On the Complexity of Finding a Potential Community

摘要

An independent 2-clique of a graph is a subset of vertices that is an independent set and such that any two vertices inside have a common neighbor outside. In this paper, we study the complexity of finding an independent 2-clique of maximum size in several graph classes and we compare its complexity with the complexity of maximum independent set. We prove that this problem is NP-hard on apex graphs, APX-hard on line graphs, not n~(1/2-є)-approximable on bipartite graphs and not n~(1-є)-approximable on split graphs, while it is polynomial-time solvable on graphs of bounded degree and their complements, graphs of bounded treewidth, planar graphs, (C_3, C_6)-free graphs, threshold graphs, interval graphs and cographs.