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Which k -trees are cover–incomparability graphs?
Discrete Applied Mathematics 167 (2014) 222–227
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Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Which k-trees are cover–incomparability graphs? Jana Maxová ∗ , Miroslava Dubcová, Pavla Pavlíková, Daniel Turzík Department of Mathematics, Institute of Chemical Technology, Prague, Technická 5, 166 28 Praha 6, Czech Republic
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Article history: Received 22 April 2013 Received in revised form 12 November 2013 Accepted 15 November 2013 Available online 28 November 2013 Keywords: Poset Cover–incomparability graph Chordal graph Simplicial vertex k-tree
abstract In this paper we deal with cover–incomparability graphs of posets. It is known that the class of cover–incomparability graphs is not closed on induced subgraphs which makes the study of structural properties of these graphs difficult. In this paper we introduce the notion of s-subgraph which enables us to define forbidden s-subgraphs (i.e. graphs that cannot appear as s-subgraphs of any cover–incomparability graph). We show that the family of minimal forbidden s-subgraphs is infinite even for cover–incomparability unit-interval graphs. Using the notion of s-subgraph we also answer the question which k-trees are cover–incomparability graphs and which chordal graphs without K4 are cover–incomparability graphs. © 2013 Elsevier B.V. All rights reserved.
1. Introduction The cover–incomparability graph of a poset P, or shortly the C–I graph, was introduced in [2] as the edge union of the cover graph of P and the incomparability graph of P. In other words, two vertices x and y are adjacent in the C–I graph G(P ) if and only if x covers y, or y covers x, or x and y are incomparable in P. The notion of C–I graph was motivated by the theory of transit functions of posets. It turns out that the C–I graph of a poset P is just the underlying graph of the standard transit function on P (see [1] for details). C–I graphs also present the only non-trivial way to obtain an associated graph of a poset as unions and/or intersections of the edge sets of the three standard associated graphs (i.e. cover, comparability and incomparability graphs). C–I graphs also belong to the class of graphs called antimatroid graphs (see [4]), more precisely C–I graphs are antimatroid graphs of the antimatroid consisting of standard order convex sets of posets. There are two basic ways how to approach the notion of cover–incomparability graphs. One possibility is to study posets whose C–I graphs have special properties. Posets whose C–I graphs are claw-free, chordal, distance-hereditary and Ptolemaic can be characterized using forbidden subposets [2]. The other direction is to try to answer questions such as: Which graphs are C–I graphs? Which chordal, interval, distancehereditary, Ptolemaic graphs are C–I graphs? These questions seem to be much harder. The problem of recognizing C–I graphs was shown to be NP-complete in general [5]. On the other hand it is clearly polynomial for instance for trees (as in any C–I graph there are at most 2 vertices of degree 1). The study of algorithmic issues related to C–I graphs is continued in [1,6]. Brešar et al. [1] concentrated on two special subclasses of chordal graphs and proved that the recognition problem is polynomial if restricted to block graphs or to split graphs. Maxová and Turzík [6] found a simple necessary condition for a graph to be a C–I graph and showed that for Ptolemaic graphs this condition is also sufficient. It follows that the problem of recognizing C–I graphs is polynomial among Ptolemaic graphs, and (with some more effort) also among all distancehereditary graphs.
∗
Corresponding author. Tel.: +420 220 443 096; fax: +420 220444477. E-mail address: [email protected] (J. Maxová).
0166-218X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.dam.2013.11.019
J. Maxová et al. / Discrete Applied Mathematics 167 (2014) 222–227
223
In this paper we solve the cover–incomparability graph recognition problem for k-trees. A k-tree is a chordal graph all of whose maximal cliques are the same size k + 1 and all of whose minimal clique separators are also all the same size k. The k-trees are exactly the maximal graphs with a given treewidth, graphs to which no more edges can be added without increasing their treewidth. The graphs that have treewidth at most k are exactly the subgraphs of k-trees. For this reason they are called partial k-trees. A k-path is a k-tree with at most two simplicial vertices (or a complete graph on k + 1 vertices). In Section 3 we show which k-trees are cover–incomparability graphs. We first observe if a k-tree G is a C–I graph then G must be a k-path. We further show that a k-path is a C–I graph whenever k is odd. For k even, a k-path on n vertices is a C–I graph if and only if n ≤ 2k + 1 (see Theorem 3). As it was observed in [5], C–I graphs are not closed on induced subgraphs. This is probably the reason that makes the recognition problem difficult. In this paper we introduce the notion of s-subgraph which seems to be crucial for the study of structural properties of C–I graphs. This notion enables us to define so-called forbidden s-subgraphs (i.e. graphs that may not appear as s-subgraphs of any C–I graph). We show that k-paths on 2k + 2 vertices are minimal forbidden s-subgraphs in the class of all chordal graphs. In Section 4 we deal with chordal graphs without K4 and show that such a graph is C–I if and only if it does not contain a triple of independent simplicial vertices and it does not contain 2-path on 6 vertices as an s-subgraph (see Theorem 4). 2. Preliminaries Let P = (V , ≤) be a poset. We will use the following notation. For u, v ∈ V we write:
• • • •
u u u u
< v if u ≤ v and u ̸= v . ▹ v if u < v and there is no z ∈ V such that u < z < v . We say that v covers u. ▹▹ v if u < v and ¬(u ▹ v). ∥ v if u and v are incomparable.
