- Email: [email protected]
Connected domination in maximal outerplanar graphs
Discrete Applied Mathematics xxx (xxxx) xxx
Contents lists available at ScienceDirect
Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Connected domination in maximal outerplanar graphs✩ Wei Zhuang School of Applied Mathematics, Xiamen University of Technology, Xiamen 361024, PR China
article
info
Article history: Received 7 March 2017 Received in revised form 28 December 2019 Accepted 30 January 2020 Available online xxxx Keywords: Outerplanar graph Connected domination Striped maximal outerplanar graph
a b s t r a c t A subset S of vertices in a graph G = (V , E) is a connected dominating set of G if every vertex of V \ S is adjacent to a vertex in S and the subgraph induced by S is connected. The minimum cardinality of a connected dominating set of G is the connected domination number γc (G). We show that the connected domination number k ⌋ − 2, ⌊ 2(n3−k) ⌋}, where k is of a maximal outerplanar graph G is bounded by min{⌊ n+ 2 the number of vertices of degree 2 in G. Moreover, we improve this upper bound for striped maximal outerplanar graphs and characterize the graphs that achieve the bound. © 2020 Elsevier B.V. All rights reserved.
1. Introduction Let G be a simple graph. The vertex set and edge set of G are denoted by V (G) and E(G), respectively. We denote the order of G by |V (G)|. If the graph G is clear from the context, we simply write |G| or n rather than |V (G)|. The open neighborhood of a vertex v ∈ V (G) is defined by NG (v ) = {u | v u ∈ E(G)}, and the closed neighborhood of v is NG [v] = NG (u) ∪ {v}. We denote the degree of v by dG (v ) = |NG (v )|. Moreover, when no confusion can arise, NG (v ), NG [v] and dG (v ) are simplified by N(v ), N [v] and d(v ), respectively. The maximum degree and minimum degree of G are denoted by ∆(G) and δ (G), respectively. The subgraph of graph G induced by a subset S ⊆ V (G) is denoted by G[S ]. For two vertices u and v in a connected graph G, the distance dG (u, v ) between u and v is the length of a shortest (u, v )-path in G. The distance between a vertex v ∈ V (G) and a set S ⊆ V (G) is defined as dG (v, S) = min{d(v, x) | x ∈ S }. A fan Fn is a graph of order n + 1 obtained by adding a vertex v to Pn with v adjacent to each vertex of Pn . A graph G is outerplanar if it can be embedded in the plane such that all vertices belong to the boundary of its outer face. An outerplanar graph G is maximal if G + uv is not outerplanar for any two nonadjacent vertices u and v . Any terminology not defined here follows that of [23]. A dominating set of a graph G is a set S ⊆ V (G) such that every vertex in G is either in S or adjacent to a vertex in S. A dominating set S is a connected dominating set if the subgraph of G induced by S is connected. The domination number (connected domination number) of G, which is denoted by γ (G) (γc (G)), is the minimum cardinality of a dominating set (connected dominating set). The concept of connected domination was introduced by Sampathkumar and Walikar [19]. More details and information are shown in [1,2,5,6,9,11,13,20]. A related problem is that of finding a spanning tree of a graph with the maximum number of leaves (maximum leaf spanning tree problem). If such a spanning tree is found, by deleting the leaves of this tree, one obtains a minimum connected dominating set. Some results on this problem have been obtained [12,18,21,24]. In 1983, Laskar and Pfaff [15] showed the NP-hardness of computing the connected domination number or the minimum ✩ The research is supported by National Natural Science Foundation of China (No. 11301440) and Natural Science Foundation of Fujian Province, China (CN)(2015J05017). E-mail address: [email protected]. https://doi.org/10.1016/j.dam.2020.01.033 0166-218X/© 2020 Elsevier B.V. All rights reserved.
Please cite this article as: W. Zhuang, Connected domination in maximal outerplanar graphs, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.01.033.
2
W. Zhuang / Discrete Applied Mathematics xxx (xxxx) xxx
connected dominating set. After that, the approximation algorithm design has become an important issue in the study of connected dominating sets [10,14,25]. In 1996, Matheson and Tarjan [17] proved that any triangulated disc G with n vertices satisfies γ (G) ≤ ⌊ 3n ⌋, and conjectured that γ (G) ≤ ⌊ 4n ⌋ for every n-vertex triangulation G with sufficiently large n. In 2013, Campos and Wakabayashi investigated this problem for maximal outerplanar graphs and showed in [4] that if G is a maximal outerplanar graph of k , where k is the number of vertices of degree 2 in G. By using a simple coloring method, Tokunaga order n, then γ (G) ≤ n+ 4 t proved the same result independently in [22]. Li et al. [16] improved the result by showing that γ (G) ≤ n+ , where t is the 4 number of pairs of consecutive 2-degree vertices with distance at least 3 on the outer cycle. In [3,7,8], the same question was investigated for total domination and secure domination. In this paper, we investigate the same question for connected domination. For outerplanar graphs of order n, which are not maximal, the upper bound n − 2 is the best possible (note that the path and cycle are both outerplanar graphs). We k ⌋ − 2, ⌊ 2(n3−k) ⌋}, show that the connected domination number of a maximal outerplanar graph G is bounded by min{⌊ n+ 2 where k is the number of vertices of degree 2 in G. Moreover, we improve this upper bound for striped maximal outerplanar graphs and characterize the graphs that achieve the bound. 2. Preliminaries A maximal outerplanar graph G can be embedded in the plane in such a way that the boundary of the outer face is a Hamiltonian cycle, and each inner face is a triangle. A maximal outerplanar graph embedded in the plane in this (unique) way is called a maximal outerplane graph (see [3,4]). For this embedding of G, we denote the Hamiltonian cycle, which is the boundary of the outer face, by HG . An inner face of a maximal outerplane graph G is an internal triangle if it is not adjacent to the outer face. A maximal outerplane graph without an internal triangle is called striped. We also say that a maximal outerplanar graph is striped (resp. contains internal triangles) when its outerplane embedding is striped (resp. contains internal triangles). To simplify notation, in most of the proofs, in which an embedding of an outerplanar graph G is considered, we assume that G is an outerplane graph (even if this is not stated explicitly). We may use the term triangle to refer to an inner face or a subgraph, which is isomorphic to K3 . The following results are given in [4,22]. Proposition 2.1 ([4]). Let G be a maximal outerplanar graph of order n ≥ 4. If G has k internal triangles, then G has k + 2 vertices of degree 2. Theorem 2.2 ([22]). Let G be a maximal outerplanar graph of order n ≥ 4, then G can be 4-colored such that every cycle of length 4 in G has all four colors. Theorem 2.3 ([22]). Let G be a maximal outerplanar graph of order n ≥ 4. Suppose that G is 4-colored such that every cycle of length 4 in G has all four colors, and let R ⊂ V (G) contain all vertices of some given color. Then R dominates all vertices of V (G) except those of degree 2. 3. Upper bounds for the connected domination number of maximal outerplanar graphs A minor of a graph G is a graph which can be obtained from G by deleting vertices and deleting or contracting edges. Given a graph H, a graph G is H-minor free if no minor of G is isomorphic to H. A K4 -minor free graph G is maximal if G + uv is not a K4 -minor free graph for any two non-adjacent vertices u and v of G. A 2-tree is recursively defined as follows. A single edge is a 2-tree. Any graph obtained from a 2-tree by adding a new vertex and making it adjacent to the end vertices of an existing edge is also a 2-tree. It is well known that the maximal K4 -minor free graphs are exactly the 2-trees. We know that a maximal outerplanar graph must be a maximal K4 -minor free graph. Hence, before giving an upper bound for the connected domination number of a maximal outerplanar graph, we prove the following stronger result. Theorem 3.1. Let G be a maximal K4 -minor free graph of order n ≥ 4. If k is the number of vertices of degree 2 in G, then γc (G) = 1 when n = 4, 5; and γc (G) ≤ ⌊ 2(n3−k) ⌋ when n ≥ 6. Proof. If n = 4 or 5, G is isomorphic to one of the three graphs in Fig. 1, and clearly, γc (G) = 1. We consider the case when n ≥ 6. Let T = {v | dG (v ) = 2}, G0 = G − T . Note that G0 is still a maximal K4 -minor free graph. Let Gi = Gi−1 − vi for i = 1, 2, . . . , n − |T | − 3, where vi is a vertex of degree 2 in Gi−1 . Since a maximal K4 -minor free graph is exactly a 2-tree, it follows from the definition of the 2-tree that Gn−|T |−3 is a K3 . We can give a proper 3-coloring to G0 by assigning color 1, 2 and 3 to the three vertices of Gn−|T |−3 respectively and color vi with the remaining color which does not appear in NGi−1 (vi ) at each stage. Denote by Vt the set of vertices assigned color t, where t = 1, 2, 3. Without loss of generality, we may assume that |V1 | ≤ |V2 | ≤ |V3 |, then |V1 | + |V2 | ≤ ⌊ 32 |G0 |⌋. Let S = V1 ∪ V2 . Next, we will show that S is a dominating set of G and G[S ] is connected. Please cite this article as: W. Zhuang, Connected domination in maximal outerplanar graphs, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.01.033.
W. Zhuang / Discrete Applied Mathematics xxx (xxxx) xxx
3
Fig. 1. The case of n = 4 or 5.
Fig. 2. Rt .
It is easy to see that S is a dominating set of G0 . Since G is a 2-tree, we have NG (u) ∩ S ̸ = ∅ for each vertex u of T . In other words, S is a dominating set of G. Finally, we only need to show that if G[V (Gi ) ∩ S ] is connected, then G[V (Gi−1 ) ∩ S ] is also connected. Assume that the subgraph of Gi induced by V (Gi ) ∩ S is connected. If vi is assigned color 3, then G[V (Gi ) ∩ S ] = G[V (Gi−1 ) ∩ S ]. If vi is assigned color 1 or 2, we note that at most one neighbor of vi is assigned color 3 in Gi−1 , thus the subgraph of Gi−1 induced by V (Gi−1 ) ∩ S is connected. Therefore, S is a connected dominating set 2(n−k) of G that satisfies |S | ≤ ⌊ 3 ⌋. □ Based on the above theorem, we have the following conclusion. Corollary 3.2. Let G be a maximal outerplanar graph of order n ≥ 4. If k is the number of vertices of degree 2 in G, then
γc (G) = 1 when n = 4, 5; and γc (G) ≤ ⌊ 2(n3−k) ⌋ when n ≥ 6.
