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

Abstract

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 [Formula: see text], [Formula: see text] such that [Formula: see text] is complete multipartite, [Formula: see text] is a multiclique and every two vertices [Formula: see text], [Formula: see text] 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.