Definition 1. For a given poset P = (V , ≤), let G(P ) = (V , E ) be a graph with E = {{u, v} | u ▹ v or v ▹ u or u ∥ v}. Then we say that G(P ) is the cover–incomparability graph of P (or the C–I graph of P for short). Note that for any u ̸= v ∈ V (G(P )), {u, v} ̸∈ E (G(P )) ⇔ u ▹▹ v or v ▹▹ u. Let us also list a few easy observations about C–I graphs [2,5]. Lemma 1. Let P = (V , ≤) be a poset and G(P ) = (V , E ) its C–I graph. Then (i) (ii) (iii) (iv) (v) (vi)
G(P ) is connected. If U ⊆ V is an antichain in P, then U induces a complete subgraph in G(P ). If I ⊆ V is an independent set in G(P ), then all points of I lie on a common chain in P. There are at most 2 vertices of degree 1 in G(P ). G(P ) contains no induced cycles of length greater than 4. G(P ) (the complement of G(P)) admits a transitive orientation.
A vertex v ∈ V (G) is called simplicial if its neighbors induce a complete subgraph of G. More formally, v ∈ V (G) is a simplicial vertex if {u, w} ̸∈ E (G) implies {{u, v}, {w, v}} ̸⊆ E (G). A graph is said to be chordal if it contains no induced cycle of length more than three. It is well known (see e.g.[3]) that every chordal graph is either a complete graph or it contains two simplicial vertices. Lemma 2 ([6]). Let P be a poset and G(P ) = (V , E ) its C–I graph. Let v be a simplicial vertex in G(P ). Then v is a maximal or a minimal element of P. Proof. Suppose v is neither a maximal nor a minimal element of P. Then there exist vertices x, y ∈ V (G) such that x ▹ v ▹ y in P. Vertices x and y are neighbors of v not connected by an edge, a contradiction with v being simplicial. Lemma 3 ([6]). If G is a C–I graph then G does not contain 3 independent simplicial vertices. Proof. Suppose there are 3 simplicial vertices in G(P ) that form an independent set. According to Lemma 1(iii) these 3 vertices lie on a common chain in the poset P. Hence one of them (the middle one) is neither a maximal nor a minimal element of P, a contradiction with Lemma 2. Lemma 3 gives us a simple necessary condition for an arbitrary graph to be a C–I graph. This condition is also sufficient if G is Ptolemaic (i.e. a chordal distance-hereditary graph) [6]. But it is not sufficient for general chordal graphs, as the graph G63 depicted in Fig. 1 is a chordal graph with only two simplicial vertices. But G63 is not a C–I graph (see Theorem 2 in Section 3). The next lemma follows immediately from the definition of C–I graphs. Lemma 4. Let P be a poset and G(P ) its C–I graph. Let v ∈ V be a minimal or a maximal element in P. Then G \ v is also a C–I graph. Proof. If v is a maximal or a minimal element of P then the poset P ′ defined as P ′ := P \ v is a subposet of P satisfying G(P ′ ) = G \ v .
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Fig. 1. G63 is not a C–I graph.
Definition 2. Let G be a graph (not necessarily chordal). We say that H is an s-subgraph of G if H can be obtained from G by a consecutive deletion of simplicial vertices. More formally, there exist graphs G = G1 , G2 , . . . , Gk = H and vertices v1 , v2 , . . . , vk−1 such that vi is a simplicial vertex in Gi and Gi+1 = Gi \ vi , for i = 1, . . . , k − 1. Combining Lemmas 2 and 4 we immediately get the following important property of s-subgraphs. Corollary 1. If G is a C–I graph then every s-subgraph of G is also a C–I graph. 3. k-trees A graph G is called a k-tree if and only if either G is a complete graph on k + 1 vertices, or G has a vertex v of degree k such that the neighbors of v form a k-clique, and G \ v is a k-tree. In other words, every k-tree can be constructed from a (k + 1)-clique by successively adding vertices of degree k connected to k vertices that form a k-clique. A k-tree G is called a k-path if either G has at most two independent simplicial vertices or G is just the complete graph Kk+1 . In this section we show which k-trees are cover–incomparability graphs. By Lemma 3 every k-tree that is a C–I graph may contain at most 2 independent simplicial vertices, thus it must be a k-path. In the rest of this section we show which k-paths are C–I graphs. Definition 3. For k and n natural numbers such that k ≤ n we define the graph Gnk by setting V (Gnk ) = {1, 2, . . . , n} and E (Gnk ) = {{i, j}; |i − j| < k}. For example, Gn2 is just a path on n vertices while Gnn is the complete graph Kn . The graph G63 is depicted in Fig. 1. Note that Gnk are unit-interval graphs for all k ≤ n. It is easy to see that Gnk is exactly the (k − 1)-path on n vertices. Theorem 1. If k is even then Gnk is a C–I graph for every n ≥ k. Proof. Let k = 2ℓ and let ≤N denote the standard ordering on natural numbers. We define a new ordering ≺ on {1, 2, . . . , n} by i ≺ j ⇔ i ≤N j − ℓ. Let P be the poset P = ({1, 2, . . . , n}, ≺). Note that in the poset P the following holds:
• i ∥ j for j ∈ {i + 1, . . . , i + ℓ − 1} • i ▹ j for j ∈ {i + ℓ, . . . , i + k − 1} • i ▹▹ j for j ∈ {i + k, . . . , n}. It is straightforward to check that Gnk = G(P ).
