The difference between the domination number and the minus domination number of a cubic graph

Abstract

The closed neighborhood of a vertex subset S of a graph G = ( V, E), denoted as N[ S], is defined as the union of S and the set of all the vertices adjacent to some vertex of S. A dominating set of a graph G = ( V, E) is defined as a set S of vertices such that N[ S] = V. The domination number of a graph G, denoted as γ( G), is the minimum possible size of a dominating set of G. A minus dominating function on a graph G = ( V, E) is a function g : V → {−1, 0, 1} such that g( N[ v]) ≥ 1 for all vertices. The weight of a minus dominating function g is defined as g( V) = Σ vϵV g( v) . The minus domination number of a graph G, denoted as γ −( G), is the minimum possible weight of a minus dominating function on G. It is well known that γ −( G) ≤ γ( G). This paper is focused on the difference between γ( G) and γ −( G) for cubic graphs. We first present a graph-theoretic description of γ −( G). Based on this, we give a necessary and sufficient condition for γ( G) − γ −( G) ≥ k. Further, we present an infinite family of cubic graphs of order 18 k + 16 and with γ( G) − γ −( G) ≥ k