A Generalization of Join and an Algorithmic Recognition Problem

We consider a new graph operation c²-join which generalizes join and co-join. We show that odd hole-free graphs (odd antihole-free graphs) are closed under c²-join and describe a polynomial time algorithm to recognize graphs that admit a c²-join. The time complexity of the (a) recognition problem, (b) maximum weight independent set (MWIS) problem, and (c) minimum coloring (MC) problem for odd hole-free graphs are still unknown. Let H be an odd hole-free graph that contains an odd antihole as an induced subgraph and 𝒢 _H be the class of all graphs generated from the induced subgraphs of H by using c²-join recursively. Then 𝒢 _H is odd hole-free, contains all P₄-free graphs, complement of all bipartite graphs, and some imperfect graphs. We show that the MWIS problem, maximum weight clique (MWC) problem, MC problem, and minimum clique cover (MCC) problem can be solved efficiently for 𝒢 _H .