Weak and Strong Reinforcement Number For a Graph

Abstract

Let G=(V(G),E(G)) be a graph.A set of vertices S in a graph G is called to be a Smarandachely dominating k-set, if each vertex of G is dominated by at least k vertices of S. Particularly, if k = 1, such a set is called a dominating set of G. The Smarandachely domination k-number k(G) of G is the minimum cardinality of a Smarandachely dominating k-set of G. S is called weak domination set if deg(u) ≤ deg(v) for every pair of (u,v) ∈ V (G) −S. The minimum cardinality of a weak domination set S is called weak domination number and denoted by w(G) . In this paper we introduce the weak reinforcement number which is the minimum number of added edges to reduce the weak dominating number. We give some boundary of this new parameter and trees. Furthermore, some boundary of strong reinforcement number has been given for a given graph G and its complemented graph G.