On the Computational Complexity of Optimization Convex Covering Problems of Graphs

摘要

In this paper we present further studies of convex covers and convex partitions of graphs. Let G be a finite simple graph. A set of vertices S of G is convex if all vertices lying on a shortest path between any pair of vertices of S are in S. If 3 = vertical bar S vertical bar = vertical bar X vertical bar - 1, then S is a nontrivial set. We prove that determining the minimum number of convex sets and the minimum number of nontrivial convex sets, which cover or partition a graph, is in general NP-hard. We also prove that it is NP-hard to determine the maximum number of nontrivial convex sets, which cover or partition a graph.