placed to page p2 (see to Fig. 3(14)); (ii) if e2 is a non-marked edge, then the edges incident to vs are placed to page p2 due to Case 4 in the placements of edges incident to C ). In Sub-Case 1.1, deg(w1 ) = 2 in Gin (Cl ), i.e., deg(w1 ) = 3 in Gin (Cr ). We modify Gin (Cl ) as follows. Remove w1 and join the edges (w2 , wi ) and (w2 , wm ). Then Gin (Cl′ ) (see Fig. 12(4)) has fewer vertices than Gin (C ). We can benefit from this by proceeding recursively, as we initially did with Gin (C ). Eventually, IP-5 should hold for some vertex wp , otherwise a graph with at most 3 vertices on its outer boundary should have a chord, a contradiction. To complete the embedding of Gin (C ), we remove (w2 , wi ) and (w2 , wm ), and connect the neighbors of wj in Gin (Cr′ ) with its copy in Gin (Cl′ ) (no crossings are introduced, since the two copies of wj in Gin (Cl′ ) and Gin (Cr′ ) are consecutive along ℓ). It remains to replace the copy of wj in Gin (Cr′ ) with w1 , and add (w1 , w2 ) and (w1 , wi ) (see Fig. 12(6)). In Sub-Case 2.1, deg(w1 ) = 3 in Gin (Cl ) and deg(w1 ) = 3 in Gin (Cr ). We modify Gin (Cl ) similarly to that of Sub-Case 1.1, we obtain a valid placement by placing exactly w1 and a1 to the right of wj and a2 to the right of wp , where a1 and a2 are adjacent to w1 through two marked edges e1 and e2 , respectively (see Fig. 12(7)). In Sub-Case 2.2, deg(w1 ) = 2 in Gin (Cl ) and deg(w1 ) = 4 in Gin (Cr ). Similarly to that of Sub-Case1.1, w1 , a1 and a2 are placed to the right of wj and the left of wj+1 , where a1 and a2 are adjacent to w1 through two marked edges e1 and e2 , respectively. Hence, we obtain a valid embedding (see Fig. 12(8)). For Case 3, we only consider this case due to the placements of anchors. Sub-Case 3.1: deg(w1 ) = 5 in Gin (C ) and w1 is incident to exactly two chords e1 and e2 of C . Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.
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Fig. 13. The non sub-hamiltonian graph of maximum degree 7.
In this case, we choose (w1 , wi2 ) ∈ P(w1 → wj ). If wq ∈ P(w1 → wj ), (wq , wx ) and (wq , wy ) are chords, x < y, then (wq , wy ) ∈ P(w1 → wj ). Hence, P(w1 → wj ) is uniquely defined, and w1 and a are placed directly to the right of wj by the argument of Sub-Case 1.1 (see Fig. 12(9)). □ Theorem 3.11. There is a quadratic-time algorithm to construct book embedding for planar graphs of maximum degree 5 on 3 pages. Proof. Given a planar graph of maximum degree 5 of n vertices, the computation of the bridgeless-subgraphs, the topological sorting of GTaux , BFS-traversals on the tangency trees Gtan and DFS-traversals on the anchored tree T can be done in linear time. Hence the algorithm runs in O(n2 ) time. □ 4. Conclusions and open problems We have noted previously that if the pagenumber of a maximal planar graph is two, then it is hamiltonian. Since there is a planar 8-graph which is non-hamiltonian in [4]. There also is a planar 7-graph requiring at least three pages [20] (the proof is rather straightforward) (see Fig. 13). Then the pagenumber of planar 8-graphs and planar 7-graphs is at least 3. This paper has presented an O(n2 ) time algorithm for embedding planar 5-graphs into books of three pages. The natural open problem is whether two pages are sufficient for planar 5-graphs. Hoffmann and Klemz [15] show that a 3-connected planar 5-graph is subhamiltonian and then it can be embedded in two pages. Acknowledgments The research is supported by NSFC (No. 11671296), Program for the Innovative Talents of Higher Learning Institutions of Shanxi, Natural Science Foundation of Shanxi Provincial (No. 201801D221193), and Shanxi Scholarship Council of China. References [1] G.A. Atneosen, On the Embeddability of Compacta in N-Books: Intrinsic and Extrinsic Properties (Ph.D. thesis), Department of Mathematics, Michigan State University, 1968. [2] M.A. Bekos, M. Gronemann, C.N. Raftopoulou, Two-page book embeddings of 4-planar graphs, Algorithmica 75 (2016) 158–185. [3] M.A. Bekos, M. Kaufmann, C. Zielke, The Book Embedding Problem from a SAT Solver Perspective, GD 2015, 2015. [4] F. Bernhart, P.C. Kainen, The book thickness of a graph, J. Combin. Theory Ser. B 27 (3) (1979) 320–331. [5] T. Bilski, Optimum embedding of complete graphs in books, Discrete Math. 182 (1998) 21–28. [6] J.A. Bondy, U.S.R. Murty, Graph Theory with Application, North-Holland, New York, 1976. [7] J.F. Buss, P.W. Shor, On the pagenumber of planar graphs, in: Proceedings of the 16th ACM Symposium on Theory of Computing, STOC84, 1984, pp. 98–100. [8] F.R.K. Chung, F.T. Leighton, A.L. Rosenberg, Embedding graphs in books a layout problem with applications to VLSI design, SIAM J. Algebr. Discrete Methods 8 (1987) 33–58. [9] V. Dujmovic, D.R. Wood, Graph treewidth and geometric thickness parameters, Discrete Comput. Geom. 37 (2007) 641–670. [10] H. Enomoto, T. Nakamigawa, K. Ota, On the pagenumber of complete bipartite graphs, J. Combin. Theory Ser. B 71 (1997) 111–120. [11] S. Even, A. Itai, Queues, stacks, and graphs, in: Z. Kohavi, A. Paz (Eds.), Theory of Machines and Computations, Academic Press, New York, 1971, pp. 71–86. [12] G. Ewald, Hamiltonian Circuits in Simplicial Complexes, D. Reidel Publishing Company, Dordrecht-Holland, 1973. [13] L. Heath, Embedding planar graphs in seven pages, in: Proceedings of the 25th IEEE Annual Symposium on Foundations of Computer Science, 1984, pp. 74–89. [14] L. Heath, Algorithms for Embedding Graphs in Books (Ph.D. thesis), University of North Carolina, Chapel Hill, 1985. [15] M. Hoffmann, B. Klemz, Triconnected planar graphs of maximum degree five are subhamiltonian, ESA-2019. https://algo2019.ak.in.tum.de/index. php/algo-program, http://drops.dagstuhl.de/opus/volltexte/2019/11179/. [16] S. Istrail, An algorithm for embedding planar graphs in six pages, 1986, manuscript.
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Please cite this article as: X. Guan and W. Yang, Embedding planar 5-graphs in three pages, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.020.