The structure of (theta, pyramid, 1‐wheel, 3‐wheel)‐free graphs

Abstract

In this paper, we study the class of graphs scriptC defined by excluding the following structures as induced subgraphs: theta, pyramid, 1‐wheel, and 3‐wheel. We describe the structure of graphs in scriptC, and we give a polynomial‐time recognition algorithm for this class. We also prove that K 4‐free graphs in scriptC are 4‐colorable. We remark that scriptC includes the class of chordal graphs, as well as the class of line graphs of triangle‐free graphs.

Three-in-a-Tree in Near Linear Time

“…Chudnovsky and Seymour are interested in thetas because they trivially imply an even hole. They developed the previously only known polynomial-time algorithm, running in O(n 11 ) time, for detecting thetas in G via solving the three-in-a-tree problem on O(n 7 ) subgraphs of G. Thus, Theorem 1.1 reduces the time to Õ(n 9 ). Moreover, we show in Lemma 6.1 that thetas in G can be detected via solving the three-in-a-tree problem on O(mn 2 ) n-vertex graphs, leading to an Õ(n 6 )-time algorithm as stated in Theorem 1.2.…”

“…See [1,4,11,14,20,21,22,23,35,44,45,47,49,51,55,70] for more related work on graph detection, recognition, and characterization. Also see [12, Appendix A] for a survey of the recognition complexity of more than 160 graph classes.…”

“…Chang et al's algorithm consists of two O(n11 )-time phases. The first phase detects beetles in O(n11 ) time, which is now reduced to O(n 7 ) time by Theorem 1.5.…”

“…Chang et al's algorithm consists of two O(n11 )-time phases. The first phase detects beetles in O(n11 ) time, which is now reduced to O(n 7 ) time by Theorem 1.5. The second phase maintains a set T of induced subgraphs of G with the property that if G is even-hole-free, then so is each graph in T until either T becomes empty or an H ∈ T is found to contain even holes.…”