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arXiv · 1901.09859

Graphs with a unique maximum open packing

Boštjan Brešar·Kirsti Kuenzel·Douglas F. Rall
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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.

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Boštjan Brešar, Kirsti Kuenzel, Douglas F. Rall. 2019-01-28. Graphs with a unique maximum open packing. https://arxiv.org/abs/1901.09859

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