Uniformly Pair-Bonded Trees

Abstract

Let \(G = (V(G), E(G))\) be a graph with \(\delta(G) \geq 1\). A set \(D \subseteq V(G)\) is a paired-dominating set if \(D\) is a dominating set and the induced subgraph \(G[D]\) contains a perfect matching. The paired domination number of \(G\), denoted by \(\gamma_p(G)\), is the minimum cardinality of a paired-dominating set of \(G\). The paired bondage number, denoted by \(b_p(G)\), is the minimum cardinality among all sets of edges \(E’ \subseteq E\) such that \(\delta(G – E’) \geq 1\) and \(\gamma_p(G – E’) > \gamma_p(G)\). For any \(b_p(G)\) edges \(E’ \subseteq E\) with \(\delta(G – E’) \geq 1\), if \(\gamma_p(G – E’) > \gamma_p(G)\), then \(G\) is called uniformly pair-bonded graph. In this paper, we prove that there exists uniformly pair-bonded tree \(T\) with \(b_p(T) = k\) for any positive integer \(k\). Furthermore, we give a constructive characterization of uniformly pair-bonded trees.