Theorem 2. Let k be an odd natural number. Then Gnk is a C–I graph if and only if n ≤ 2k − 1. −1 Proof. First we show that Gnk is a C–I graph for n = 2k − 1. In G2k the vertex k is adjacent to every other vertex. We consider k 2k−1 2k−1 the graph Gk \ {k}. It is easy to check that Gk \ {k} is isomorphic to Gk2k−−12 . Thus by Theorem 1, as k − 1 is even, there
−2 ′ exists a poset P such that G(P ) = G2k k−1 . We modify the poset P to a new poset P by adding a new element k incomparable 2k−1 with all other elements in P. It is easy to check that Gk = G(P ′ ). −1 If n < 2k − 1 then Gnk is an s-subgraph of G2k . Thus by Corollary 1, Gnk is a C–I graph also for n < 2k − 1 and one k implication is proved. n For the reverse implication it is enough to show that G2k k is not a C–I graph. By Corollary 1, the same holds for Gk if n > 2k. 2k Suppose for contradiction that there exists a poset P such that G(P ) = Gk . There are two simplicial vertices in G2k k , namely 1 and 2k. As they are not connected by an edge without loss of generality we may suppose that vertex 1 is a minimal element in P and vertex 2k is a maximal element in P.
Claim 1. If i
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Claim 2. For every i = 1, . . . , k there exists a vertex x such that i ▹P x ▹P i + k and i i, such that {i, j} ̸∈ E (G2k k ). We call the pairs [i, i + k], i = 1, . . . , k minimal non-edges in G2k k and the vertex x a separating vertex of [i, i + k]. Claim 3. Distinct minimal non-edges have distinct separating vertices. More formally, if i ▹P xi ▹P i + k and j ▹P xj ▹P j + k for 1 ≤N i
Claim 4. There are no i, j, ℓ ∈ {1, . . . , k} such that both i and i + k are separating vertices for minimal non-edges [j, j + k] and [ℓ, ℓ + k], respectively. More formally, for every i, j ∈ {1, . . . , k} the following holds: (i) If j ▹P i ▹P j + k then there is no ℓ ∈ {1, . . . , k} such that ℓ ▹P i + k ▹P ℓ + k. (ii) If j ▹P i + k ▹P j + k then there is no ℓ ∈ {1, . . . , k} such that ℓ ▹P i ▹P ℓ + k. Suppose for contradiction that j ▹P i ▹P j + k and ℓ ▹P i + k ▹P ℓ + k for some j, ℓ ∈ {1, . . . , k}, not necessarily j ̸= ℓ. 2k Let x be the separating vertex for [i, i + k], that is i ▹P x ▹P i + k. It follows that {j, x} ̸∈ E (G2k k ) and {ℓ + k, x} ̸∈ E (Gk ). On 2k 2k the other hand, if x ∈ {1, . . . , k} then by the definition of Gk , the pair {j, x} must form an edge in Gk , a contradiction. If 2k x ∈ {k + 1, . . . , 2k} then by the definition of G2k k , the pair {ℓ + k, x} ∈ E (Gk ), a contradiction. Let us define a mapping π : {1, . . . , k} → {1, . . . , k} by π (i) = j if and only if j ▹P i ▹P j + k or j ▹P i + k ▹P j + k. By Claims 3 and 4, π is well defined. By Claim 2, π is surjective. Hence π is a permutation on {1, . . . , k}. We consider the decomposition of π into cycles. As k is odd, there must be at least one odd cycle in the decomposition. Let C = (x1 , x2 , . . . , xr ) be an odd cycle. We color the pairs {xi , xi+1 } of vertices along the cycle C by red and blue in the following way (all indices are taken mod r, where r is the length of C ). If xi+1 ▹P xi ▹P xi+1 + k we color the pair {xi , xi+1 } by red. If xi+1 ▹P xi + k ▹P xi+1 + k we color the pair {xi , xi+1 } by blue. Suppose that there are two consecutive red pairs {xi−1 , xi } and {xi , xi+1 } in C . It follows that xi+1 ▹P xi ▹P xi−1 and thus {xi+1 , xi−1 } ̸∈ E (G2k k ), a contradiction with {xi−1 , xi , xi+1 } ⊆ {1, . . . , k}. Suppose there are two consecutive blue pairs {xi−1 , xi } and {xi , xi+1 } in C . It follows that xi−1 + k ▹P xi + k ▹P xi+1 + k and thus {xi−1 + k, xi+1 + k} ̸∈ E (G2k k ), a contradiction with {xi−1 + k, xi + k, xi+1 + k} ⊆ {k + 1, . . . , 2k}. Hence, the colors along the cycle C must alternate, but this is not possible as the cycle C has odd length. Recall that Gnk is exactly the (k − 1)-path on n vertices and that there are no k-trees that are C–I graphs other than k-paths. Hence we can reformulate Theorems 1 and 2 in the following way. Theorem 3. For k odd, a k-tree G is a C–I graph if and only if G is a k-path. For k even, a k-tree G is a C–I graph if and only if G is a k-path and n ≤ 2(k + 1) − 1 = 2k + 1 where n is the number of vertices of G. The concept of s-subgraphs also enables us to define ‘‘minimal’’ graphs which are forbidden to appear as s-subgraphs in any C–I graph: Definition 4. We say that G is a minimal forbidden s-subgraph if the following two conditions are satisfied 1. G is not a C–I graph. 2. Every proper s-subgraph of G is a C–I graph. A natural question arises whether there are infinitely many forbidden s-subgraphs e.g. for all chordal graphs. An affirmative answer to this question was in fact already given in this section. For k odd, the graphs G2k k are minimal forbidden s-subgraphs, they are chordal graphs (even unit-interval graphs) and there are infinitely many of them. 4. Chordal graphs without K4 As G63 is not a C–I graph it may not be an s-subgraph of any C–I graph. This together with Lemma 3 gives us a necessary and sufficient condition for chordal graphs without K4 to be C–I graphs: Theorem 4. Let G be a connected chordal graph without K4 . Then G is a C–I graph if and only if the following two conditions are satisfied: (C1) G does not contain a triple of independent simplicial vertices. (C2) G does not contain G63 as an s-subgraph. Before we prove Theorem 4 let us introduce one more notion. A k-fan is a graph on k + 2 vertices obtained from a path on k + 1 vertices by adding a vertex of degree k + 1 connected to every vertex of the path. It is easy to see that a k fan is a C–I graph for every k ≥ 1 (see e.g. [2] or Fig. 2).
