The bondage and reinforcement numbers of γf for some graphs Original Research Article

For any graph G, a real-valued function g: V(G) → [0, 1] is called a dominating function if for every v ϵ V(G), ΣwϵN[v]g(w) ⩾ 1 The fractional domination number is defined to be γf(G) =min{ΣvϵV(G) g(v): g is a dominating function of G}. In this paper, we initiate the study of bondage and reinforcement associated with fractional domination. The bondage number of γf,bf(G), is defined to be the minimum cardinality of a set of edges whose removal from G results in a graph G1 satisfying γf(G1) > γf(G). The reinforcement number of γf,rf(G), is defined to be the minimum cardinality of a set of edges which when added to G results in a graph G″ satisfying γf(G″) < γf(G). We will give exact values of bf(G) and rf(G) for some classes of graphs.