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

  • Authors:

  • Ton Kloks;Haiko Müller;Kristina Vušković

  • Affiliations:

  • School of Computing, University of Leeds, Leeds LS2 9JT, UK;School of Computing, University of Leeds, Leeds LS2 9JT, UK;School of Computing, University of Leeds, Leeds LS2 9JT, UK

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2009

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Abstract

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 (even-hole, diamond)-free. For this class of graphs we first obtain a decomposition theorem, using clique cutsets, bisimplicial cutsets (which is a special type of a star cutset) and 2-joins. This decomposition theorem is then used to prove that every graph that is (even-hole, diamond)-free contains a simplicial extreme (i.e., a vertex that is either of degree 2 or whose neighborhood induces a clique). This characterization implies that for every (even-hole, diamond)-free graph G, @g(G)=