Stable sets and graphs with no even holes

Abstract

We develop decomposition/composition tools for efficiently solving maximum weight stable sets problems as well as for describing them as polynomially sized linear programs (using "compact systems"). Some of these are well-known but need some extra work to yield polynomial "decomposition schemes".We apply the tools to graphs with no even hole and no cap. A hole is a chordless cycle of length greater than three and a cap is a hole together with an additional node that is adjacent to two adjacent nodes of the hole and that has no other neighbors on the hole.

χ‐bounds, operations, and chords

“…We say that the type of is , according to the types and of the two holes with respect to the amalgam. Based on Lemma 7.1, and the property that these two holes share only one edges, there are 13 possible types for : 11,14,16,18,22,25,27,29,46,48,59,57,89. If belongs to type 89 then any vertex of 2 (this vertex is exist since 2 ≠ ∅) with hole type 8 induces a cycle with a long unichord in in 1 , a contradiction.…”

“…This is the case because is not basic, so it is not chordal and contains a chordless cycle of length at least 4 (that provides at least two nonadjacent pairs). ■ The next lemma is implicitly proved in [11] (as Corollary 2.16), but the machinery there is much heavier and relies on many definitions, so we prefer to give our own simple proof. Note that in [12], it is claimed without proof that any graph with an amalgam should have an amalgam such that at least one block of decomposition has no amalgam.…”

“…Note that in [12], it is claimed without proof that any graph with an amalgam should have an amalgam such that at least one block of decomposition has no amalgam. Such a result would imply the existence of a decomposition tree of linear size, but unfortunately, in [11], a counterexample to the claim is provided. Lemma 7.4.…”

The world of hereditary graph classes viewed through Truemper configurations

“…On the other hand, it is easy to see how to use Theorem 8.1 to obtain a polynomial time algorithm to solve the maximum weight clique problem for cap-free graphs, see [51].…”

“…Conforti and Gerards [51] show how to obtain a polynomial time algorithm for solving maximum weight independent set problem, on any class of graphs that is decomposable by amalgams into basic graphs for which one can solve the maximum weight independent set problem in polynomial time. In particular, using Theorem 8.1, they obtain a polynomial time algorithm for solving the maximum weight independent set problem for (cap, odd-hole)-free graphs (i.e.…”