The Signed Domination Numbers Of Graphs

Let G=(V(G),E(G)) be an undirected graph. For any v€V(G), assign a value f(v) to v, where f(u) ∈{-1,+1}. Let N(v) be the neighbourhood of u and let N[v]= N(u)∪{v}. Denote If f[u]≥1 for any u∈V(G), then f is a signed dominating function. Denote The signed domination number is defined as γs(G)-min{f(V(G))|f is a signed dominating function of G}. The signed dominating function f with f(V(G))=γs(G) is called the minimum signed dominating function of G. Let Wn be a graph obtained from a cycle of length n by adding an isolated vertex v0 satisfying that all vertices of the cycle are adjacent to u0.Wn is called a wheel and u0 is called the central vertex of Wn. This paper mainly focus on the signed domination numbers for graphs. The paper consists of the three chapters.The first chapter introduces the basic concepts on graphs used in this paper.The second chapter gives the lower bound on the signed domination number of a graph. The result is as follows.For a graph G of order n, let Δ and δ be the maximum degree and the minimum degree of G, respectively. ThenThe third chapter gives the signed domination number for two special classes of graphs. The result is as follows.(1) Let y(n,m)={G|G consists of n copies of Wm which have exactly one common noncentral vertex}. Then(2) Let H(n,m)={G|G consists of n copies of Wm which have exactly one common edge}. Then γs(H(n,m))=(m-1-2[m/3])n+2.