In what follows, we are ready to introduce a family of graphs depicted in [8]. We note that the connected domination number of each graph in this family achieves the bounds given by Theorem 3.1 and Corollary 3.2. Let Rt be the graph shown in Fig. 2, and suppose V (Rt ) = 6t, where t ≥ 1. Let x be a 2-degree vertex of Rt , y and z be its neighbors. Join x and y to a new vertex a, and x and z to a new vertex b. Repeat this operation on all 2-degree vertices of Rt , and call the resulting graph R′t . Note that V (R′t ) = 6t + 4t = 10t, and R′t has 4t vertices of degree 2. From the definition of 2-tree, it is easy to see that R′t is a 2-tree. That is, it is a maximal K4 -minor free graph. Moreover, 2(|R′ |−k)
it is also a maximal outerplanar graph. It can be easily checked that γc (R′t ) = ⌊ t3 ⌋. Hence, we have that the bounds given by Theorem 3.1 and Corollary 3.2 are both tight. Let G be a maximal outerplanar graph and HG be the Hamiltonian cycle of the outer face of G. Suppose that x and y are two vertices on HG . We call the path on HG that goes in the clockwise direction from x to y the xy-segment of HG , and the other path the yx-segment of HG . Observation 3.3. Let G be a maximal outerplanar graph and HG be the Hamiltonian cycle of the outer face of G, xy ∈ E(G). If the xy-segment (yx-segment) of HG contains at least one vertex other than x and y, then the xy-segment (yx-segment) of HG contains a vertex of N(x) ∩ N(y). Moreover, we can obtain another upper bound by applying Tokunaga’s method [22]. Theorem 3.4. Let G be a maximal outerplanar graph of order n ≥ 3. If k is the number of vertices of degree 2 in G, then γc (G) ≤ ⌊ n+2 k ⌋ − 2. k Proof. If n = 3, G is a triangle, then k = 3. Thus, γc (G) = 1 = ⌊ n+ ⌋ − 2. We assume that n ≥ 4. 2 Let S = {v1 , v2 , . . . , vk } be the set of vertices of G having degree 2, and let ui be one of the two vertices adjacent to vi for each i ∈ {1, 2, . . . , k}. We take a set of k additional vertices S ′ = {v1′ , v2′ , . . . , vk′ } and construct a graph G′ such that V (G′ ) = V (G) ∪ S ′ and E(G′ ) = E(G) ∪ {v1′ v1 , v2′ v2 , . . . , vk′ vk } ∪ {v1′ u1 , v2′ u2 , . . . , vk′ uk }. Clearly, G′ is also a maximal outerplanar graph, so G′ can be 4-colored such that every cycle of length 4 in G′ has all four colors. Let Vi (i = 1, 2, 3, 4) be the subset of V (G′ ) which is colored by color i. Without loss of generality, we may assume that |V1 | ≤ |V2 | ≤ |V3 | ≤ |V4 |. Let ki be the number of vertices of Vi with degree 2 in G′ . Note that k1 + k2 + k3 + k4 = k.
Please cite this article as: W. Zhuang, Connected domination in maximal outerplanar graphs, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.01.033.
4
W. Zhuang / Discrete Applied Mathematics xxx (xxxx) xxx
Suppose that |V1 | = 1 and v is the unique vertex in V1 . Let w be any vertex of G. Then, w has degree at least three in G′ . Let w1 , w2 and w3 be consecutive vertices of NG′ (v ) in this order. Note that ww1 w2 w3 forms a cycle of length 4. It follows from Theorem 2.2 that v ∈ {w, w1 , w2 , w3 }, i.e., each vertex of G is dominated by v . In addition, if v ∈ V (G′ ) \ V (G), then v is a 2-degree vertex of G′ , and it can only dominate its two neighbors, which contradicts the condition that n ≥ 4. Thus, v ∈ V (G), and {v} is a connected dominating set of G. The proof is complete. Next, we consider the case when |V1 | ≥ 2. Let HG′ be the Hamiltonian cycle of the outer face of G′ . Since |V1 | ≤ |V2 | ≤ |V3 | ≤ |V4 |, |G′ | ≥ 8. We recursively construct a sequence of subsets S1 , S2 , . . . of V1 as follows. First take a vertex v ∈ V1 , and let S1 = {v}. For i ≥ 2, choose any vertex v ′ ∈ V1 \ Si−1 with dG′ (v ′ , Si−1 ) ≤ 2, and let Si = Si−1 ∪ {v ′ }. We end up the procedure at a set Sp when Sp = V1 or dG′ (x, Sp ) ≥ 3 for any x ∈ V1 \ Sp . Next, we are ready to show that Sp = V1 . By contradiction, let z be a vertex of V1 \ Sp , then we have that dG′ (z , x) ≥ 3 for each x ∈ Sp . Let z ′ ∈ Sp and z ′′ ∈ NG′ (z ′ ). It is easy to see that z is in the z ′ z ′′ -segment or z ′′ z ′ -segment of HG′ . Denote by lz (z ′ , z ′′ ) the length of the segment between z ′ and z ′′ containing z. We select s ∈ Sp and s′ ∈ NG′ (s) such that lz (s, s′ ) ≤ lz (z ′ , z ′′ ) for any z ′ ∈ Sp and z ′′ ∈ NG′ (z ′ ). Without loss of generality, we assume that z is in the ss′ -segment of HG′ , and s′ ∈ V2 . By Observation 3.3, a vertex z1 ∈ NG′ (s) ∩ NG′ (s′ ) is in the ss′ -segment of HG′ . Moreover, z1 belongs to V3 or V4 , say V3 . So, z1 ̸ = z. We have that z1 is in the sz-segment of HG′ (if z1 is in the zs′ -segment of HG′ , then lz (s, z1 ) < lz (s, s′ ), a contradiction). By Observation 3.3, a vertex z2 ∈ NG′ (z1 ) ∩ NG′ (s′ ) is in the z1 s′ -segment of HG′ . Note that sz1 z2 s′ forms a cycle of length 4. It follows from Theorem 2.2 that z2 ∈ V4 , and so z2 ̸ = z. z2 is in the z1 z-segment or zs′ -segment of HG′ . Case a1. z2 is in the z1 z-segment of HG′ . By Observation 3.3, a vertex z3 ∈ NG′ (z2 ) ∩ NG′ (s′ ) is in the z2 s′ -segment of HG′ . z1 z2 z3 s′ forms a cycle of length 4. By Theorem 2.2, z3 ∈ V1 . It follows from the definition of Sp and dG′ (s, z3 ) = 2 that z3 ∈ Sp , and so z3 ̸ = z. z3 is in the z2 zsegment or zs′ -segment of HG′ . In the former case, lz (z3 , s′ ) < lz (s, s′ ), a contradiction. In the latter case, lz (z2 , z3 ) < lz (s, s′ ), a contradiction. Case a2. z2 is in the zs′ -segment of HG′ . By Observation 3.3, a vertex z3 ∈ NG′ (z1 ) ∩ NG′ (z2 ) is in the z1 z2 -segment of HG′ . z1 z3 z2 s′ forms a cycle of length 4. By Theorem 2.2, z3 ∈ V1 . Similar to the argument of Case a1, we have z3 ∈ Sp and z3 ̸ = z. z3 is in the z1 z-segment or zz2 -segment of HG′ . In the former case, lz (z3 , z2 ) < lz (s, s′ ), a contradiction. In the latter case, lz (z3 , z1 ) < lz (s, s′ ), a contradiction. Hence, Sp = V1 . By Theorem 2.3, we know that each Vi (i = 1, 2, 3, 4) dominates all vertices of V (G). Combining the definition of Sp , there exists a connected subgraph of G′ with order 2|V1 | − 1, say G1 , which contains all vertices of V1 . Let x be a 2-degree vertex of G′ , if x ∈ V1 , then x belongs to V (G1 ), and one of its neighbors also belongs to V (G1 ). It implies that the set D′1 which is obtained from V (G1 ) by removing all 2-degree vertices of G′ belonging to V1 , can still dominate all vertices of V (G), the subgraph induced by D′1 is still connected. That is, D′1 is a connected dominating set of G with cardinality 2|V1 | − 1 − k1 . Similarly, for each i ∈ {2, 3, 4}, there exists a connected subgraph of G′ with order 2|Vi |− 1, say Gi , which contains all vertices of Vi . Moreover, we can always obtain a connected dominating set D′i of G with cardinality 2|Vi | − 1 − ki , which is obtained from V (Gi ) by removing all 2-degree vertices of G′ belonging to Vi . Assume that |G′ | = 4t + h, where h = 0, 1, 2, 3. Since |V1 | ≤ |V2 | ≤ |V3 | ≤ |V4 | and |G′ | ≥ 8, we have t ≥ 2 and |V1 | ≤ t. Next, we consider three cases for the four possible values of h. Case b1. h ∈ {2, 3}. |G′ | k ⌋ − 2. In this case, γc (G) ≤ 2|V1 | − 1 ≤ 2t − 1 = ⌊ 2 ⌋ − 2 = ⌊ n+ 2 Case b2. h = 0. |G′ | We have that |V1 | < t or |V1 | = |V2 | = |V3 | = |V4 | = t. In the former case, γc (G) ≤ 2|V1 | − 1 ≤ 2t − 3 = ⌊ 2 ⌋ − 3 = n+k ′ ⌊ 2 ⌋ − 3. In the latter case, suppose that w is a 2-degree vertex of G , without loss of generality, let w ∈ V1 . Then, ′
γc (G) ≤ 2|V1 | − 1 − k1 ≤ 2|V1 | − 1 − 1 ≤ 2t − 2 = ⌊ |G2 | ⌋ − 2 = ⌊ n+2 k ⌋ − 2.
Case b3. h = 1. |G′ | k ⌋ − 3. The remaining case is |V1 | = |V2 | = |V3 | = t and If |V1 | < t, γc (G) ≤ 2|V1 | − 1 ≤ 2t − 3 = ⌊ 2 ⌋ − 3 = ⌊ n+ 2 ′ |V4 | = t + 1. If there exists a 2-degree vertex of G which belongs to V1 ∪ V2 ∪ V3 , we are done. So, we assume that all 2-degree vertices of G′ belong to V4 . Let y be a 2-degree vertex of G′ , and y′ be the neighbor of y which has degree two in G. Since y ∈ V4 , y′ ∈ V1 ∪ V2 ∪ V3 . Without loss of generality, let y′ ∈ V2 . From the previous analysis, the connected dominating set D′2 of G has cardinality 2|V2 | − 1, and it contains all vertices of V2 . Note that D′2 \ {y′ } is still a connected |G′ | k dominating set of G. Hence, γc (G) ≤ 2|V2 | − 1 − 1 = 2t − 2 = ⌊ 2 ⌋ − 2 = ⌊ n+ ⌋ − 2. □ 2 Let Ft be the graph shown in Fig. 3(a) with V (Ft ) = {u, u1 , u2 , . . . , ut , v1 , v2 , . . . , vt } as indicated, where t ≥ 2. Let each square that appears in Ft be triangulated arbitrarily, and all of those resulting graphs constitute a family Ft . Let Lt be the graph shown in Fig. 3(b) with V (Lt ) = {u, v, u1 , u2 , . . . , ut , v1 , v2 , . . . , vt } as indicated, where t ≥ 1. Let each square that appears in Lt be triangulated arbitrarily, and all of those resulting graphs constitute a family Lt . |F ′ |+2 |L′ |+2 It can be easily checked that γc (Ft′ ) = ⌊|Ft′ |/2⌋ − 1 = ⌊ t 2 ⌋ − 2, γc (L′t ) = ⌊|L′t |/2⌋ − 1 = ⌊ t 2 ⌋ − 2, where Ft′ ∈ Ft and L′t ∈ Lt . It means that the bound given by Theorem 3.4 is also best possible. The following conclusion is immediate from Corollary 3.2 and Theorem 3.4. Please cite this article as: W. Zhuang, Connected domination in maximal outerplanar graphs, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.01.033.
