The (theta, wheel)-free graphs Part III: Cliques, stable sets and coloring

Abstract

A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three paths between the same pair of distinct vertices so that the union of any two of the paths induces a hole. A wheel is a graph formed by a hole and a node that has at least 3 neighbors in the hole. In this series of papers we study the class of graphs that do not contain as an induced subgraph a theta nor a wheel. In Part II of the series we prove a decomposition theorem for this class, that uses clique cutsets and 2-joins, and consequently obtain a polynomial time recognition algorithm for the class. In this paper we further use this decomposition theorem to obtain polynomial time algorithms for maximum weight clique, maximum weight stable set and coloring problems. We also show that for a graph G in the class, if its maximum clique size is ω, then its chromatic number is bounded by max{ω, 3}, and that the class is 3-clique-colorable.

The (theta, wheel)-free graphs Part IV: Induced paths and cycles

“…Lemma 2.6 (Lemma 2.10 in [21]) Let G be a graph from D. Let (X 1 , X 2 ) be a 2-join of G, and G 1 , G 2 the blocks of decomposition with respect to this 2-join whose marker paths are of length at least 2. Then G 1 and G 2 are in D and they do not have star cutsets.…”

The (theta, wheel)-free graphs Part II: Structure theorem

“…This will be discussed in more details in Section 4. Lemma 2.6 (Lemma 2.10 in [21]) Let G be a graph from D. Let (X 1 , X 2 ) be a 2-join of G, and G 1 , G 2 the blocks of decomposition with respect to this 2-join whose marker paths are of length at least 2. Then G 1 and G 2 are in D and they do not have star cutsets.…”

“…In [25] it is shown that one can decompose a graph with no star cutset using a sequence of 'non-crossing' 2-joins into graphs with no star cutset and no 2-join (which will in our case be basic). In this paper we will use minimally-sided 2-joins (as opposed to [25] and [21], where minimally-sided 2-joins were 'moved' to allow marker paths to be disjoint). This will be particularly important when solving the induced paths problem.…”

“…In Part II of the series [20], we prove a decomposition theorem for graphs in C that uses clique cutsets and 2-joins, and use it to obtain a polynomial time recognition algorithm for the class. In Part III of the series [21] we use the decomposition theorem from [20] to obtain further properties of graphs in the class and to construct polynomial time algorithms for maximum weight clique, maximum weight stable set, and coloring problems. In this part we use the decomposition theorem from [20] to obtain polynomial time algorithms for several problems related to finding induced paths and cycles.…”

Triangle-free graphs with large chromatic number and no induced wheel

“…It remains open whether or not there is a polynomial χ-bounding function for graphs with no induced theta. The class of graphs with no induced theta or wheel has χ-bounding function max{ω, 3} [10] (where ω is the size of the largest complete subgraph).…”