Clique Separator Decomposition of Hole- and Diamond-Free Graphs and Algorithmic Consequences

Clique separator decomposition introduced by Tarjan and Whitesides is one of the most important graph decompositions. A graph is an atom if it has no clique separator. A hole is a chordless cycle with at least five vertices, and an antihole is the complement graph of a hole. A graph is weakly chordal if it is holeand antihole-free. K4−e is also called diamond. Paraglider has five vertices four of which induce a diamond, and the fifth vertex sees exactly the two vertices of degree two in the diamond. In this paper we show that atoms of holeand diamond-free graphs (of holeand paraglider-free graphs, respectively) are either weakly chordal or of a very specific structure. Holeand paraglider-free graphs are perfect graphs. The structure of their atoms leads to efficient algorithms for various problems.