W. Zhuang / Discrete Applied Mathematics xxx (xxxx) xxx
5
Fig. 3. Ft and Lt .
Corollary 3.5. Let G be a maximal outerplanar graph of order n ≥ 3. If k is the number of vertices of degree 2 in G, then
{ γc (G) ≤
⌊ 2(n3−k) ⌋, ⌊
n+k 2
⌋ − 2,
if k ≥
n+12 7
and n ≥ 6;
otherwise
We know that the upper bounds for Corollary 3.5 are tight. It is natural to consider the following question: can the upper bound for Corollary 3.5 be improved for the striped maximal outerplanar graph? From the above result, we know k that ⌊ n+ ⌋ − 2 < ⌊ 2(n3−k) ⌋ when k is far less than n. Then combining Proposition 2.1, we have the following. 2 Corollary 3.6. Let G be a striped maximal outerplanar graph with n ≥ 4 vertices. Then, γc (G) ≤ ⌊n/2⌋ − 1. In the next section, we will show that a striped ⋃∞ ⋃∞maximal outerplanar graph G satisfies γc (G) = ⌊|G|/2⌋ − 1 if and only if G ∈ F ∪ L , where F = t =2 Ft and L = t =1 Lt . These families of graphs show that the upper bound stated in Corollary 3.6 is the best possible when other parameters of the graphs are not taken into consideration. 4. Upper bound for the connected domination number of striped maximal outerplanar graphs In the previous section we have obtained a tight upper bound for the connected domination number of striped outerplanar graphs. Next, we will characterize the graphs that achieve the bound stated in Corollary 3.6. Later, we prove an upper bound in terms of the degree of the vertices of the graph. This new upper bound is much better than ⌊n/2⌋ − 1 when the graph has many vertices of degree larger than 5 (and if the maximum degree is at most 5, this bound simplifies to the bound stated in Corollary 3.6). Lemma 4.1 ([16]). Let G be a striped maximal outerplanar graph of order n > 4. Then for any vertex v ∈ V (G) of degree 2, the degrees of two neighbors of v are 3 and p (p ≥ 4). For a striped maximal outerplanar graph G of order n ≥ 6 and a vertex u ∈ V (G) adjacent to a 2-degree vertex and a 3-degree vertex, we define a graph Gu and the DELETED VERTEX SEQUENCE (v1 , v2 , . . .) as follows. Procedure CREATE_GRAPH(G, u); input: a graph G and a vertex u ∈ V (G). begin i := 1; S := ∅; T := {w | w ∈ N [u], dG−S (w ) = 2}; while T ̸ = ∅ do select a vertex v ∈ T ; vi := v ; S := S ∪ {vi }; i := i + 1; T := {w | w ∈ N [u], dG−S (w ) = 2}; end while Gu := G \ (S \ {vi−1 }); end. In fact, Gu is a subgraph of G obtained by repeatedly removing the vertices of degree 2 in N [u] from G, and vi is the ith removal of the procedure CREATE_GRAPH. See Figs. 4 and 5. Let d(u) = p. If n = p + 1, then G is a p-fan and γc (G) = 1. If n ≥ p + 2, by Proposition 2.1 and Lemma 4.1, it can be seen that u = vp−1 and the DELETED VERTEX SEQUENCE is (v1 , v2 , . . . , vp−1 (= u)) or (v1 , v2 , . . . , vp−1 (= u), vp ). Then Gu has n − p + 1 or n − p vertices. Moreover, if n ≤ p + 3, then G is isomorphic to one of the graphs shown in Fig. 6. In either case, γc (G) = 2. If n ≥ p + 4, then Gu is still a striped maximal outerplanar graph. Theorem 4.2. Let G be a striped maximal outerplanar graph of order n ≥ 8 and u be a vertex of G which is adjacent to a 2-degree vertex and a 3-degree vertex. Suppose that |G| ≥ d(u) + 4, then if |Gu | = |G| − d(u), γc (G) = γc (Gu ) + 2; otherwise, γc (G) = γc (Gu ) + 1. Please cite this article as: W. Zhuang, Connected domination in maximal outerplanar graphs, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.01.033.
6
W. Zhuang / Discrete Applied Mathematics xxx (xxxx) xxx
Fig. 4. The case of Gu with n − p vertices.
Fig. 5. The case of Gu with n − p + 1 vertices.
Fig. 6. The striped maximal outerplanar graphs with n ≤ p + 3.
Proof. Let d(u) = p and (v1 , v2 , . . .) be the DELETED VERTEX SEQUENCE of the procedure CREATE_GRAPH. If |Gu | = |G| − d(u) ≥ 4, it follows that vp ̸ ∈ Gu , vp+1 ∈ Gu and dGu (vp+1 ) = 2. Note that for any minimum connected dominating set D1 of Gu , D1 ∪ {u, vp+1 } is a connected dominating set of G. Therefore, γc (G) ≤ |D1 | + 2 = γc (Gu ) + 2. Next, we prove that γc (G) ≥ γc (Gu ) + 2. Let D be a minimum connected dominating set of G. Since D contains at least one vertex of N [v1 ] and N [vi ] ⊆ N [u] for any i ∈ {1, 2, . . . , p − 2}, (D \ {v1 , v2 , . . . , vp−2 }) ∪ {u} is also a minimum connected dominating set of G. Without loss of generality, we assume that u ∈ D and vi ̸ ∈ D for any i ∈ {1, 2, . . . , p − 2}. It follows from D be a minimum connected dominating set of G that either vp+1 or vp is in D. Because of dGu (vp+1 ) = 2, in either case, NGu (vp+1 ) ∩ D ̸ = ∅. It implies that the set D \ {u, vp , vp+1 } is a connected dominating set of Gu . Because at least one of vp+1 and vp is in D, we conclude that γc (Gu ) ≤ γc (G) − 2. If |Gu | ̸ = |G| − d(u), then |Gu | = |G| − d(u) + 1, and we have vp , vp+1 ∈ Gu and |NGu (vp+1 )| ≥ 4. It is easy to see that Gu is also a striped maximal outerplanar graph and |NGu (vp )| = 2. By Lemma 4.1, there is a 3-degree vertex t which is the common neighbor of vp and vp+1 in Gu . Let t1 be the neighbor of t other than vp and vp+1 . Let D2 be a minimum connected dominating set of Gu . If vp or vp+1 is in D2 , the set D2 ∪ {u} is a connected dominating set of G. Suppose that neither vp Please cite this article as: W. Zhuang, Connected domination in maximal outerplanar graphs, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.01.033.
W. Zhuang / Discrete Applied Mathematics xxx (xxxx) xxx
7
Fig. 7. The striped maximal outerplanar graphs with n ≤ p + 3 and n = 6, 7.
Fig. 8. (v1 , v2 , u) is the DELETED VERTEX SEQUENCE of the procedure CREATE_GRAPH.
nor vp+1 are in D2 , then t ∈ D2 . Moreover, t1 ∈ D2 . Hence, the set (D2 \ {t }) ∪ {u, vp+1 } is a connected dominating set of G. In summary, γc (G) ≤ |D2 | + 1 = γc (Gu ) + 1. Next, we prove that γc (G) ≥ γc (Gu ) + 1. Let D′ be a minimum connected dominating set of G. Similar to the above discussion, without loss of generality, we may assume that u ∈ D′ and vi ̸ ∈ D′ for any i ∈ {1, 2, . . . , p − 2}. Then, either vp or vp+1 is in D′ . In either case, the set D′ \ {u} is a connected dominating set of Gu . Therefore, γc (Gu ) ≤ γc (G) − 1. □ By Theorem 4.2, one can present a linear time algorithm for computing the connected domination number of a striped maximal outerplanar graph (this linear time algorithm is not difficult, and it is not the core of this article, so we omit it), and it is natural to consider the case of the maximal outerplanar graphs with internal triangles. Thus, we intend to provide the following issue as an open question. Problem 4.3. Is there a polynomial-time algorithm for computing the connected domination number of a maximal outerplanar graph with internal triangles? By Theorem 4.2, we may obtain the following conclusion. Theorem 4.4. Let G be a striped maximal outerplanar graph of order n ≥ 4. Then, γc (G) = ⌊n/2⌋− 1 if and only if G ∈ F ∪ L . Proof. The sufficiency is easy to verify. So we prove the necessity only. Let G be a striped maximal outerplanar graph of order n ≥ 4, which satisfies γc (G) = ⌊n/2⌋ − 1. Suppose that n ≥ 8, as otherwise the proof is simple. The proof is by induction on n. Assume that for any striped maximal outerplanar graph of order n′ < n, the result holds. Take a vertex of degree 2 in G, say v1 . By Lemma 4.1, one of the two neighbors of v1 has degree at least 4, say u. Let N(u) = {v1 , v2 , . . .}. If n ≤ d(u) + 3, as mentioned above, G is isomorphic to one of the graphs shown in Fig. 6. In either case, γc (G) = 2. Since γc (G) = ⌊n/2⌋ − 1, n = 6 or 7. It means that G is isomorphic to one of the graphs shown in Fig. 7, and each of those graphs is in F ∪ L . Next, we consider the case when n ≥ d(u) + 4. By Theorem 4.2, if |Gu | = n − d(u) + 1, we have γc (Gu ) = γc (G) − 1 = ⌊n/2⌋ − 2 = ⌊ n−2 4 ⌋. Note that Gu is also a striped maximal outerplanar n−d(u)+1 graph with |Gu | ≥ 4. So by Corollary 3.6, γc (Gu ) ≤ ⌊|Gu |/2⌋ − 1 = ⌊ ⌋ − 1 ≤ ⌊ n−2 3 ⌋ − 1 = ⌊ n−2 5 ⌋. Therefore, n 2 u u is odd, d(u) = 4, and γc (G ) = ⌊|G |/2⌋ − 1. Without loss of generality, we have that (v1 , v2 , u) is the DELETED VERTEX SEQUENCE of the procedure CREATE_GRAPH (we have the situation depicted in Fig. 8). Note that |Gu | is an even number greater than 4 and Gu is still a striped maximal outerplanar graph, and γc (Gu ) = ⌊|Gu |/2⌋ − 1. It means that Gu ∈ L . Note that E(G) \ E(Gu ) = {v1 u, v2 u, v3 u, v4 u, v1 v2 , v2 v3 } or E(G) \ E(Gu ) = {v1 u, v2 u, v3 u, v4 u, v1 v2 , v2 v4 }. Thus, it is easy to verify that G ∈ F . If |Gu | = n − d(u), analogously to the previous discussion, we have d(u) = 4. Without loss of generality, (v1 , v2 , u, v3 ) is the DELETED VERTEX SEQUENCE of the procedure CREATE_GRAPH (we have the situation depicted in Fig. 9). Note that |Gu | ≥ 4 and Gu is still a striped maximal outerplanar graph, and γc (Gu ) = γc (G)−2 = ⌊n/2⌋−3 = ⌊ n−2 4 ⌋−1 = ⌊|Gu |/2⌋−1. It means that Gu ∈ F ∪ L . Note that E(G) \ E(Gu ) = {v1 u, v2 u, v3 u, v4 u, v1 v2 , v2 v3 , v3 v4 , v3 t } or E(G) \ E(Gu ) = Please cite this article as: W. Zhuang, Connected domination in maximal outerplanar graphs, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.01.033.