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Fig. 2. A k-fan Bi and its poset Pi .
Fig. 3. Blocks of G and the constructed poset P.
Proof. One implication is easy. According to Lemma 3, condition (C1) is a necessary condition for a graph G to be a C–I graph. Theorem 2 and Corollary 1 imply that condition (C2) is also necessary. Thus every C–I graph satisfies both (C1) and (C2). For the proof of the reverse implication let G be a chordal graph without K4 satisfying both (C1) and (C2). We distinguish two cases. Case 1: G is 2-connected Let us consider a perfect elimination scheme v1 , v2 , . . . , vn of G. This means that for every i = 1, . . . , n the vertex vi is a simplicial vertex in G|{v1 , . . . , vi } (i.e. the subgraph of G induced by {v1 , . . . , vi }). Such a scheme exists for every chordal graph (see e.g. [3]). We prove by induction on i that for every i = 3, . . . , n the graph G|{v1 , . . . , vi } is always isomorphic to an (i − 2)-fan. This statement is clear for i = 3 and i = 4. Now let n > i ≥ 4, let G|{v1 , . . . , vi } be a (i − 2)-fan and let vi+1 be the next vertex in the perfect elimination scheme of G. As G does not contain K4 the degree deg(vi+1 ) < 3 in G|{v1 , . . . , vi+1 }. As G is 2-connected deg(vi+1 ) = 2. Further note that v1 and vi are the only simplicial vertices of the (i − 2)-fan G|{v1 , . . . , vi }. Obviously one of the vertices v1 , vi must be a neighbor of vi+1 otherwise there are 3 independent simplicial vertices v1 , vi , vi+1 in G|{v1 , . . . , vi+1 }. As adding a simplicial vertex cannot decrease the number of independent simplicial vertices there are at least three independent simplicial vertices also in G, a contradiction. Thus vi+1 is connected by an edge either to v1 or to vi . The other neighbor of vi+1 must be the vertex of degree 2, otherwise we get G63 as an s-subgraph of G|{v1 , . . . , vi+1 }, and hence also of G, a contradiction. Thus G|{v1 , . . . , vi+1 } is an (i − 1)-fan and the induction is finished. Hence, for every i = 3, . . . , n the graph G|{v1 , . . . , vi } is isomorphic to an (i − 2)-fan. It follows that G is a C–I graph (see Fig. 2 for an example of a suitable poset P for a k-fan). Case 2: G is not 2-connected We consider the blocks (maximal 2-connected components) of G. Every block is isomorphic to a k-fan for some k. It is a well-known fact (see e.g. [3]) that the blocks of G form a tree-like structure. In our case, according to condition (C1) the tree must be a path. Let B1 , . . . , Bb be the blocks of G or single vertices not contained in any block of G. We number the Bi ’s from 1 to b along this path (see Fig. 3). Note that Bi cannot be a triangle (a 1-fan) for i = 2, . . . , b − 1, otherwise there are 3 independent simplicial vertices in G. For k ≥ 2 and every k-fan Bi let Pi be a poset formed by a chain on k + 1 points and another point comparable only with the maximal and the minimal element of the chain (see Fig. 2). If B1 is a 1-fan let P1 ∼ = (V , <1 ) for V = {a, b, c }, <1 = {a < c , b < c } where c is the only non-simplicial vertex of B1 in G. If Bb is a 1-fan let Pb ∼ = (V ,
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Now we construct a poset P = (V ,
(∃i s.t. x, y ∈ V (Bi ) ∧ x
An example of the graph G and the poset P is depicted in Fig. 3. It is easy to check that G = G(P ).
Acknowledgment This work was partially supported by means of Specific research funds of the Institute of Chemical Technology, Prague. References [1] B. Brešar, M. Changat, T. Gologranc, J. Mathews, A. Mathews, Cover-incomparability graphs and chordal graphs, Discrete Appl. Math. 158 (2010) 1752–1759. [2] B. Brešar, M. Changat, S. Klavžar, M. Kovše, J. Mathews, A. Mathews, Cover-incomparability graphs of posets, Order 25 (2008) 335–347. [3] R. Diestel, Graph Theory, in: Graduate Texts in Mathematics, Springer-Verlag, 2005. [4] R.E. Jamison-Waldner, Convexity and block graphs, Congr. Numer. 33 (1981) 129–142. [5] J. Maxová, P. Pavlíková, D. Turzík, On the complexity of cover-incomparability graphs of posets, Order 26 (2009) 229–236. [6] J. Maxová, D. Turzík, Which distance-hereditary graphs are cover-incomparability graphs? Discrete Appl. Math. 161 (13–14) (2013) 2095–2100.
