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

Abstract

A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three internally vertex-disjoint paths of length at least 2 between the same pair of distinct vertices. 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. In this paper we use this decomposition theorem to solve several problems related to finding induced paths and cycles in our class.

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

“…Theorem 2.4 ( [11]) If G is (theta, wheel)-free, then G is a line graph of a triangle-free chordless graph or a P-graph, or G has a clique cutset or a 2-join.…”

“…In [11] the blocks of decomposition w.r.t. a 2-join that we used in construction of a recognition algorithm had marker paths of length 2.…”

“…This project is motivated and explained in more detail in Part I of the series [3], where two subclasses of (theta, wheel)-free graphs are studied. In Part II of the series [11], we prove a decomposition theorem for (theta, wheel)-free graphs that uses clique cutsets and 2-joins, and use it to obtain a polynomial time recognition algorithm for the class. In this part we use the decomposition theorem from [11] to obtain further properties of the graphs in the class and to construct polynomial time algorithms for maximum weight clique, maximum weight stable set, and coloring problems.…”

“…In Part II of the series [11], we prove a decomposition theorem for (theta, wheel)-free graphs that uses clique cutsets and 2-joins, and use it to obtain a polynomial time recognition algorithm for the class. In this part we use the decomposition theorem from [11] to obtain further properties of the graphs in the class and to construct polynomial time algorithms for maximum weight clique, maximum weight stable set, and coloring problems. In Part IV of the series [12] we show that the induced version of the k-linkage problem can be solved in polynomial time for (theta, wheel)-free graphs.…”

“…In this part we use the decomposition theorem from [11] to obtain further properties of the graphs in the class and to construct polynomial time algorithms for maximum weight clique, maximum weight stable set, and coloring problems. In Part IV of the series [12] we show that the induced version of the k-linkage problem can be solved in polynomial time for (theta, wheel)-free graphs.…”