Structure and algorithms for (cap, even hole)-free graphs

Abstract

A graph is even-hole-free if it has no induced even cycles of length 4 or more. A cap is a cycle of length at least 5 with exactly one chord and that chord creates a triangle with the cycle. In this paper, we consider (cap, even hole)-free graphs, and more generally, (cap, 4-hole)-free odd-signable graphs. We give an explicit construction of these graphs. We prove that every such graph G has a vertex of degree at most 3 2 ω(G) − 1, and hencewhere ω(G) denotes the size of a largest clique in G and χ(G) denotes the chromatic number of G. We give an O(nm) algorithm for q-coloring these graphs for fixed q and an O(nm) algorithm for maximum weight stable set. We also give a polynomial-time algorithm for minimum coloring.Our algorithms are based on our results that triangle-free odd-signable graphs have treewidth at most 5 and thus have clique-width at most 48, and that (cap, 4-hole)-free odd-signable graphs G without clique cutsets have treewidth at most 6ω(G) − 1 and clique-width at most 48.

Clique‐cutsets beyond chordal graphs

“…Thus, can be partitioned into true twin classes of in a unique way, and clearly, every true twin class of is a clique of . An exercise from states that, given an input graph , all true twin classes of can be found in time; a detailed proof of this result can be found in . Given a graph and a partition of into true twin classes of , we define the graph (called the quotient graph of with respect to ) to be the graph whose vertex set is , and in which distinct are adjacent if and only if and are complete to each other in .…”

“…Proof We test for (b) from Lemma . We first check in time whether is (, cap)‐free (to test whether is ‐free, we simply examine all five‐tuples of vertices of , and to check whether is cap‐free, we use the time algorithm from ). If is not (, cap)‐free, then the algorithm returns the answer that and stops.…”

“…Note that α K ( ) = 3 2,3 , and that holes of length at least 6 have stability number at least 3. Thus, graphs of stability number at most 2 contain no K 2,3 and no holes of length at least 6; consequently, C C ( , ) 5 6 -free graphs of stability number at most 2 are in fact (long hole, K C , 2,3 6 )-free. Let U be the class of all graphs G that satisfy one of the following:…”

“…Let G be a 3PC. Then G contains three induced paths, say P x y P x y = ,…, , = ,…, 1 1 1 2 2 2 , and P x y = ,…, 3 3…”

“…Suppose that the first statement is false, and fix ∈ y X 4 4 such that y 4 is nonadjacent to at least one of y 1 and y 3 ; by symmetry, we may assume that y 4 is nonadjacent to y 3 , and consequently (since x 3 is complete to X 4 , and x 4 is complete to X 3 ), we have that ≠ y x 3 3 and ≠ y x 4 4 . By the choice of y 3 , it follows that ∉ xx E G ( ) 3 .…”