New Min-Max Theorems for Weakly Chordal and Dually Chordal Graphs

A distance-k matching in a graph G is matching M in which the distance between any two edges of M is at least k. A distance-2 matching is more commonly referred to as an induced matching. In this paper, we show that when G is weakly chordal, the size of the largest induced matching in G is equal to the minimum number of co-chordal subgraphs of G needed to cover the edges of G, and that the co-chordal subgraphs of a minimum cover can be found in polynomial time. Using similar techniques, we show that the distance-k matching problem for k > 1 is tractable for weakly chordal graphs when k is even, and is NP-hard when k is odd. For dually chordal graphs, we use properties of hypergraphs to show that the distance-k matching problem is solvable in polynomial time whenever k is odd, and NP-hard when k is even. Motivated by our use of hypergraphs, we define a class of hypergraphs which lies strictly in between the well studied classes of acyclic hypergraphs and normal hypergraphs. 1 Background and Motivation In this paper, all graphs are undirected, simple, and finite. That is, a graph G = (V,E) where V is a finite set whose elements are called vertices together with a set E of unordered pairs of vertices. We say H = (V ′, E′) is a subgraph of G = (V,E) if V ′ ⊆ V and E′ ⊆ E, and we say that H is an induced subgraph if E′ = {uv ∈ E | {u, v} ⊆ V ′}. We use Pk to denote an induced path on k vertices and Ck is an induced cycle on k vertices. A graph is chordal if it does not contain any Ck, k ≥ 4. A graph is co-chordal if its complement is chordal. A graph G is weakly chordal if neither G nor G contains any Ck, k ≥ 5. For background on these and other graph classes referenced below, we refer the interested reader to [6]. An induced matching in a graph is a matching that is also an induced subgraph, i.e., no two edges of the matching are joined by an edge in the graph. Acknowledges support from the University of Dayton Research Institute. Acknowledges support from The National Security Agency, USA. W. Wu and O. Daescu (Eds.): COCOA 2010, Part II, LNCS 6509, pp. 207–218, 2010. c © Springer-Verlag Berlin Heidelberg 2010 208 A.H. Busch, F.F. Dragan, and R. Sritharan The size of an induced matching is the number of edges in it. Let im(G) denote the size of a largest induced matching in G. Given G and positive integer k, the problem of deciding whether im(G) ≥ k is NP-complete [8] even when G is bipartite. For vertices x and y of G, let distG(x, y) be the number of edges on a shortest path between x and y in G. For edges ei and ej of G let distG(ei, ej) = min{distG(x, y) | x ∈ ei and y ∈ ej}. For M ⊆ E(G), M is a distance-k matching for a positive integer k ≥ 1 if for every ei, ej ∈ M with i = j, distG(ei, ej) ≥ k. For k = 1, this gives the usual notion of matching in graphs. For k = 2, this gives the notion of induced matching. The distance-k matching problem is to find, for a given graph G and an integer k ≥ 1, a distance-k matching with the largest possible number of edges. A bipartite graph G = (X,Y,E) is a chain graph if it does not have a 2K2 as an induced subgraph. Bipartite graph G′ = (X ′, Y ′, E′) is a chain subgraph of bipartite graph G = (X,Y,E), if G′ is a subgraph of G and G′ contains no 2K2. For a bipartite graph G = (X,Y,E), let ch(G) denote the fewest number of chain subgraphs of G the union of whose edge-sets is E. A set of ch(G) chain subgraphs of bipartite graph G = (V,E) whose edge-sets cover E is a minimum chain subgraph cover for G. Yannakakis showed [23] that when k ≥ 3, deciding whether ch(G) ≤ k for a given bipartite graph G is NP-complete. An efficient algorithm to determine whether ch(G) ≤ 2 for a given bipartite graph G is known [16]. It is clear that for any bipartite graph G, im(G) ≤ ch(G). Families of bipartite graphs where equality holds have been considered in literature. For example, it was shown in [24] that when G is a convex bipartite graph, im(G) = ch(G). A bipartite graph is chordal bipartite if it does not contain any induced cycles on 6 or more vertices. It is known that every convex bipartite graph is also chordal bipartite. The following more general result was recently shown: Proposition 1. [1] For a chordal bipartite graph G, im(G) = ch(G). Let us move away from the setting of bipartite graphs and consider graphs in general. We say H is a co-chordal subgraph of G if H is a subgraph of G and also H is co-chordal. Let coc(G) be the minimum number of co-chordal subgraphs of G needed to cover all the edges of G. As a chain subgraph of a bipartite graph G is a co-chordal subgraph of G and vice versa, the parameter coc(G) when restricted to a bipartite graph G is essentially the same as ch(G). Again, it is clear from the definitions that for any graph G, im(G) ≤ coc(G). In Section 2, we show