ON THE COMPUTATIONAL COMPLEXITY OF OPTIMIZATION CONVEX COVERING PROBLEMS OF GRAPHS

Abstract

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 ≤ | S | ≤ | X | − 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.