Contents lists available at ScienceDirect
Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Which k-trees are cover–incomparability graphs? Jana Maxová ∗ , Miroslava Dubcová, Pavla Pavlíková, Daniel Turzík Department of Mathematics, Institute of Chemical Technology, Prague, Technická 5, 166 28 Praha 6, Czech Republic
article
info
Article history: Received 22 April 2013 Received in revised form 12 November 2013 Accepted 15 November 2013 Available online 28 November 2013 Keywords: Poset Cover–incomparability graph Chordal graph Simplicial vertex k-tree
abstract In this paper we deal with cover–incomparability graphs of posets. It is known that the class of cover–incomparability graphs is not closed on induced subgraphs which makes the study of structural properties of these graphs difficult. In this paper we introduce the notion of s-subgraph which enables us to define forbidden s-subgraphs (i.e. graphs that cannot appear as s-subgraphs of any cover–incomparability graph). We show that the family of minimal forbidden s-subgraphs is infinite even for cover–incomparability unit-interval graphs. Using the notion of s-subgraph we also answer the question which k-trees are cover–incomparability graphs and which chordal graphs without K4 are cover–incomparability graphs. © 2013 Elsevier B.V. All rights reserved.
1. Introduction The cover–incomparability graph of a poset P, or shortly the C–I graph, was introduced in [2] as the edge union of the cover graph of P and the incomparability graph of P. In other words, two vertices x and y are adjacent in the C–I graph G(P ) if and only if x covers y, or y covers x, or x and y are incomparable in P. The notion of C–I graph was motivated by the theory of transit functions of posets. It turns out that the C–I graph of a poset P is just the underlying graph of the standard transit function on P (see [1] for details). C–I graphs also present the only non-trivial way to obtain an associated graph of a poset as unions and/or intersections of the edge sets of the three standard associated graphs (i.e. cover, comparability and incomparability graphs). C–I graphs also belong to the class of graphs called antimatroid graphs (see [4]), more precisely C–I graphs are antimatroid graphs of the antimatroid consisting of standard order convex sets of posets. There are two basic ways how to approach the notion of cover–incomparability graphs. One possibility is to study posets whose C–I graphs have special properties. Posets whose C–I graphs are claw-free, chordal, distance-hereditary and Ptolemaic can be characterized using forbidden subposets [2]. The other direction is to try to answer questions such as: Which graphs are C–I graphs? Which chordal, interval, distancehereditary, Ptolemaic graphs are C–I graphs? These questions seem to be much harder. The problem of recognizing C–I graphs was shown to be NP-complete in general [5]. On the other hand it is clearly polynomial for instance for trees (as in any C–I graph there are at most 2 vertices of degree 1). The study of algorithmic issues related to C–I graphs is continued in [1,6]. Brešar et al. [1] concentrated on two special subclasses of chordal graphs and proved that the recognition problem is polynomial if restricted to block graphs or to split graphs. Maxová and Turzík [6] found a simple necessary condition for a graph to be a C–I graph and showed that for Ptolemaic graphs this condition is also sufficient. It follows that the problem of recognizing C–I graphs is polynomial among Ptolemaic graphs, and (with some more effort) also among all distancehereditary graphs.
∗
Corresponding author. Tel.: +420 220 443 096; fax: +420 220444477. E-mail address: [email protected] (J. Maxová).
0166-218X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.dam.2013.11.019
J. Maxová et al. / Discrete Applied Mathematics 167 (2014) 222–227
223
In this paper we solve the cover–incomparability graph recognition problem for k-trees. A k-tree is a chordal graph all of whose maximal cliques are the same size k + 1 and all of whose minimal clique separators are also all the same size k. The k-trees are exactly the maximal graphs with a given treewidth, graphs to which no more edges can be added without increasing their treewidth. The graphs that have treewidth at most k are exactly the subgraphs of k-trees. For this reason they are called partial k-trees. A k-path is a k-tree with at most two simplicial vertices (or a complete graph on k + 1 vertices). In Section 3 we show which k-trees are cover–incomparability graphs. We first observe if a k-tree G is a C–I graph then G must be a k-path. We further show that a k-path is a C–I graph whenever k is odd. For k even, a k-path on n vertices is a C–I graph if and only if n ≤ 2k + 1 (see Theorem 3). As it was observed in [5], C–I graphs are not closed on induced subgraphs. This is probably the reason that makes the recognition problem difficult. In this paper we introduce the notion of s-subgraph which seems to be crucial for the study of structural properties of C–I graphs. This notion enables us to define so-called forbidden s-subgraphs (i.e. graphs that may not appear as s-subgraphs of any C–I graph). We show that k-paths on 2k + 2 vertices are minimal forbidden s-subgraphs in the class of all chordal graphs. In Section 4 we deal with chordal graphs without K4 and show that such a graph is C–I if and only if it does not contain a triple of independent simplicial vertices and it does not contain 2-path on 6 vertices as an s-subgraph (see Theorem 4). 2. Preliminaries Let P = (V , ≤) be a poset. We will use the following notation. For u, v ∈ V we write:
• • • •
u u u u
< v if u ≤ v and u ̸= v . ▹ v if u < v and there is no z ∈ V such that u < z < v . We say that v covers u. ▹▹ v if u < v and ¬(u ▹ v). ∥ v if u and v are incomparable.
