Finding a smallest odd hole in a claw-free graph using global structure

A lemma of Fouquet implies that a claw-free graph contains an induced C5, contains no odd hole, or is quasi-line. In this paper we use this result to give an improved shortest-oddhole algorithm for claw-free graphs by exploiting the structural relationship between line graphs and quasi-line graphs suggested by Chudnovsky and Seymour’s structure theorem for quasi-line graphs. Our approach involves reducing the problem to that of finding a shortest odd cycle of length ≥ 5 in a graph. Our algorithm runs in O(m + n log n) time, improving upon Shrem, Stern, and Golumbic’s recent O(nm) algorithm, which uses a local approach. The best known recognition algorithms for claw-free graphs run in O(m)∩O(n) time, or O(m)∩O(n) without fast matrix multiplication. 1 Background and motivation A hole in a graph is an induced cycle Ck of length k ≥ 4. Odd holes are fundamental to the study of perfect graphs [5]; although there are polynomial-time algorithms that decide whether or not either a graph or its complement contains an odd hole [9, 2], no general algorithm for detecting an odd hole in a graph is known. Odd holes are also fundamental to the study of claw-free graphs, i.e. graphs containing no induced copy of K1,3. Every neighbourhood v in a claw-free graph has stability number α(G[N(v)]) ≤ 2. So if G[N(v)] is perfect then v is bisimplicial (i.e. its neighbours can be partitioned into two cliques, i.e. G[N(v)] is cobipartite), and if G[N(v)] is imperfect then G[N(v)] contains the complement of an odd hole. Fouquet proved something stronger: Lemma 1 (Fouquet [11]). Let G be a connected claw-free graph with α(G) ≥ 3. Then every vertex of G is bisimplicial or contains an induced C5 in its neighbourhood. It follows that a claw-free graphG has α(G) ≤ 2, or contains an induced C5 in the neighbourhood of some vertex, or is quasi-line, meaning every vertex is bisimplicial. Chvátal and Sbihi proved a decomposition theorem for perfect claw-free graphs that yields a polynomial-time recognition algorithm [8]. More recently, Shrem, Stern, and Golumbic gave ∗School of Computer Science, McGill University, Montreal. †Corresponding author: [email protected], IEOR Department, Columbia University, New York. Research supported by an NSERC postdoctoral fellowship. 1 ar X iv :1 10 3. 62 22 v2 [ cs .D M ] 2 4 M ay 2 01 1 an O(nm2) algorithm for finding a shortest odd hole in a claw-free graph based on a variant of breadth-first search in an auxiliary graph [17]. We solve the same problem, but instead of using local structure we use global structure and take advantage of the similarities between claw-free graphs, quasi-line graphs, and line graphs. We prove the following: Theorem 2. There exists an algorithm that, given a claw-free graph G on n vertices and m edges, finds a smallest odd hole in G or determines that none exists in O(m2 + n2 log n) time. Fouquet’s lemma allows us to focus on quasi-line graphs. Their global structure, described by Chudnovsky and Seymour [6], resembles that of line graphs closely enough that we can reduce the shortest odd hole problem on quasi-line graphs to a set of shortest path problems in underlying multigraphs. Our algorithm is not much slower than the fastest known recognition algorithms for claw-free graphs: Alon and Boppana gave an O(n3.5) recognition algorithm [1]. Kloks, Kratsch, and Müller gave an O(m1.69) recognition algorithm that relies on impractical fast matrix multiplication [14]. Their approach takes O(m2) time using naïve matrix multiplication, and more generally O(m(β+1)/2) time using O(nβ) matrix multiplication. 2 The easy cases: Finding a C5 We begin by taking advantage of Fouquet’s lemma in order to reduce the problem to quasi-line graphs. We denote the closed neighbourhood of a vertex v by Ñ(v). Theorem 3. Let G be graph with α(G) ≤ 2. In O(m2) time we can find an induced C5 in G or determine that none exists. Proof. For each edge uv we do the following. First, we construct sets X = N(u) \ Ñ(v), Y = N(v) \ Ñ(u), and Z = V (G) \ (N(u) ∪ N(v)). If u and v are in an induced C5 together then all three must be nonempty. Since α(G) ≤ 2, we know that both X and Y are complete to Z. Second, we search for x ∈ X and y ∈ Y which are nonadjacent – if such x and y exist then this clearly gives us a C5. It is easy to see that we can construct the sets in O(n) time, and that we can search for a non-edge between X and Y in O(m) time, since we can terminate once we find one. Thus it takes O(m2) time to do this for every edge, and if an induced C5 exists in G we will identify it as uvyzx for any z ∈ Z. Kloks, Kratsch, and Müller observed that as a consequence of Turán’s theorem, every vertex in a claw-free graph has at most 2 √ m neighbours [14]. We make repeated use of this fact, starting with a consequence of the previous lemma: Corollary 4. Let G be a claw-free graph with α(G) ≥ 3. Then in O(m2) time we can find an induced W5 in G or determine that G is quasi-line. Proof. By Fouquet’s lemma, any vertex of G is either bisimplicial or contains an induced C5 in its neighbourhood. For any v ∈ V (G), we can easily check whether or not G[N(v)] is cobipartite in O(d(v)2) time. Since G is claw-free, d(v)2 = O(m). Thus in O(nm) time we can determine that G is quasi-line or find a vertex v which is not bisimplicial.