THE SIGNED k-DOMINATION NUMBERS IN GRAPHS

For any integer k >= 1, a signed (total) k-dominating function is a function f : V(G) -> {-1, 1} satisfying E-w is an element of,V([v])f(w) >= k (Sigma(w is an element of N(v)) f(w) >= k) for every v is an element of V(G), where N(v) = {u is an element of V(G)vertical bar uv is an element of E(G)} and N[v] = N(v)boolean OR{v}. The minimum of the values of Sigma(v is an element of V(G)) f(u), taken over all signed (total) k-dominating functions f, is called the signed (total) k-domination number and is denoted by (gamma kS)(G) (gamma(')(kS)(G), resp.) In this paper, several sharp lower bounds of these numbers for general graphs are presented.