Definition 1. For a given poset P = (V , ≤), let G(P ) = (V , E ) be a graph with E = {{u, v} | u ▹ v or v ▹ u or u ∥ v}. Then we say that G(P ) is the cover–incomparability graph of P (or the C–I graph of P for short). Note that for any u ̸= v ∈ V (G(P )), {u, v} ̸∈ E (G(P )) ⇔ u ▹▹ v or v ▹▹ u. Let us also list a few easy observations about C–I graphs [2,5]. Lemma 1. Let P = (V , ≤) be a poset and G(P ) = (V , E ) its C–I graph. Then (i) (ii) (iii) (iv) (v) (vi)
G(P ) is connected. If U ⊆ V is an antichain in P, then U induces a complete subgraph in G(P ). If I ⊆ V is an independent set in G(P ), then all points of I lie on a common chain in P. There are at most 2 vertices of degree 1 in G(P ). G(P ) contains no induced cycles of length greater than 4. G(P ) (the complement of G(P)) admits a transitive orientation.
A vertex v ∈ V (G) is called simplicial if its neighbors induce a complete subgraph of G. More formally, v ∈ V (G) is a simplicial vertex if {u, w} ̸∈ E (G) implies {{u, v}, {w, v}} ̸⊆ E (G). A graph is said to be chordal if it contains no induced cycle of length more than three. It is well known (see e.g.[3]) that every chordal graph is either a complete graph or it contains two simplicial vertices. Lemma 2 ([6]). Let P be a poset and G(P ) = (V , E ) its C–I graph. Let v be a simplicial vertex in G(P ). Then v is a maximal or a minimal element of P. Proof. Suppose v is neither a maximal nor a minimal element of P. Then there exist vertices x, y ∈ V (G) such that x ▹ v ▹ y in P. Vertices x and y are neighbors of v not connected by an edge, a contradiction with v being simplicial. Lemma 3 ([6]). If G is a C–I graph then G does not contain 3 independent simplicial vertices. Proof. Suppose there are 3 simplicial vertices in G(P ) that form an independent set. According to Lemma 1(iii) these 3 vertices lie on a common chain in the poset P. Hence one of them (the middle one) is neither a maximal nor a minimal element of P, a contradiction with Lemma 2. Lemma 3 gives us a simple necessary condition for an arbitrary graph to be a C–I graph. This condition is also sufficient if G is Ptolemaic (i.e. a chordal distance-hereditary graph) [6]. But it is not sufficient for general chordal graphs, as the graph G63 depicted in Fig. 1 is a chordal graph with only two simplicial vertices. But G63 is not a C–I graph (see Theorem 2 in Section 3). The next lemma follows immediately from the definition of C–I graphs. Lemma 4. Let P be a poset and G(P ) its C–I graph. Let v ∈ V be a minimal or a maximal element in P. Then G \ v is also a C–I graph. Proof. If v is a maximal or a minimal element of P then the poset P ′ defined as P ′ := P \ v is a subposet of P satisfying G(P ′ ) = G \ v .
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Fig. 1. G63 is not a C–I graph.
Definition 2. Let G be a graph (not necessarily chordal). We say that H is an s-subgraph of G if H can be obtained from G by a consecutive deletion of simplicial vertices. More formally, there exist graphs G = G1 , G2 , . . . , Gk = H and vertices v1 , v2 , . . . , vk−1 such that vi is a simplicial vertex in Gi and Gi+1 = Gi \ vi , for i = 1, . . . , k − 1. Combining Lemmas 2 and 4 we immediately get the following important property of s-subgraphs. Corollary 1. If G is a C–I graph then every s-subgraph of G is also a C–I graph. 3. k-trees A graph G is called a k-tree if and only if either G is a complete graph on k + 1 vertices, or G has a vertex v of degree k such that the neighbors of v form a k-clique, and G \ v is a k-tree. In other words, every k-tree can be constructed from a (k + 1)-clique by successively adding vertices of degree k connected to k vertices that form a k-clique. A k-tree G is called a k-path if either G has at most two independent simplicial vertices or G is just the complete graph Kk+1 . In this section we show which k-trees are cover–incomparability graphs. By Lemma 3 every k-tree that is a C–I graph may contain at most 2 independent simplicial vertices, thus it must be a k-path. In the rest of this section we show which k-paths are C–I graphs. Definition 3. For k and n natural numbers such that k ≤ n we define the graph Gnk by setting V (Gnk ) = {1, 2, . . . , n} and E (Gnk ) = {{i, j}; |i − j| < k}. For example, Gn2 is just a path on n vertices while Gnn is the complete graph Kn . The graph G63 is depicted in Fig. 1. Note that Gnk are unit-interval graphs for all k ≤ n. It is easy to see that Gnk is exactly the (k − 1)-path on n vertices. Theorem 1. If k is even then Gnk is a C–I graph for every n ≥ k. Proof. Let k = 2ℓ and let ≤N denote the standard ordering on natural numbers. We define a new ordering ≺ on {1, 2, . . . , n} by i ≺ j ⇔ i ≤N j − ℓ. Let P be the poset P = ({1, 2, . . . , n}, ≺). Note that in the poset P the following holds:
• i ∥ j for j ∈ {i + 1, . . . , i + ℓ − 1} • i ▹ j for j ∈ {i + ℓ, . . . , i + k − 1} • i ▹▹ j for j ∈ {i + k, . . . , n}. It is straightforward to check that Gnk = G(P ).
