Affiliations: Indian Institute of Information Technology Design & Manufacturing (IIITD&M) Kancheepuram, Chennai, India. [email protected]; [email protected]
Correspondence:
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Address for correspondence: Indian Institute of Information Technology Design & Manufacturing (IIITD&M) Kancheepuram, Chennai - 600 127, India.
Abstract: 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 𝒢H be the class of all graphs generated from the induced subgraphs of H by using c2-join recursively. Then 𝒢H 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 𝒢H.
Keywords: c2-join, odd hole-free graphs, modular decomposition, maximum weight independent set problem, maximum weight clique problem, minimum coloring problem, minimum clique cover problem
