Clique separator decomposition of hole-free and diamond-free graphs and algorithmic consequences

Clique separator decomposition, introduced by Whitesides and Tarjan, is one of the most important graph decompositions. A hole is a chordless cycle with at least five vertices. A paraglider is a graph with five vertices a,b,c,d,e and edges ab,ac,bc,bd,cd,ae,de. We show that every (hole, paraglider)-free graph admits a clique separator decomposition into graphs of three very specific types. This yields efficient algorithms for various optimization problems in this class of graphs.