8
W. Zhuang / Discrete Applied Mathematics xxx (xxxx) xxx
Fig. 9. (v1 , v2 , u, v3 ) is the DELETED VERTEX SEQUENCE of the procedure CREATE_GRAPH.
Fig. 10. The vertex u and its neighbors.
Fig. 11. The process of constructing the graph F .
{v1 u, v2 u, v3 u, v4 u, v1 v2 , v2 v4 , v3 v4 , v3 t }, where t is the neighbor of v4 of degree 3 in Gu , then it is easy to verify that G ∈ F ∪ L. □ At the end of this article, we prove an upper bound in terms of the degree of the vertices of a striped maximal outerplanar graph. This new upper bound is much better than ⌊n/2⌋ − 1 when the graph has many vertices of degree larger than 5. Lemma 4.5. Let G be a striped maximal outerplanar graph with ∆(G) ≥ 5. Then there is always a minimum connected dominating set of G, which contains all vertices of degree more than 4. Proof. Let u be any vertex of G having degree more than 4, and let u1 , u2 , . . . , ut −1 , ut be consecutive vertices of NG (u) in this order, as depicted in Fig. 10. Assume that S is a minimum connected dominating set of G. If u ̸ ∈ S, then u3 ∈ S (note that d(u3 ) = 3). The vertex set (S \ {u3 }) ∪ {u} is also a minimum connected dominating set of G. It implies that there is always a minimum connected dominating set of G, say S ′ , which contains all vertices of degree more than 4. □ Take a graph G ∈ F ∪ L , let u be any vertex of degree 5 in G, and let u1 , u2 , u3 , u4 , u5 be consecutive vertices of NG (u) in this order (see Fig. 11(a)). Subdivide u2 u3 and u3 u4 by inserting a few new vertices respectively, and join u to all the new vertices (see Fig. 11(b)). For each vertex of degree 5 in G, we repeat the operation mentioned above. We denote the resulting graph by F , and let H be the family of graphs consisting of all such graphs F arising in the way. Note that (F ∪ L ) ⊂ H . Please cite this article as: W. Zhuang, Connected domination in maximal outerplanar graphs, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.01.033.
W. Zhuang / Discrete Applied Mathematics xxx (xxxx) xxx
9
Theorem 4.6. If G is a striped maximal outerplanar graph of order n ≥ 4, then γc (G) ≤ ⌊ equality holds if and only if G ∈ H .
∑ n− ni=1 max{di −5,0} 2
⌋ − 1. Moreover,
Proof. If ∆(G) ≤ 5, then by Corollary 3.6, the first result holds. So we consider the case when ∆(G) > 5. Let v be any vertex of G having degree more than 5, i.e., |NG (v )| ≥ 6. Let v1 , v2 , . . . , vs−1 , vs be consecutive vertices of NG (v ) in this order. Note that the induced subgraph G[{v3 , v4 , . . . , vs−2 }] is a path. Let G′ be obtained from G by contracting ∑n each of all such paths into a single vertex. The resulting graph is a striped maximal outerplanar graph with n′ = n − i=1 max{di − 5, 0} vertices ′ (clearly, |G′ | ≥ 4) and ∆(G′ ) ≤ 5. By Corollary 3.6, γc (G′ ) ≤ ⌊ n2 ⌋ − 1. By Lemma 4.5, we know that there is one minimum ′ connected dominating set of G , which contains all vertices of degree more than 4.∑Note that this minimum connected n−
n
max{d −5,0}
i i=1 ⌋ − 1. This proves the dominating set is also the connected dominating set of G. Therefore, γc (G) ≤ ⌊ 2 desired bound, and we turn our attention to proving the second part of Theorem 4.6, i.e., ∑ the characterization of graphs
that achieve this bound. First, if a graph G ∈ H , it is easy to verify that γc (G) = ⌊
n−
n i=1 max{di −5,0}
2
∑n
n−
⌋ − 1. Conversely,
max{d −5,0}
i i=1 ⌋ − 1. We assume that G is a striped maximal outerplanar graph of order n ≥ 4, which satisfies γc (G) = ⌊ 2 will show that G ∈ H . If ∆(G) ≤ 5, then by Theorem 4.4, G ∈ (F ∪ L ) ⊂ H . So we consider the case when ∆(G) > 5. By Lemma 4.5, there is a minimum connected dominating set of G, say S, which contains all vertices of degree more than 4. We construct a new graph G′ from G by the operation mentioned at the beginning of this proof. Then, S is also a minimum connected dominating set of G′ . (if not, we must be able to find a connected dominating set of G′ , say S ′ , such that |S ′ | < |S |, and it contains all vertices of degree more than 4 in G′ . Note that S ′ is also a connected dominating set of G, a contradiction.) By Theorem 4.4, G′ ∈ F ∪ L . It follows that G ∈ H . □
CRediT authorship contribution statement Wei Zhuang: Conceptualization, Methodology, Validation, Writing - original draft, Writing - review & editing. Acknowledgments The author would like to thank Jianxiang Li, Zepeng Li, Baoyindureng Wu and the referees for many helpful comments and suggestions. References [1] N. Ananchuen, W. Ananchuen, M.D. Plummer, Matching properties in connected domination critical graphs, Discrete Appl. Math. 308 (2008) 1260–1267. [2] W. Ananchuen, N. Ananchuen, M.D. Plummer, Connected domination: vertex criticality and matchings, Util. Math. 89 (2012) 141–159. [3] T. Araki, I. Yumoto, On the secure domination numbers of maximal outerplanar graphs, Discrete Appl. Math. 236 (2018) 23–29. [4] C.N. Campos, Y. Wakabayashi, On dominating sets of maximal outerplanar graphs, Discrete Appl. Math. 161 (2013) 330–335. [5] W.J. Desormeaux, T.H. Haynes, M.A. Henning, Bounds on the connected domination number of a graph, Discrete Appl. Math. 161 (2013) 2925–2931. [6] W.J. Desormeaux, T.H. Haynes, L. van der Merwe, Connected domination stable graphs upon edge addition, Quaest. Math. 38 (2015) 841–848. [7] M. Dorfling, J.H. Hattingh, E. Jonck, Total domination in maximal outerplanar graphs II, Discrete Math. 339 (2016) 1180–1188. [8] M. Dorfling, J.H. Hattingh, E. Jonck, Total domination in maximal outerplanar graphs, Discrete Appl. Math. 217 (2017) 506–511. [9] W. Duckworth, B. Mans, Connected domination of regular graphs, Discrete Math. 309 (2009) 2305–2322. [10] B. Gao, Y. Yang, H. Ma, A new distributed approximation algorithm for constructing minimum connected dominating set in wireless ad hoc networks, Int. J. Commun. Syst. 18 (2005) 743–762. [11] T.W. Haynes, S.T. Hedetniemi, P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, New York, 1998. [12] S. Kamei, H. Kakugawa, S. Devismes, S. Tixeuil, A self-stabilizing 3-approximation for the maximum leaf spanning tree problem in arbitrary networks, J. Comb. Optim. 25 (2013) 430–459. [13] H. Karami, S.M. Sheikholeslami, A. Khodkar, Connected domination number of a graph and its complement, Graphs Comb. 28 (2012) 123–131. [14] D. Kim, Z. Zhang, X. Li, W. Wang, W. Wu, D. Du, A better approximation algorithm for computing connected dominating sets in unit ball graphs, IEEE Trans. Mob. Comput. 9 (2010) 1108–1118. [15] R. Laskar, J. Pfaff, Domination and Irredundance in Split Graphs, Technical Report 430, Deparment of Mathematical Sciences, Clemson University, 1983. [16] Z. Li, E. Zhu, Z. Shao, J. Xu, On dominating sets of maximal outerplanar and planar graphs, Discrete Appl. Math. 198 (2016) 164–169. [17] L.R. Matheson, R.E. Tarjan, Dominating sets in planar graphs, European J. Combin. 17 (1996) 565–568. [18] A. Reich, Complexity of the maximum leaf spanning tree problem on planar and regular graphs, Theoret. Comput. Sci. 626 (2016) 134–143. [19] E. Sampathkumar, H.B. Walikar, The connected domination number of a graph, J. Math. Phys. Sci. 13 (1979) 607–613. [20] O. Schaudt, On graphs for which the connected domination number is at most the total domination number, Discrete Appl. Math. 160 (2012) 1281–1284. [21] R. Solis-Oba, P. Bonsma, S. Lowski, A 2-approximation algorithm for finding a spanning tree with maximum number of leaves, Algorithmica 77 (2017) 374–388. [22] Shin-ichi Tokunaga, Dominating sets of maximal outerplanar graphs, Discrete Appl. Math. 161 (2013) 3097–3099. [23] D.B. West, Introduction to Graph Theory, second ed., Pearson Education, Beijing, 2004. [24] X. Xia, Y. Zhou, X. Lai, On the analysis of the (1 + 1) evolutionary algorithm for the maximum leaf spanning tree problem, Int. J. Comput. Math. 92 (2015) 2023–2035. [25] J. Zhou, Z. Zhang, S. Tang, X. Huang, D. Du, Breaking the O(ln n) barrier: an enhanced approximation algorithm for fault-tolerant minimum weight connected dominating set, Informs J. Comput. 30 (2018) 225–235.
