Graphs with a unique maximum open packing

Abstract

A set S of vertices in a graph is an open packing if (open) neighborhoods of any two distinct vertices in S are disjoint. In this paper, we consider the graphs that have a unique maximum open packing. We characterize the trees with this property by using four local operations such that any nontrivial tree with a unique maximum open packing can be obtained by a sequence of these operations starting from P 2 . We also prove that the decision version of the open packing number is NP-complete even when restricted to graphs of girth at least 6. Finally, we show that the recognition of the graphs with a unique maximum open packing is polynomially equivalent to the recognition of the graphs with a unique maximum independent set, and we prove that the complexity of both problems is not polynomial, unless P=NP.