Theorem 2. Let k be an odd natural number. Then Gnk is a C–I graph if and only if n ≤ 2k − 1. −1 Proof. First we show that Gnk is a C–I graph for n = 2k − 1. In G2k the vertex k is adjacent to every other vertex. We consider k 2k−1 2k−1 the graph Gk \ {k}. It is easy to check that Gk \ {k} is isomorphic to Gk2k−−12 . Thus by Theorem 1, as k − 1 is even, there
−2 ′ exists a poset P such that G(P ) = G2k k−1 . We modify the poset P to a new poset P by adding a new element k incomparable 2k−1 with all other elements in P. It is easy to check that Gk = G(P ′ ). −1 If n < 2k − 1 then Gnk is an s-subgraph of G2k . Thus by Corollary 1, Gnk is a C–I graph also for n < 2k − 1 and one k implication is proved. n For the reverse implication it is enough to show that G2k k is not a C–I graph. By Corollary 1, the same holds for Gk if n > 2k. 2k Suppose for contradiction that there exists a poset P such that G(P ) = Gk . There are two simplicial vertices in G2k k , namely 1 and 2k. As they are not connected by an edge without loss of generality we may suppose that vertex 1 is a minimal element in P and vertex 2k is a maximal element in P.
Claim 1. If i
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Claim 2. For every i = 1, . . . , k there exists a vertex x such that i ▹P x ▹P i + k and i
Claim 4. There are no i, j, ℓ ∈ {1, . . . , k} such that both i and i + k are separating vertices for minimal non-edges [j, j + k] and [ℓ, ℓ + k], respectively. More formally, for every i, j ∈ {1, . . . , k} the following holds: (i) If j ▹P i ▹P j + k then there is no ℓ ∈ {1, . . . , k} such that ℓ ▹P i + k ▹P ℓ + k. (ii) If j ▹P i + k ▹P j + k then there is no ℓ ∈ {1, . . . , k} such that ℓ ▹P i ▹P ℓ + k. Suppose for contradiction that j ▹P i ▹P j + k and ℓ ▹P i + k ▹P ℓ + k for some j, ℓ ∈ {1, . . . , k}, not necessarily j ̸= ℓ. 2k Let x be the separating vertex for [i, i + k], that is i ▹P x ▹P i + k. It follows that {j, x} ̸∈ E (G2k k ) and {ℓ + k, x} ̸∈ E (Gk ). On 2k 2k the other hand, if x ∈ {1, . . . , k} then by the definition of Gk , the pair {j, x} must form an edge in Gk , a contradiction. If 2k x ∈ {k + 1, . . . , 2k} then by the definition of G2k k , the pair {ℓ + k, x} ∈ E (Gk ), a contradiction. Let us define a mapping π : {1, . . . , k} → {1, . . . , k} by π (i) = j if and only if j ▹P i ▹P j + k or j ▹P i + k ▹P j + k. By Claims 3 and 4, π is well defined. By Claim 2, π is surjective. Hence π is a permutation on {1, . . . , k}. We consider the decomposition of π into cycles. As k is odd, there must be at least one odd cycle in the decomposition. Let C = (x1 , x2 , . . . , xr ) be an odd cycle. We color the pairs {xi , xi+1 } of vertices along the cycle C by red and blue in the following way (all indices are taken mod r, where r is the length of C ). If xi+1 ▹P xi ▹P xi+1 + k we color the pair {xi , xi+1 } by red. If xi+1 ▹P xi + k ▹P xi+1 + k we color the pair {xi , xi+1 } by blue. Suppose that there are two consecutive red pairs {xi−1 , xi } and {xi , xi+1 } in C . It follows that xi+1 ▹P xi ▹P xi−1 and thus {xi+1 , xi−1 } ̸∈ E (G2k k ), a contradiction with {xi−1 , xi , xi+1 } ⊆ {1, . . . , k}. Suppose there are two consecutive blue pairs {xi−1 , xi } and {xi , xi+1 } in C . It follows that xi−1 + k ▹P xi + k ▹P xi+1 + k and thus {xi−1 + k, xi+1 + k} ̸∈ E (G2k k ), a contradiction with {xi−1 + k, xi + k, xi+1 + k} ⊆ {k + 1, . . . , 2k}. Hence, the colors along the cycle C must alternate, but this is not possible as the cycle C has odd length. Recall that Gnk is exactly the (k − 1)-path on n vertices and that there are no k-trees that are C–I graphs other than k-paths. Hence we can reformulate Theorems 1 and 2 in the following way. Theorem 3. For k odd, a k-tree G is a C–I graph if and only if G is a k-path. For k even, a k-tree G is a C–I graph if and only if G is a k-path and n ≤ 2(k + 1) − 1 = 2k + 1 where n is the number of vertices of G. The concept of s-subgraphs also enables us to define ‘‘minimal’’ graphs which are forbidden to appear as s-subgraphs in any C–I graph: Definition 4. We say that G is a minimal forbidden s-subgraph if the following two conditions are satisfied 1. G is not a C–I graph. 2. Every proper s-subgraph of G is a C–I graph. A natural question arises whether there are infinitely many forbidden s-subgraphs e.g. for all chordal graphs. An affirmative answer to this question was in fact already given in this section. For k odd, the graphs G2k k are minimal forbidden s-subgraphs, they are chordal graphs (even unit-interval graphs) and there are infinitely many of them. 4. Chordal graphs without K4 As G63 is not a C–I graph it may not be an s-subgraph of any C–I graph. This together with Lemma 3 gives us a necessary and sufficient condition for chordal graphs without K4 to be C–I graphs: Theorem 4. Let G be a connected chordal graph without K4 . Then G is a C–I graph if and only if the following two conditions are satisfied: (C1) G does not contain a triple of independent simplicial vertices. (C2) G does not contain G63 as an s-subgraph. Before we prove Theorem 4 let us introduce one more notion. A k-fan is a graph on k + 2 vertices obtained from a path on k + 1 vertices by adding a vertex of degree k + 1 connected to every vertex of the path. It is easy to see that a k fan is a C–I graph for every k ≥ 1 (see e.g. [2] or Fig. 2).