Please cite this article as: W. Zhuang, Connected domination in maximal outerplanar graphs, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.01.033.
Contents lists available at ScienceDirect
Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Connected domination in maximal outerplanar graphs✩ Wei Zhuang School of Applied Mathematics, Xiamen University of Technology, Xiamen 361024, PR China
article
info
Article history: Received 7 March 2017 Received in revised form 28 December 2019 Accepted 30 January 2020 Available online xxxx Keywords: Outerplanar graph Connected domination Striped maximal outerplanar graph
a b s t r a c t A subset S of vertices in a graph G = (V , E) is a connected dominating set of G if every vertex of V \ S is adjacent to a vertex in S and the subgraph induced by S is connected. The minimum cardinality of a connected dominating set of G is the connected domination number γc (G). We show that the connected domination number k ⌋ − 2, ⌊ 2(n3−k) ⌋}, where k is of a maximal outerplanar graph G is bounded by min{⌊ n+ 2 the number of vertices of degree 2 in G. Moreover, we improve this upper bound for striped maximal outerplanar graphs and characterize the graphs that achieve the bound. © 2020 Elsevier B.V. All rights reserved.
1. Introduction Let G be a simple graph. The vertex set and edge set of G are denoted by V (G) and E(G), respectively. We denote the order of G by |V (G)|. If the graph G is clear from the context, we simply write |G| or n rather than |V (G)|. The open neighborhood of a vertex v ∈ V (G) is defined by NG (v ) = {u | v u ∈ E(G)}, and the closed neighborhood of v is NG [v] = NG (u) ∪ {v}. We denote the degree of v by dG (v ) = |NG (v )|. Moreover, when no confusion can arise, NG (v ), NG [v] and dG (v ) are simplified by N(v ), N [v] and d(v ), respectively. The maximum degree and minimum degree of G are denoted by ∆(G) and δ (G), respectively. The subgraph of graph G induced by a subset S ⊆ V (G) is denoted by G[S ]. For two vertices u and v in a connected graph G, the distance dG (u, v ) between u and v is the length of a shortest (u, v )-path in G. The distance between a vertex v ∈ V (G) and a set S ⊆ V (G) is defined as dG (v, S) = min{d(v, x) | x ∈ S }. A fan Fn is a graph of order n + 1 obtained by adding a vertex v to Pn with v adjacent to each vertex of Pn . A graph G is outerplanar if it can be embedded in the plane such that all vertices belong to the boundary of its outer face. An outerplanar graph G is maximal if G + uv is not outerplanar for any two nonadjacent vertices u and v . Any terminology not defined here follows that of [23]. A dominating set of a graph G is a set S ⊆ V (G) such that every vertex in G is either in S or adjacent to a vertex in S. A dominating set S is a connected dominating set if the subgraph of G induced by S is connected. The domination number (connected domination number) of G, which is denoted by γ (G) (γc (G)), is the minimum cardinality of a dominating set (connected dominating set). The concept of connected domination was introduced by Sampathkumar and Walikar [19]. More details and information are shown in [1,2,5,6,9,11,13,20]. A related problem is that of finding a spanning tree of a graph with the maximum number of leaves (maximum leaf spanning tree problem). If such a spanning tree is found, by deleting the leaves of this tree, one obtains a minimum connected dominating set. Some results on this problem have been obtained [12,18,21,24]. In 1983, Laskar and Pfaff [15] showed the NP-hardness of computing the connected domination number or the minimum ✩ The research is supported by National Natural Science Foundation of China (No. 11301440) and Natural Science Foundation of Fujian Province, China (CN)(2015J05017). E-mail address: [email protected]. https://doi.org/10.1016/j.dam.2020.01.033 0166-218X/© 2020 Elsevier B.V. All rights reserved.
Please cite this article as: W. Zhuang, Connected domination in maximal outerplanar graphs, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.01.033.
2
W. Zhuang / Discrete Applied Mathematics xxx (xxxx) xxx
connected dominating set. After that, the approximation algorithm design has become an important issue in the study of connected dominating sets [10,14,25]. In 1996, Matheson and Tarjan [17] proved that any triangulated disc G with n vertices satisfies γ (G) ≤ ⌊ 3n ⌋, and conjectured that γ (G) ≤ ⌊ 4n ⌋ for every n-vertex triangulation G with sufficiently large n. In 2013, Campos and Wakabayashi investigated this problem for maximal outerplanar graphs and showed in [4] that if G is a maximal outerplanar graph of k , where k is the number of vertices of degree 2 in G. By using a simple coloring method, Tokunaga order n, then γ (G) ≤ n+ 4 t proved the same result independently in [22]. Li et al. [16] improved the result by showing that γ (G) ≤ n+ , where t is the 4 number of pairs of consecutive 2-degree vertices with distance at least 3 on the outer cycle. In [3,7,8], the same question was investigated for total domination and secure domination. In this paper, we investigate the same question for connected domination. For outerplanar graphs of order n, which are not maximal, the upper bound n − 2 is the best possible (note that the path and cycle are both outerplanar graphs). We k ⌋ − 2, ⌊ 2(n3−k) ⌋}, show that the connected domination number of a maximal outerplanar graph G is bounded by min{⌊ n+ 2 where k is the number of vertices of degree 2 in G. Moreover, we improve this upper bound for striped maximal outerplanar graphs and characterize the graphs that achieve the bound. 2. Preliminaries A maximal outerplanar graph G can be embedded in the plane in such a way that the boundary of the outer face is a Hamiltonian cycle, and each inner face is a triangle. A maximal outerplanar graph embedded in the plane in this (unique) way is called a maximal outerplane graph (see [3,4]). For this embedding of G, we denote the Hamiltonian cycle, which is the boundary of the outer face, by HG . An inner face of a maximal outerplane graph G is an internal triangle if it is not adjacent to the outer face. A maximal outerplane graph without an internal triangle is called striped. We also say that a maximal outerplanar graph is striped (resp. contains internal triangles) when its outerplane embedding is striped (resp. contains internal triangles). To simplify notation, in most of the proofs, in which an embedding of an outerplanar graph G is considered, we assume that G is an outerplane graph (even if this is not stated explicitly). We may use the term triangle to refer to an inner face or a subgraph, which is isomorphic to K3 . The following results are given in [4,22]. Proposition 2.1 ([4]). Let G be a maximal outerplanar graph of order n ≥ 4. If G has k internal triangles, then G has k + 2 vertices of degree 2. Theorem 2.2 ([22]). Let G be a maximal outerplanar graph of order n ≥ 4, then G can be 4-colored such that every cycle of length 4 in G has all four colors. Theorem 2.3 ([22]). Let G be a maximal outerplanar graph of order n ≥ 4. Suppose that G is 4-colored such that every cycle of length 4 in G has all four colors, and let R ⊂ V (G) contain all vertices of some given color. Then R dominates all vertices of V (G) except those of degree 2. 3. Upper bounds for the connected domination number of maximal outerplanar graphs A minor of a graph G is a graph which can be obtained from G by deleting vertices and deleting or contracting edges. Given a graph H, a graph G is H-minor free if no minor of G is isomorphic to H. A K4 -minor free graph G is maximal if G + uv is not a K4 -minor free graph for any two non-adjacent vertices u and v of G. A 2-tree is recursively defined as follows. A single edge is a 2-tree. Any graph obtained from a 2-tree by adding a new vertex and making it adjacent to the end vertices of an existing edge is also a 2-tree. It is well known that the maximal K4 -minor free graphs are exactly the 2-trees. We know that a maximal outerplanar graph must be a maximal K4 -minor free graph. Hence, before giving an upper bound for the connected domination number of a maximal outerplanar graph, we prove the following stronger result. Theorem 3.1. Let G be a maximal K4 -minor free graph of order n ≥ 4. If k is the number of vertices of degree 2 in G, then γc (G) = 1 when n = 4, 5; and γc (G) ≤ ⌊ 2(n3−k) ⌋ when n ≥ 6. Proof. If n = 4 or 5, G is isomorphic to one of the three graphs in Fig. 1, and clearly, γc (G) = 1. We consider the case when n ≥ 6. Let T = {v | dG (v ) = 2}, G0 = G − T . Note that G0 is still a maximal K4 -minor free graph. Let Gi = Gi−1 − vi for i = 1, 2, . . . , n − |T | − 3, where vi is a vertex of degree 2 in Gi−1 . Since a maximal K4 -minor free graph is exactly a 2-tree, it follows from the definition of the 2-tree that Gn−|T |−3 is a K3 . We can give a proper 3-coloring to G0 by assigning color 1, 2 and 3 to the three vertices of Gn−|T |−3 respectively and color vi with the remaining color which does not appear in NGi−1 (vi ) at each stage. Denote by Vt the set of vertices assigned color t, where t = 1, 2, 3. Without loss of generality, we may assume that |V1 | ≤ |V2 | ≤ |V3 |, then |V1 | + |V2 | ≤ ⌊ 32 |G0 |⌋. Let S = V1 ∪ V2 . Next, we will show that S is a dominating set of G and G[S ] is connected. Please cite this article as: W. Zhuang, Connected domination in maximal outerplanar graphs, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.01.033.
W. Zhuang / Discrete Applied Mathematics xxx (xxxx) xxx
3
Fig. 1. The case of n = 4 or 5.
Fig. 2. Rt .
It is easy to see that S is a dominating set of G0 . Since G is a 2-tree, we have NG (u) ∩ S ̸ = ∅ for each vertex u of T . In other words, S is a dominating set of G. Finally, we only need to show that if G[V (Gi ) ∩ S ] is connected, then G[V (Gi−1 ) ∩ S ] is also connected. Assume that the subgraph of Gi induced by V (Gi ) ∩ S is connected. If vi is assigned color 3, then G[V (Gi ) ∩ S ] = G[V (Gi−1 ) ∩ S ]. If vi is assigned color 1 or 2, we note that at most one neighbor of vi is assigned color 3 in Gi−1 , thus the subgraph of Gi−1 induced by V (Gi−1 ) ∩ S is connected. Therefore, S is a connected dominating set 2(n−k) of G that satisfies |S | ≤ ⌊ 3 ⌋. □ Based on the above theorem, we have the following conclusion. Corollary 3.2. Let G be a maximal outerplanar graph of order n ≥ 4. If k is the number of vertices of degree 2 in G, then
γc (G) = 1 when n = 4, 5; and γc (G) ≤ ⌊ 2(n3−k) ⌋ when n ≥ 6.