that when G is weakly chordal, im(G) = coc(G) and that the co-chordal subgraphs of a minimum cover can be found in polynomial time. As every chordal bipartite graph is weakly chordal and as a chain subgraph of a bipartite graph is a co-chordal subgraph and vice versa, our result generalizes Proposition 1. In Section 3, we use similar techniques to show that the distancek matching problem for k > 1 is tractable for weakly chordal graphs when k is even, and NP-hard when k is odd. Next, in Section 4 we use techniques from the study of hypergraphs to show that the opposite holds for the class of dually New Min-Max Theorems for Weakly Chordal and Dually Chordal Graphs 209 chordal graphs; the distance-k matching problem can be solved in polynomial time for dually chordal graphs if a k is odd, and is NP-hard for all even k. Motivated by our results and by the use of hypergraphs in Section 4, we define a class of hypergraphs in Section 5 which lies strictly in between the well studied classes of acyclic hypergraphs and normal hypergraphs. 2 A Min-Max Theorem for Weakly Chordal Graphs For a graph G, let G∗ denote the square of the line graph of G. More explicitly, vertices of G∗ are edges of G. Edges ei, ej of G are nonadjacent in G∗ if and only if they form a 2K2 in G. It is clear from the construction of G∗ that the set of edges of a co-chordal subgraph of G maps to a clique of G∗. Further, im(G) = α(G∗), where α(G∗) is the size of a largest independent set in G∗. The following is known: Proposition 2. [9] If G is weakly chordal, then G∗ is weakly chordal. Also, it is well known that every weakly chordal graph is perfect [13]. Therefore, when G is weakly chordal, im(G) = α(G∗) = θ(G∗), where θ(G∗) is the minimum clique cover number of G∗. Thus, when G is weakly chordal θ(G∗) ≤ coc(G). We will show that when G is weakly chordal, coc(G) ≤ θ(G∗) also holds and therefore we have the following: Proposition 3. If G is weakly chordal, then coc(G) = im(G). The proof of Proposition 3 utilizes the following edge elimination scheme for weakly chordal graphs. Edge xy is a co-pair of graph G, if vertices x and y are not the endpoints of any Pk, k ≥ 4, in G. Proposition 4. [19] Suppose e is a co-pair of graph G. Then, G is weakly chordal if and only if G− e is weakly chordal. The following is implied by Corollary 2 in [12]: Proposition 5. [12] Suppose G is a weakly chordal graph that contains a 2K2. Then, G contains co-pairs e and f such that e and f form a 2K2 in G. Lemma 6. If e is a co-pair of a weakly chordal graph G, then G∗−e = (G−e)∗. Proof. Deleting an edge xy from G will never destroy a 2K2, unless it is one of the edges of the 2K2. If deleting xy creates a new 2K2 then xy must be the middle edge of a P4 in G, or equivalently, x and y are the end vertices of a P4 in G. Thus, when e is a co-pair, two edges form a 2K2 in G − e if and only if they form a 2K2 in G that does not include the edge e. Since the vertices of (G∗ − e) and (G − e)∗ both consist of the edges of G − e, this guarantees that the edge sets of (G∗ − e) and (G − e)∗, are identical as well. Hence the graphs are identical. 210 A.H. Busch, F.F. Dragan, and R. Sritharan In order to establish that when G is weakly chordal, coc(G) ≤ θ(G∗), first observe that every member of a clique cover of G∗ can be assumed to be a maximal clique of G∗. We have the following: Theorem 7. Let G be weakly chordal. Then, every maximal clique of G∗ is the edge-set of a maximal co-chordal subgraph of G. Proof. Proof is by induction on the number of edges in the graph. Clearly, the statement is true when G has no edges. Assume the statement is true for all weakly chordal graphs with up to k − 1 edges, and let G be a weakly chordal graph with k edges. If G contains no 2K2, then G is co-chordal and G∗ is a clique and the theorem holds. Now, suppose G contains a 2K2. Then, from Proposition 5, G contains a 2K2 e1, e2 each of which is a co-pair of G. Let M be a maximal clique of G∗. As no maximal clique of G∗ contains both e1 and e2, we can choose i ∈ {1, 2} such that ei / ∈ M . As a result,M is a maximal clique of G∗ − ei which equals (G − ei)∗ by Lemma 6. Also, by Proposition 4, G− ei is weakly chordal. It then follows by the induction hypothesis that M is the edge set of a maximal co-chordal subgraph of G− ei. Clearly, this subgraph remains co-chordal in G, so it remains to show that this subgraph is, in fact, maximal. If this is not the case, then there exists a co-chordal subgraph M ′ of G such that M ⊂ M ′. As every co-chordal subgraph of G maps to a clique of G∗, it follows that M and M ′ are cliques of G∗ such that M ⊂ M ′; this contradicts M being a maximal clique of G∗. Thus, θ(G∗) = coc(G), establishing Proposition 3. As an efficient algorithm exists [14] to compute a minimum clique cover of a weakly chordal graph, we have the following: Corollary 