Efficient sets in graphs

Abstract

The efficiency of a set S of vertices in an undirected graph G=( V, E) is defined to be ε( S)=|{ v| vϵV— S and v is adjacent to exactly one vertex in S}|, i.e., the number of vertices in V—S that are dominated by exactly one vertex in S. The efficiency of a graph G=( V, E) equals the maximum efficiency of any subset S of vertices of V. A linear time algorithm is presented for computing the efficiency of an arbitrary tree and an NP-completeness proof is given for the problem of deciding if an arbitrary planar bipartite graph has a set S such that ε( S)≥ k, for some positive integer k.