ABSTRACT
Let denote the domination number of a graph . A set is called a bondage edge set of if . The bondage number of is the cardinality of a minimum bondage edge set of . A set is called a k-packing of graph G if for every pair of distinct vertices . A vertex v of G is called critical if . In this paper, we prove that for any nontrivial tree T, if and only if the set composed of all the critical vertices of T is a maximum 2-packing of T. Moreover, as the main work of this paper, we obtain several results of some sharp upper bounds of the bondage number of the strong product of a nonempty graph G and a nontrivial tree T under different conditions.
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Disclosure statement
No potential conflict of interest was reported by the authors.
ORCID
Heping Zhang http://orcid.org/0000-0001-5385-6687