On the structure and clique-width of ( 4 K 1 , C 4 , C 6 , C 7 )-free graphs

We give a complete structural description of ( 4 K 1 , C 4 , C 6 , C 7 )-free graphs that do not contain a simplicial vertex, and we prove that such graphs have bounded clique-width. Together with the results of Foley et al., this implies that ( 4 K 1 , C 4 , C 6 )-free graphs that do not contain a simplicial vertex have bounded clique-width. Consequently, Graph Coloring can be solved in polynomial time for ( 4 K 1 , C 4 , C 6 )-free graphs, that is, for even-hole-free graphs of stability number at most three.