In what follows, we are ready to introduce a family of graphs depicted in [8]. We note that the connected domination number of each graph in this family achieves the bounds given by Theorem 3.1 and Corollary 3.2. Let Rt be the graph shown in Fig. 2, and suppose V (Rt ) = 6t, where t ≥ 1. Let x be a 2-degree vertex of Rt , y and z be its neighbors. Join x and y to a new vertex a, and x and z to a new vertex b. Repeat this operation on all 2-degree vertices of Rt , and call the resulting graph R′t . Note that V (R′t ) = 6t + 4t = 10t, and R′t has 4t vertices of degree 2. From the definition of 2-tree, it is easy to see that R′t is a 2-tree. That is, it is a maximal K4 -minor free graph. Moreover, 2(|R′ |−k)
it is also a maximal outerplanar graph. It can be easily checked that γc (R′t ) = ⌊ t3 ⌋. Hence, we have that the bounds given by Theorem 3.1 and Corollary 3.2 are both tight. Let G be a maximal outerplanar graph and HG be the Hamiltonian cycle of the outer face of G. Suppose that x and y are two vertices on HG . We call the path on HG that goes in the clockwise direction from x to y the xy-segment of HG , and the other path the yx-segment of HG . Observation 3.3. Let G be a maximal outerplanar graph and HG be the Hamiltonian cycle of the outer face of G, xy ∈ E(G). If the xy-segment (yx-segment) of HG contains at least one vertex other than x and y, then the xy-segment (yx-segment) of HG contains a vertex of N(x) ∩ N(y). Moreover, we can obtain another upper bound by applying Tokunaga’s method [22]. Theorem 3.4. Let G be a maximal outerplanar graph of order n ≥ 3. If k is the number of vertices of degree 2 in G, then γc (G) ≤ ⌊ n+2 k ⌋ − 2. k Proof. If n = 3, G is a triangle, then k = 3. Thus, γc (G) = 1 = ⌊ n+ ⌋ − 2. We assume that n ≥ 4. 2 Let S = {v1 , v2 , . . . , vk } be the set of vertices of G having degree 2, and let ui be one of the two vertices adjacent to vi for each i ∈ {1, 2, . . . , k}. We take a set of k additional vertices S ′ = {v1′ , v2′ , . . . , vk′ } and construct a graph G′ such that V (G′ ) = V (G) ∪ S ′ and E(G′ ) = E(G) ∪ {v1′ v1 , v2′ v2 , . . . , vk′ vk } ∪ {v1′ u1 , v2′ u2 , . . . , vk′ uk }. Clearly, G′ is also a maximal outerplanar graph, so G′ can be 4-colored such that every cycle of length 4 in G′ has all four colors. Let Vi (i = 1, 2, 3, 4) be the subset of V (G′ ) which is colored by color i. Without loss of generality, we may assume that |V1 | ≤ |V2 | ≤ |V3 | ≤ |V4 |. Let ki be the number of vertices of Vi with degree 2 in G′ . Note that k1 + k2 + k3 + k4 = k.
Please cite this article as: W. Zhuang, Connected domination in maximal outerplanar graphs, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.01.033.
4
W. Zhuang / Discrete Applied Mathematics xxx (xxxx) xxx
Suppose that |V1 | = 1 and v is the unique vertex in V1 . Let w be any vertex of G. Then, w has degree at least three in G′ . Let w1 , w2 and w3 be consecutive vertices of NG′ (v ) in this order. Note that ww1 w2 w3 forms a cycle of length 4. It follows from Theorem 2.2 that v ∈ {w, w1 , w2 , w3 }, i.e., each vertex of G is dominated by v . In addition, if v ∈ V (G′ ) \ V (G), then v is a 2-degree vertex of G′ , and it can only dominate its two neighbors, which contradicts the condition that n ≥ 4. Thus, v ∈ V (G), and {v} is a connected dominating set of G. The proof is complete. Next, we consider the case when |V1 | ≥ 2. Let HG′ be the Hamiltonian cycle of the outer face of G′ . Since |V1 | ≤ |V2 | ≤ |V3 | ≤ |V4 |, |G′ | ≥ 8. We recursively construct a sequence of subsets S1 , S2 , . . . of V1 as follows. First take a vertex v ∈ V1 , and let S1 = {v}. For i ≥ 2, choose any vertex v ′ ∈ V1 \ Si−1 with dG′ (v ′ , Si−1 ) ≤ 2, and let Si = Si−1 ∪ {v ′ }. We end up the procedure at a set Sp when Sp = V1 or dG′ (x, Sp ) ≥ 3 for any x ∈ V1 \ Sp . Next, we are ready to show that Sp = V1 . By contradiction, let z be a vertex of V1 \ Sp , then we have that dG′ (z , x) ≥ 3 for each x ∈ Sp . Let z ′ ∈ Sp and z ′′ ∈ NG′ (z ′ ). It is easy to see that z is in the z ′ z ′′ -segment or z ′′ z ′ -segment of HG′ . Denote by lz (z ′ , z ′′ ) the length of the segment between z ′ and z ′′ containing z. We select s ∈ Sp and s′ ∈ NG′ (s) such that lz (s, s′ ) ≤ lz (z ′ , z ′′ ) for any z ′ ∈ Sp and z ′′ ∈ NG′ (z ′ ). Without loss of generality, we assume that z is in the ss′ -segment of HG′ , and s′ ∈ V2 . By Observation 3.3, a vertex z1 ∈ NG′ (s) ∩ NG′ (s′ ) is in the ss′ -segment of HG′ . Moreover, z1 belongs to V3 or V4 , say V3 . So, z1 ̸ = z. We have that z1 is in the sz-segment of HG′ (if z1 is in the zs′ -segment of HG′ , then lz (s, z1 ) < lz (s, s′ ), a contradiction). By Observation 3.3, a vertex z2 ∈ NG′ (z1 ) ∩ NG′ (s′ ) is in the z1 s′ -segment of HG′ . Note that sz1 z2 s′ forms a cycle of length 4. It follows from Theorem 2.2 that z2 ∈ V4 , and so z2 ̸ = z. z2 is in the z1 z-segment or zs′ -segment of HG′ . Case a1. z2 is in the z1 z-segment of HG′ . By Observation 3.3, a vertex z3 ∈ NG′ (z2 ) ∩ NG′ (s′ ) is in the z2 s′ -segment of HG′ . z1 z2 z3 s′ forms a cycle of length 4. By Theorem 2.2, z3 ∈ V1 . It follows from the definition of Sp and dG′ (s, z3 ) = 2 that z3 ∈ Sp , and so z3 ̸ = z. z3 is in the z2 zsegment or zs′ -segment of HG′ . In the former case, lz (z3 , s′ ) < lz (s, s′ ), a contradiction. In the latter case, lz (z2 , z3 ) < lz (s, s′ ), a contradiction. Case a2. z2 is in the zs′ -segment of HG′ . By Observation 3.3, a vertex z3 ∈ NG′ (z1 ) ∩ NG′ (z2 ) is in the z1 z2 -segment of HG′ . z1 z3 z2 s′ forms a cycle of length 4. By Theorem 2.2, z3 ∈ V1 . Similar to the argument of Case a1, we have z3 ∈ Sp and z3 ̸ = z. z3 is in the z1 z-segment or zz2 -segment of HG′ . In the former case, lz (z3 , z2 ) < lz (s, s′ ), a contradiction. In the latter case, lz (z3 , z1 ) < lz (s, s′ ), a contradiction. Hence, Sp = V1 . By Theorem 2.3, we know that each Vi (i = 1, 2, 3, 4) dominates all vertices of V (G). Combining the definition of Sp , there exists a connected subgraph of G′ with order 2|V1 | − 1, say G1 , which contains all vertices of V1 . Let x be a 2-degree vertex of G′ , if x ∈ V1 , then x belongs to V (G1 ), and one of its neighbors also belongs to V (G1 ). It implies that the set D′1 which is obtained from V (G1 ) by removing all 2-degree vertices of G′ belonging to V1 , can still dominate all vertices of V (G), the subgraph induced by D′1 is still connected. That is, D′1 is a connected dominating set of G with cardinality 2|V1 | − 1 − k1 . Similarly, for each i ∈ {2, 3, 4}, there exists a connected subgraph of G′ with order 2|Vi |− 1, say Gi , which contains all vertices of Vi . Moreover, we can always obtain a connected dominating set D′i of G with cardinality 2|Vi | − 1 − ki , which is obtained from V (Gi ) by removing all 2-degree vertices of G′ belonging to Vi . Assume that |G′ | = 4t + h, where h = 0, 1, 2, 3. Since |V1 | ≤ |V2 | ≤ |V3 | ≤ |V4 | and |G′ | ≥ 8, we have t ≥ 2 and |V1 | ≤ t. Next, we consider three cases for the four possible values of h. Case b1. h ∈ {2, 3}. |G′ | k ⌋ − 2. In this case, γc (G) ≤ 2|V1 | − 1 ≤ 2t − 1 = ⌊ 2 ⌋ − 2 = ⌊ n+ 2 Case b2. h = 0. |G′ | We have that |V1 | < t or |V1 | = |V2 | = |V3 | = |V4 | = t. In the former case, γc (G) ≤ 2|V1 | − 1 ≤ 2t − 3 = ⌊ 2 ⌋ − 3 = n+k ′ ⌊ 2 ⌋ − 3. In the latter case, suppose that w is a 2-degree vertex of G , without loss of generality, let w ∈ V1 . Then, ′
γc (G) ≤ 2|V1 | − 1 − k1 ≤ 2|V1 | − 1 − 1 ≤ 2t − 2 = ⌊ |G2 | ⌋ − 2 = ⌊ n+2 k ⌋ − 2.
Case b3. h = 1. |G′ | k ⌋ − 3. The remaining case is |V1 | = |V2 | = |V3 | = t and If |V1 | < t, γc (G) ≤ 2|V1 | − 1 ≤ 2t − 3 = ⌊ 2 ⌋ − 3 = ⌊ n+ 2 ′ |V4 | = t + 1. If there exists a 2-degree vertex of G which belongs to V1 ∪ V2 ∪ V3 , we are done. So, we assume that all 2-degree vertices of G′ belong to V4 . Let y be a 2-degree vertex of G′ , and y′ be the neighbor of y which has degree two in G. Since y ∈ V4 , y′ ∈ V1 ∪ V2 ∪ V3 . Without loss of generality, let y′ ∈ V2 . From the previous analysis, the connected dominating set D′2 of G has cardinality 2|V2 | − 1, and it contains all vertices of V2 . Note that D′2 \ {y′ } is still a connected |G′ | k dominating set of G. Hence, γc (G) ≤ 2|V2 | − 1 − 1 = 2t − 2 = ⌊ 2 ⌋ − 2 = ⌊ n+ ⌋ − 2. □ 2 Let Ft be the graph shown in Fig. 3(a) with V (Ft ) = {u, u1 , u2 , . . . , ut , v1 , v2 , . . . , vt } as indicated, where t ≥ 2. Let each square that appears in Ft be triangulated arbitrarily, and all of those resulting graphs constitute a family Ft . Let Lt be the graph shown in Fig. 3(b) with V (Lt ) = {u, v, u1 , u2 , . . . , ut , v1 , v2 , . . . , vt } as indicated, where t ≥ 1. Let each square that appears in Lt be triangulated arbitrarily, and all of those resulting graphs constitute a family Lt . |F ′ |+2 |L′ |+2 It can be easily checked that γc (Ft′ ) = ⌊|Ft′ |/2⌋ − 1 = ⌊ t 2 ⌋ − 2, γc (L′t ) = ⌊|L′t |/2⌋ − 1 = ⌊ t 2 ⌋ − 2, where Ft′ ∈ Ft and L′t ∈ Lt . It means that the bound given by Theorem 3.4 is also best possible. The following conclusion is immediate from Corollary 3.2 and Theorem 3.4. Please cite this article as: W. Zhuang, Connected domination in maximal outerplanar graphs, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.01.033.