8. When G is weakly chordal, coc(G) and a minimum cover of G by co-chordal subgraphs of G can be found in polynomial time. We recently learned of a surprising application of this result: the parameters coc(G) and im(G) yield upper and lower bounds, respectively, on the CastelnuovoMumford regularity of the edge ideal of G [22]. Thus, when G is weakly chordal, this parameter can be computed efficiently. Another application of Corollary 8 utilizes the complement of G. As the complement of a weakly chordal graph remains weakly chordal, after taking the complement of each graph in a cover by co-chordal subgraphs, we have a set of chordal graphs whose edge-intersection is the edge-set of a weakly chordal graph. The study of a variety of similar parameters, known as the intersection dimension of a graph G with respect to a graph class A, was introduced in [15]. The problem when A is the set of chordal graphs was termed the chordality of G in [17]. We use dimCH(G) to denote the chordality or chordal dimension of a graph G. In this context, we have another corollary of Theorem 7. Corollary 9. When G is weakly chordal, dimCH(G) = im(G) and a minimum set of chordal graphs whose edge-intersection give the edge-set of G can be found in polynomial time. New Min-Max Theorems for Weakly Chordal and Dually Chordal Graphs 211 A chordal graph that does not contain a 2K2 is a split graph, and it has been shown in [8] that a split graph cover of a chordal graph can be computed in polynomial time. Proposition 10. [8] Let G be a chordal graph. Then, a minimum cover of edges of G by split subgraphs of G can be found in polynomial time. The proof of Proposition 10 in [8] utilizes the clique tree of a chordal graph G and the Helly property. An alternate proof can be given by showing that the edges referred to in Proposition 5 can be chosen so that each edge is incident with a simplicial vertex of G. Since no such edge is the only chord of a cycle, this guarantees that G − e will be chordal whenever G is chordal. As every chordal graph is also weakly chordal, Lemma 6 and a slightly modified version of Theorem 7 then imply Proposition 10. 3 Distance-k Matchings in Weakly Chordal Graphs In this section, we observe that the correspondence between maximal cliques of G∗ and maximal co-chordal subgraphs of G can be adapted to find maximum a distance-k matching in a weakly chordal graph G for any positive even integer k. We then show that finding a largest distance-k matching when k is odd and k ≥ 3 is NP-hard. We begin by noting the fundamental connection between distance-k matchings in a graph G and independent sets in the kth power of the line graph of G, which we denote L(G). Proposition 11. [7] For k ≥ 1 and graph G, the edge set M is a distance-k matching in G if and only if M is an independent vertex set in L(G). As a result, identifying a largest distance-k matching in a graph G is no more difficult than constructing the kth power of the line graph of G and finding a maximum independent set in L(G). Clearly, for any edge e, the set of edges within distance k of e can be computed in linear time, and as a result a polynomial time algorithm exists for the distance-k matching problem whenever an efficient algorithm exists for finding a largest independent set in L(G). Proposition 2 guarantees that such an efficient algorithm exists for induced matchings, as efficient algorithms to compute the largest independent sets of weakly chordal graphs are well known. For k > 1, the existence of a polynomial algorithm for computing distance-2k matchings in weakly chordal graphs is guaranteed by combining Proposition 2 with the following result. Proposition 12. [5] Let G be a graph and k ≥ 1 be a fixed integer. If G is weakly chordal, then so is G. For distance-k matchings when k is odd, we note that the case k = 3 was recently shown to be NP-complete for the class of chordal graphs, which is properly contained in the class of weakly chordal graphs. 212 A.H. Busch, F.F. Dragan, and R. Sritharan Proposition 13. [7] The largest distance-3 matching problem is NP-hard for chordal graphs. We will extend this result to distance-(2k+1)matchings for every positive integer k. This extension is done by showing that the distance-(2k+1) matching problem can be transformed into the distance-(2k + 3) matching problem in polynomial time, for any positive integer k. Theorem 14. For any positive integer k, there exists a polynomial time transformation from the distance-(2k + 1) matching problem to the distance-(2k + 3) matching problem. Proof. Let G = (V,E) be a graph, and let k be a positive integer. We will define the graph G = (V , E) from G as follows. For each edge e = uv of G, we introduce two new vertices xe and ye and add edges to make the subgraph induced by {u, v, xe, ye} a clique. Formally, V + = V ∪ {xe, ye | e ∈ E}, and