Valency-based domination number

Let [Formula: see text] be a connected graph. For two distinct vertices [Formula: see text] and [Formula: see text] in [Formula: see text], the effect of [Formula: see text] on [Formula: see text] is defined as [Formula: see text] when [Formula: see text] and [Formula: see text] otherwise, where [Formula: see text] is the degree of [Formula: see text] in [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is said to be a valency-based dominating set of [Formula: see text], if for any vertex [Formula: see text] not in [Formula: see text], [Formula: see text], where [Formula: see text] is the minimum degree of [Formula: see text]. The vb-domination number [Formula: see text] is the minimum cardinality of a valency-based dominating set of [Formula: see text]. In this paper, we introduced [Formula: see text] and initiated its study and established some bounds.