Decomposition of even-hole-free graphs with star cutsets and 2-joins

Even-hole-free graphs part I: Decomposition theorem

We prove a decomposition theorem for even-hole-free graphs. The decompositions used are 2-joins and star, double-star and triple-star cutsets. This theorem is used in the second part of this paper to obtain a polytime recognition algorithm for even-hole-free graphs.

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

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 paper we obtain a decomposition theorem for the class of graphs that do not contain an induced s...

Decomposition of odd-hole-free graphs by double star cutsets and 2-joins

Even-hole-free graphs that do not contain diamonds: A structure theorem and its consequences

In this paper we consider the class of simple graphs defined by excluding, as induced subgraphs, even holes (i.e., chordless cycles of even length) and diamonds (i.e., a graph obtained from a clique of size 4 by removing an edge). We say that such graphs are (evenhole, diamond)-free. For this class of graphs we first obtain a decomposition theorem, using clique cutsets, bisimplicial cutsets (wh...

Combinatorial optimization with 2-joins

A 2-join is an edge cutset that naturally appears in decomposition of several classes of graphs closed under taking induced subgraphs, such as perfect graphs and claw-free graphs. In this paper we construct combinatorial polynomial time algorithms for finding a maximum weighted clique, a maximum weighted stable set and an optimal coloring for a class of perfect graphs decomposable by 2-joins: t...