W. Zhuang / Discrete Applied Mathematics xxx (xxxx) xxx
5
Fig. 3. Ft and Lt .
Corollary 3.5. Let G be a maximal outerplanar graph of order n ≥ 3. If k is the number of vertices of degree 2 in G, then
{ γc (G) ≤
⌊ 2(n3−k) ⌋, ⌊
n+k 2
⌋ − 2,
if k ≥
n+12 7
and n ≥ 6;
otherwise
We know that the upper bounds for Corollary 3.5 are tight. It is natural to consider the following question: can the upper bound for Corollary 3.5 be improved for the striped maximal outerplanar graph? From the above result, we know k that ⌊ n+ ⌋ − 2 < ⌊ 2(n3−k) ⌋ when k is far less than n. Then combining Proposition 2.1, we have the following. 2 Corollary 3.6. Let G be a striped maximal outerplanar graph with n ≥ 4 vertices. Then, γc (G) ≤ ⌊n/2⌋ − 1. In the next section, we will show that a striped ⋃∞ ⋃∞maximal outerplanar graph G satisfies γc (G) = ⌊|G|/2⌋ − 1 if and only if G ∈ F ∪ L , where F = t =2 Ft and L = t =1 Lt . These families of graphs show that the upper bound stated in Corollary 3.6 is the best possible when other parameters of the graphs are not taken into consideration. 4. Upper bound for the connected domination number of striped maximal outerplanar graphs In the previous section we have obtained a tight upper bound for the connected domination number of striped outerplanar graphs. Next, we will characterize the graphs that achieve the bound stated in Corollary 3.6. Later, we prove an upper bound in terms of the degree of the vertices of the graph. This new upper bound is much better than ⌊n/2⌋ − 1 when the graph has many vertices of degree larger than 5 (and if the maximum degree is at most 5, this bound simplifies to the bound stated in Corollary 3.6). Lemma 4.1 ([16]). Let G be a striped maximal outerplanar graph of order n > 4. Then for any vertex v ∈ V (G) of degree 2, the degrees of two neighbors of v are 3 and p (p ≥ 4). For a striped maximal outerplanar graph G of order n ≥ 6 and a vertex u ∈ V (G) adjacent to a 2-degree vertex and a 3-degree vertex, we define a graph Gu and the DELETED VERTEX SEQUENCE (v1 , v2 , . . .) as follows. Procedure CREATE_GRAPH(G, u); input: a graph G and a vertex u ∈ V (G). begin i := 1; S := ∅; T := {w | w ∈ N [u], dG−S (w ) = 2}; while T ̸ = ∅ do select a vertex v ∈ T ; vi := v ; S := S ∪ {vi }; i := i + 1; T := {w | w ∈ N [u], dG−S (w ) = 2}; end while Gu := G \ (S \ {vi−1 }); end. In fact, Gu is a subgraph of G obtained by repeatedly removing the vertices of degree 2 in N [u] from G, and vi is the ith removal of the procedure CREATE_GRAPH. See Figs. 4 and 5. Let d(u) = p. If n = p + 1, then G is a p-fan and γc (G) = 1. If n ≥ p + 2, by Proposition 2.1 and Lemma 4.1, it can be seen that u = vp−1 and the DELETED VERTEX SEQUENCE is (v1 , v2 , . . . , vp−1 (= u)) or (v1 , v2 , . . . , vp−1 (= u), vp ). Then Gu has n − p + 1 or n − p vertices. Moreover, if n ≤ p + 3, then G is isomorphic to one of the graphs shown in Fig. 6. In either case, γc (G) = 2. If n ≥ p + 4, then Gu is still a striped maximal outerplanar graph. Theorem 4.2. Let G be a striped maximal outerplanar graph of order n ≥ 8 and u be a vertex of G which is adjacent to a 2-degree vertex and a 3-degree vertex. Suppose that |G| ≥ d(u) + 4, then if |Gu | = |G| − d(u), γc (G) = γc (Gu ) + 2; otherwise, γc (G) = γc (Gu ) + 1. Please cite this article as: W. Zhuang, Connected domination in maximal outerplanar graphs, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.01.033.
6
W. Zhuang / Discrete Applied Mathematics xxx (xxxx) xxx
Fig. 4. The case of Gu with n − p vertices.
Fig. 5. The case of Gu with n − p + 1 vertices.
Fig. 6. The striped maximal outerplanar graphs with n ≤ p + 3.
Proof. Let d(u) = p and (v1 , v2 , . . .) be the DELETED VERTEX SEQUENCE of the procedure CREATE_GRAPH. If |Gu | = |G| − d(u) ≥ 4, it follows that vp ̸ ∈ Gu , vp+1 ∈ Gu and dGu (vp+1 ) = 2. Note that for any minimum connected dominating set D1 of Gu , D1 ∪ {u, vp+1 } is a connected dominating set of G. Therefore, γc (G) ≤ |D1 | + 2 = γc (Gu ) + 2. Next, we prove that γc (G) ≥ γc (Gu ) + 2. Let D be a minimum connected dominating set of G. Since D contains at least one vertex of N [v1 ] and N [vi ] ⊆ N [u] for any i ∈ {1, 2, . . . , p − 2}, (D \ {v1 , v2 , . . . , vp−2 }) ∪ {u} is also a minimum connected dominating set of G. Without loss of generality, we assume that u ∈ D and vi ̸ ∈ D for any i ∈ {1, 2, . . . , p − 2}. It follows from D be a minimum connected dominating set of G that either vp+1 or vp is in D. Because of dGu (vp+1 ) = 2, in either case, NGu (vp+1 ) ∩ D ̸ = ∅. It implies that the set D \ {u, vp , vp+1 } is a connected dominating set of Gu . Because at least one of vp+1 and vp is in D, we conclude that γc (Gu ) ≤ γc (G) − 2. If |Gu | ̸ = |G| − d(u), then |Gu | = |G| − d(u) + 1, and we have vp , vp+1 ∈ Gu and |NGu (vp+1 )| ≥ 4. It is easy to see that Gu is also a striped maximal outerplanar graph and |NGu (vp )| = 2. By Lemma 4.1, there is a 3-degree vertex t which is the common neighbor of vp and vp+1 in Gu . Let t1 be the neighbor of t other than vp and vp+1 . Let D2 be a minimum connected dominating set of Gu . If vp or vp+1 is in D2 , the set D2 ∪ {u} is a connected dominating set of G. Suppose that neither vp Please cite this article as: W. Zhuang, Connected domination in maximal outerplanar graphs, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.01.033.
W. Zhuang / Discrete Applied Mathematics xxx (xxxx) xxx
7
Fig. 7. The striped maximal outerplanar graphs with n ≤ p + 3 and n = 6, 7.
Fig. 8. (v1 , v2 , u) is the DELETED VERTEX SEQUENCE of the procedure CREATE_GRAPH.
nor vp+1 are in D2 , then t ∈ D2 . Moreover, t1 ∈ D2 . Hence, the set (D2 \ {t }) ∪ {u, vp+1 } is a connected dominating set of G. In summary, γc (G) ≤ |D2 | + 1 = γc (Gu ) + 1. Next, we prove that γc (G) ≥ γc (Gu ) + 1. Let D′ be a minimum connected dominating set of G. Similar to the above discussion, without loss of generality, we may assume that u ∈ D′ and vi ̸ ∈ D′ for any i ∈ {1, 2, . . . , p − 2}. Then, either vp or vp+1 is in D′ . In either case, the set D′ \ {u} is a connected dominating set of Gu . Therefore, γc (Gu ) ≤ γc (G) − 1. □ By Theorem 4.2, one can present a linear time algorithm for computing the connected domination number of a striped maximal outerplanar graph (this linear time algorithm is not difficult, and it is not the core of this article, so we omit it), and it is natural to consider the case of the maximal outerplanar graphs with internal triangles. Thus, we intend to provide the following issue as an open question. Problem 4.3. Is there a polynomial-time algorithm for computing the connected domination number of a maximal outerplanar graph with internal triangles? By Theorem 4.2, we may obtain the following conclusion. Theorem 4.4. Let G be a striped maximal outerplanar graph of order n ≥ 4. Then, γc (G) = ⌊n/2⌋− 1 if and only if G ∈ F ∪ L . Proof. The sufficiency is easy to verify. So we prove the necessity only. Let G be a striped maximal outerplanar graph of order n ≥ 4, which satisfies γc (G) = ⌊n/2⌋ − 1. Suppose that n ≥ 8, as otherwise the proof is simple. The proof is by induction on n. Assume that for any striped maximal outerplanar graph of order n′ < n, the result holds. Take a vertex of degree 2 in G, say v1 . By Lemma 4.1, one of the two neighbors of v1 has degree at least 4, say u. Let N(u) = {v1 , v2 , . . .}. If n ≤ d(u) + 3, as mentioned above, G is isomorphic to one of the graphs shown in Fig. 6. In either case, γc (G) = 2. Since γc (G) = ⌊n/2⌋ − 1, n = 6 or 7. It means that G is isomorphic to one of the graphs shown in Fig. 7, and each of those graphs is in F ∪ L . Next, we consider the case when n ≥ d(u) + 4. By Theorem 4.2, if |Gu | = n − d(u) + 1, we have γc (Gu ) = γc (G) − 1 = ⌊n/2⌋ − 2 = ⌊ n−2 4 ⌋. Note that Gu is also a striped maximal outerplanar n−d(u)+1 graph with |Gu | ≥ 4. So by Corollary 3.6, γc (Gu ) ≤ ⌊|Gu |/2⌋ − 1 = ⌊ ⌋ − 1 ≤ ⌊ n−2 3 ⌋ − 1 = ⌊ n−2 5 ⌋. Therefore, n 2 u u is odd, d(u) = 4, and γc (G ) = ⌊|G |/2⌋ − 1. Without loss of generality, we have that (v1 , v2 , u) is the DELETED VERTEX SEQUENCE of the procedure CREATE_GRAPH (we have the situation depicted in Fig. 8). Note that |Gu | is an even number greater than 4 and Gu is still a striped maximal outerplanar graph, and γc (Gu ) = ⌊|Gu |/2⌋ − 1. It means that Gu ∈ L . Note that E(G) \ E(Gu ) = {v1 u, v2 u, v3 u, v4 u, v1 v2 , v2 v3 } or E(G) \ E(Gu ) = {v1 u, v2 u, v3 u, v4 u, v1 v2 , v2 v4 }. Thus, it is easy to verify that G ∈ F . If |Gu | = n − d(u), analogously to the previous discussion, we have d(u) = 4. Without loss of generality, (v1 , v2 , u, v3 ) is the DELETED VERTEX SEQUENCE of the procedure CREATE_GRAPH (we have the situation depicted in Fig. 9). Note that |Gu | ≥ 4 and Gu is still a striped maximal outerplanar graph, and γc (Gu ) = γc (G)−2 = ⌊n/2⌋−3 = ⌊ n−2 4 ⌋−1 = ⌊|Gu |/2⌋−1. It means that Gu ∈ F ∪ L . Note that E(G) \ E(Gu ) = {v1 u, v2 u, v3 u, v4 u, v1 v2 , v2 v3 , v3 v4 , v3 t } or E(G) \ E(Gu ) = Please cite this article as: W. Zhuang, Connected domination in maximal outerplanar graphs, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.01.033.
