Maximum weight induced multicliques and complete multipartite subgraphs in directed path overlap graphs

A graph is a directed path overlap graph if it is the overlap graph of family of directed paths in a rooted directed tree. A graph is a multiclique if its connected components are cliques. A graph is a complete multipartite graph if it is the complement of a multiclique. A graph is a multiclique-multipartite graph if its vertex set has a partition U, W such that G[U] is complete multipartite, G[W] is a multiclique and every two vertices u ∈ U, v ∈ W are adjacent. We describe a polynomial time algorithm to find a maximum weight induced complete multipartite (MWICM) subgraph in a directed path overlap graph. In addition, we describe polynomial time algorithms to find maximum weight induced (restricted) multicliques (MWIM) and multiclique-multipartite (MWIMM) subgraphs in directed path overlap graphs.

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