Stability of certified domination upon edge addition

Abstract

A dominating set [Formula: see text] of a graph [Formula: see text] is said to be certified, if every vertex [Formula: see text] has either 0 neighbors or at least two neighbors in [Formula: see text]. The cardinality of a minimum certified dominating set of [Formula: see text] is called the certified domination number of [Formula: see text] and is denoted by [Formula: see text]. A graph [Formula: see text] is said to be [Formula: see text]-[Formula: see text] stable if for any two vertices [Formula: see text] such that [Formula: see text], we have [Formula: see text]. In this paper, we provide upper bounds for the [Formula: see text]-value of [Formula: see text]-[Formula: see text] stable graph. We also characterize the trees and unicyclic graphs that are [Formula: see text]-[Formula: see text] stable.