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Fig. 2. A k-fan Bi and its poset Pi .
Fig. 3. Blocks of G and the constructed poset P.
Proof. One implication is easy. According to Lemma 3, condition (C1) is a necessary condition for a graph G to be a C–I graph. Theorem 2 and Corollary 1 imply that condition (C2) is also necessary. Thus every C–I graph satisfies both (C1) and (C2). For the proof of the reverse implication let G be a chordal graph without K4 satisfying both (C1) and (C2). We distinguish two cases. Case 1: G is 2-connected Let us consider a perfect elimination scheme v1 , v2 , . . . , vn of G. This means that for every i = 1, . . . , n the vertex vi is a simplicial vertex in G|{v1 , . . . , vi } (i.e. the subgraph of G induced by {v1 , . . . , vi }). Such a scheme exists for every chordal graph (see e.g. [3]). We prove by induction on i that for every i = 3, . . . , n the graph G|{v1 , . . . , vi } is always isomorphic to an (i − 2)-fan. This statement is clear for i = 3 and i = 4. Now let n > i ≥ 4, let G|{v1 , . . . , vi } be a (i − 2)-fan and let vi+1 be the next vertex in the perfect elimination scheme of G. As G does not contain K4 the degree deg(vi+1 ) < 3 in G|{v1 , . . . , vi+1 }. As G is 2-connected deg(vi+1 ) = 2. Further note that v1 and vi are the only simplicial vertices of the (i − 2)-fan G|{v1 , . . . , vi }. Obviously one of the vertices v1 , vi must be a neighbor of vi+1 otherwise there are 3 independent simplicial vertices v1 , vi , vi+1 in G|{v1 , . . . , vi+1 }. As adding a simplicial vertex cannot decrease the number of independent simplicial vertices there are at least three independent simplicial vertices also in G, a contradiction. Thus vi+1 is connected by an edge either to v1 or to vi . The other neighbor of vi+1 must be the vertex of degree 2, otherwise we get G63 as an s-subgraph of G|{v1 , . . . , vi+1 }, and hence also of G, a contradiction. Thus G|{v1 , . . . , vi+1 } is an (i − 1)-fan and the induction is finished. Hence, for every i = 3, . . . , n the graph G|{v1 , . . . , vi } is isomorphic to an (i − 2)-fan. It follows that G is a C–I graph (see Fig. 2 for an example of a suitable poset P for a k-fan). Case 2: G is not 2-connected We consider the blocks (maximal 2-connected components) of G. Every block is isomorphic to a k-fan for some k. It is a well-known fact (see e.g. [3]) that the blocks of G form a tree-like structure. In our case, according to condition (C1) the tree must be a path. Let B1 , . . . , Bb be the blocks of G or single vertices not contained in any block of G. We number the Bi ’s from 1 to b along this path (see Fig. 3). Note that Bi cannot be a triangle (a 1-fan) for i = 2, . . . , b − 1, otherwise there are 3 independent simplicial vertices in G. For k ≥ 2 and every k-fan Bi let Pi be a poset formed by a chain on k + 1 points and another point comparable only with the maximal and the minimal element of the chain (see Fig. 2). If B1 is a 1-fan let P1 ∼ = (V , <1 ) for V = {a, b, c }, <1 = {a < c , b < c } where c is the only non-simplicial vertex of B1 in G. If Bb is a 1-fan let Pb ∼ = (V ,
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Now we construct a poset P = (V ,
(∃i s.t. x, y ∈ V (Bi ) ∧ x
An example of the graph G and the poset P is depicted in Fig. 3. It is easy to check that G = G(P ).
Acknowledgment This work was partially supported by means of Specific research funds of the Institute of Chemical Technology, Prague. References [1] B. Brešar, M. Changat, T. Gologranc, J. Mathews, A. Mathews, Cover-incomparability graphs and chordal graphs, Discrete Appl. Math. 158 (2010) 1752–1759. [2] B. Brešar, M. Changat, S. Klavžar, M. Kovše, J. Mathews, A. Mathews, Cover-incomparability graphs of posets, Order 25 (2008) 335–347. [3] R. Diestel, Graph Theory, in: Graduate Texts in Mathematics, Springer-Verlag, 2005. [4] R.E. Jamison-Waldner, Convexity and block graphs, Congr. Numer. 33 (1981) 129–142. [5] J. Maxová, P. Pavlíková, D. Turzík, On the complexity of cover-incomparability graphs of posets, Order 26 (2009) 229–236. [6] J. Maxová, D. Turzík, Which distance-hereditary graphs are cover-incomparability graphs? Discrete Appl. Math. 161 (13–14) (2013) 2095–2100.