8
W. Zhuang / Discrete Applied Mathematics xxx (xxxx) xxx
Fig. 9. (v1 , v2 , u, v3 ) is the DELETED VERTEX SEQUENCE of the procedure CREATE_GRAPH.
Fig. 10. The vertex u and its neighbors.
Fig. 11. The process of constructing the graph F .
{v1 u, v2 u, v3 u, v4 u, v1 v2 , v2 v4 , v3 v4 , v3 t }, where t is the neighbor of v4 of degree 3 in Gu , then it is easy to verify that G ∈ F ∪ L. □ At the end of this article, we prove an upper bound in terms of the degree of the vertices of a striped maximal outerplanar graph. This new upper bound is much better than ⌊n/2⌋ − 1 when the graph has many vertices of degree larger than 5. Lemma 4.5. Let G be a striped maximal outerplanar graph with ∆(G) ≥ 5. Then there is always a minimum connected dominating set of G, which contains all vertices of degree more than 4. Proof. Let u be any vertex of G having degree more than 4, and let u1 , u2 , . . . , ut −1 , ut be consecutive vertices of NG (u) in this order, as depicted in Fig. 10. Assume that S is a minimum connected dominating set of G. If u ̸ ∈ S, then u3 ∈ S (note that d(u3 ) = 3). The vertex set (S \ {u3 }) ∪ {u} is also a minimum connected dominating set of G. It implies that there is always a minimum connected dominating set of G, say S ′ , which contains all vertices of degree more than 4. □ Take a graph G ∈ F ∪ L , let u be any vertex of degree 5 in G, and let u1 , u2 , u3 , u4 , u5 be consecutive vertices of NG (u) in this order (see Fig. 11(a)). Subdivide u2 u3 and u3 u4 by inserting a few new vertices respectively, and join u to all the new vertices (see Fig. 11(b)). For each vertex of degree 5 in G, we repeat the operation mentioned above. We denote the resulting graph by F , and let H be the family of graphs consisting of all such graphs F arising in the way. Note that (F ∪ L ) ⊂ H . Please cite this article as: W. Zhuang, Connected domination in maximal outerplanar graphs, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.01.033.
W. Zhuang / Discrete Applied Mathematics xxx (xxxx) xxx
9
Theorem 4.6. If G is a striped maximal outerplanar graph of order n ≥ 4, then γc (G) ≤ ⌊ equality holds if and only if G ∈ H .
∑ n− ni=1 max{di −5,0} 2
⌋ − 1. Moreover,
Proof. If ∆(G) ≤ 5, then by Corollary 3.6, the first result holds. So we consider the case when ∆(G) > 5. Let v be any vertex of G having degree more than 5, i.e., |NG (v )| ≥ 6. Let v1 , v2 , . . . , vs−1 , vs be consecutive vertices of NG (v ) in this order. Note that the induced subgraph G[{v3 , v4 , . . . , vs−2 }] is a path. Let G′ be obtained from G by contracting ∑n each of all such paths into a single vertex. The resulting graph is a striped maximal outerplanar graph with n′ = n − i=1 max{di − 5, 0} vertices ′ (clearly, |G′ | ≥ 4) and ∆(G′ ) ≤ 5. By Corollary 3.6, γc (G′ ) ≤ ⌊ n2 ⌋ − 1. By Lemma 4.5, we know that there is one minimum ′ connected dominating set of G , which contains all vertices of degree more than 4.∑Note that this minimum connected n−
n
max{d −5,0}
i i=1 ⌋ − 1. This proves the dominating set is also the connected dominating set of G. Therefore, γc (G) ≤ ⌊ 2 desired bound, and we turn our attention to proving the second part of Theorem 4.6, i.e., ∑ the characterization of graphs
that achieve this bound. First, if a graph G ∈ H , it is easy to verify that γc (G) = ⌊
n−
n i=1 max{di −5,0}
2
∑n
n−
⌋ − 1. Conversely,
max{d −5,0}
i i=1 ⌋ − 1. We assume that G is a striped maximal outerplanar graph of order n ≥ 4, which satisfies γc (G) = ⌊ 2 will show that G ∈ H . If ∆(G) ≤ 5, then by Theorem 4.4, G ∈ (F ∪ L ) ⊂ H . So we consider the case when ∆(G) > 5. By Lemma 4.5, there is a minimum connected dominating set of G, say S, which contains all vertices of degree more than 4. We construct a new graph G′ from G by the operation mentioned at the beginning of this proof. Then, S is also a minimum connected dominating set of G′ . (if not, we must be able to find a connected dominating set of G′ , say S ′ , such that |S ′ | < |S |, and it contains all vertices of degree more than 4 in G′ . Note that S ′ is also a connected dominating set of G, a contradiction.) By Theorem 4.4, G′ ∈ F ∪ L . It follows that G ∈ H . □
CRediT authorship contribution statement Wei Zhuang: Conceptualization, Methodology, Validation, Writing - original draft, Writing - review & editing. Acknowledgments The author would like to thank Jianxiang Li, Zepeng Li, Baoyindureng Wu and the referees for many helpful comments and suggestions. References [1] N. Ananchuen, W. Ananchuen, M.D. Plummer, Matching properties in connected domination critical graphs, Discrete Appl. Math. 308 (2008) 1260–1267. [2] W. Ananchuen, N. Ananchuen, M.D. Plummer, Connected domination: vertex criticality and matchings, Util. Math. 89 (2012) 141–159. [3] T. Araki, I. Yumoto, On the secure domination numbers of maximal outerplanar graphs, Discrete Appl. Math. 236 (2018) 23–29. [4] C.N. Campos, Y. Wakabayashi, On dominating sets of maximal outerplanar graphs, Discrete Appl. Math. 161 (2013) 330–335. [5] W.J. Desormeaux, T.H. Haynes, M.A. Henning, Bounds on the connected domination number of a graph, Discrete Appl. Math. 161 (2013) 2925–2931. [6] W.J. Desormeaux, T.H. Haynes, L. van der Merwe, Connected domination stable graphs upon edge addition, Quaest. Math. 38 (2015) 841–848. [7] M. Dorfling, J.H. Hattingh, E. Jonck, Total domination in maximal outerplanar graphs II, Discrete Math. 339 (2016) 1180–1188. [8] M. Dorfling, J.H. Hattingh, E. Jonck, Total domination in maximal outerplanar graphs, Discrete Appl. Math. 217 (2017) 506–511. [9] W. Duckworth, B. Mans, Connected domination of regular graphs, Discrete Math. 309 (2009) 2305–2322. [10] B. Gao, Y. Yang, H. Ma, A new distributed approximation algorithm for constructing minimum connected dominating set in wireless ad hoc networks, Int. J. Commun. Syst. 18 (2005) 743–762. [11] T.W. Haynes, S.T. Hedetniemi, P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, New York, 1998. [12] S. Kamei, H. Kakugawa, S. Devismes, S. Tixeuil, A self-stabilizing 3-approximation for the maximum leaf spanning tree problem in arbitrary networks, J. Comb. Optim. 25 (2013) 430–459. [13] H. Karami, S.M. Sheikholeslami, A. Khodkar, Connected domination number of a graph and its complement, Graphs Comb. 28 (2012) 123–131. [14] D. Kim, Z. Zhang, X. Li, W. Wang, W. Wu, D. Du, A better approximation algorithm for computing connected dominating sets in unit ball graphs, IEEE Trans. Mob. Comput. 9 (2010) 1108–1118. [15] R. Laskar, J. Pfaff, Domination and Irredundance in Split Graphs, Technical Report 430, Deparment of Mathematical Sciences, Clemson University, 1983. [16] Z. Li, E. Zhu, Z. Shao, J. Xu, On dominating sets of maximal outerplanar and planar graphs, Discrete Appl. Math. 198 (2016) 164–169. [17] L.R. Matheson, R.E. Tarjan, Dominating sets in planar graphs, European J. Combin. 17 (1996) 565–568. [18] A. Reich, Complexity of the maximum leaf spanning tree problem on planar and regular graphs, Theoret. Comput. Sci. 626 (2016) 134–143. [19] E. Sampathkumar, H.B. Walikar, The connected domination number of a graph, J. Math. Phys. Sci. 13 (1979) 607–613. [20] O. Schaudt, On graphs for which the connected domination number is at most the total domination number, Discrete Appl. Math. 160 (2012) 1281–1284. [21] R. Solis-Oba, P. Bonsma, S. Lowski, A 2-approximation algorithm for finding a spanning tree with maximum number of leaves, Algorithmica 77 (2017) 374–388. [22] Shin-ichi Tokunaga, Dominating sets of maximal outerplanar graphs, Discrete Appl. Math. 161 (2013) 3097–3099. [23] D.B. West, Introduction to Graph Theory, second ed., Pearson Education, Beijing, 2004. [24] X. Xia, Y. Zhou, X. Lai, On the analysis of the (1 + 1) evolutionary algorithm for the maximum leaf spanning tree problem, Int. J. Comput. Math. 92 (2015) 2023–2035. [25] J. Zhou, Z. Zhang, S. Tang, X. Huang, D. Du, Breaking the O(ln n) barrier: an enhanced approximation algorithm for fault-tolerant minimum weight connected dominating set, Informs J. Comput. 30 (2018) 225–235.
Please cite this article as: W. Zhuang, Connected domination in maximal outerplanar graphs, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.01.033.