International Journal of Mathematics And Computer Research

Abstract

A set D is a subset of V (G) is called dominating (or total dominating) set in G, if D ∩ N[v] ≠ ɸ (or D ∩ N(v) ≠ ɸ , respectively) for every vertex v ϵ V(G). The minimum number of vertices of a dominating set (or of a total dominating set) in G is called the domination number γ(G) (or the total domination number γt(G), respectively) of G. If v is a vertex of a graph G, then N(v) is its open neighbourhood, (ie) the set of all vertices adjacent to v in G. A mapping f : V(G) → { -2,1 }, where V(G) is the vertex set of G, is called a Bold Signed Total Dominating Function (BSTDF) on G, if w(f) = ΣxϵN(v) f(x) ≥ 1 for each v ϵ V(G). minf{ΣxϵV(G) f(x): f is a BSTDF } is called the bold signed total domination number of G and is denoted by γ bst(G). The bold signed total domination number of a graph is a certain variant of the domination number. The lower bounds of γbst(G) are found for the case of regular graphs, and γbst(G) are found for complete graphs, circuits and complete bipatite graphs. The independent proofs are seen.