Independent strong weak domination: A mathematical programming approach

Abstract

Let [Formula: see text] be a graph. A subset [Formula: see text] of vertices is a dominating set if every vertex in [Formula: see text] is adjacent to at least one vertex of [Formula: see text]. The domination number is the minimum cardinality of a dominating set. Let [Formula: see text]. Then, [Formula: see text] strongly dominates [Formula: see text] and [Formula: see text] weakly dominates [Formula: see text] if (i) [Formula: see text] and (ii) [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is a strong (weak) dominating set of [Formula: see text] if every vertex in [Formula: see text] is strongly (weakly) dominated by at least one vertex in [Formula: see text]. The strong (weak) domination number of [Formula: see text] is the minimum cardinality of a strong (weak) dominating set. A set [Formula: see text] is an independent (or stable) set if no two vertices of [Formula: see text] are adjacent. The independent domination number of [Formula: see text] (independent strong domination number, independent weak domination number, respectively) is the minimum size of an independent dominating set (independent strong dominating set, independent weak dominating set, respectively) of [Formula: see text]. In this paper, mathematical models are developed for the problems of independent domination and independent strong (weak) domination of a graph. Then test problems are solved by the GAMS software, the optima and execution times are implemented. To the best of our knowledge, this is the first mathematical programming formulations for the problems, and computational results show that the proposed models are capable of finding optimal solutions within a reasonable amount of time.