Geodesic
One-Three Join: A Graph Operation and Its Consequences
Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 3, pp. 633-647

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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.
Keywords: one-three join, bipartite-join, homogeneous set, odd hole-free graphs 
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Shalu, M.A.; Devi Yamini, S. One-Three Join: A Graph Operation and Its Consequences. Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 3, pp. 633-647. http://geodesic.mathdoc.fr/item/DMGT_2017_37_3_a10/