One-Three Join: A Graph Operation and Its Consequences

• Main Authors: Shalu M.A., Devi Yamini S.
• Format: Article
• Language: English
• Published: Sciendo 2017-08-01
• Series: Discussiones Mathematicae Graph Theory
• Subjects: one-three join; bipartite-join; homogeneous set; odd hole-free graphs; 05c75; 05c76
• Online Access: https://doi.org/10.7151/dmgt.1948

In this paper, we introduce a graph operation, namely one-three join. We show that the graph G admits a one-three join if and only if either G is one of the basic graphs (bipartite, complement of bipartite, split graph) or G admits a constrained homogeneous set or a bipartite-join or a join. Next, we define ℳH as the class of all graphs generated from the induced subgraphs of an odd hole-free graph H that contains an odd anti-hole as an induced subgraph by using one-three join and co-join recursively and show that the maximum independent set problem, the maximum clique problem, the minimum coloring problem, and the minimum clique cover problem can be solved efficiently for ℳH.