New bounds on the signed domination numbers of graphs

A signed dominating function of a graph G with vertex set V is a function f : V -> such that for every vertex v in V the sum of the values of f at v and at every vertex a adjacent to v is at least -I. The weight of f is the sum of the values of f at every vertex of V. The signed domination number of G is the minimum weight of a signed dominating function of G. In t his paper, we study the signed domination numbers of graphs and present new sharp lower and upper bounds for this parameter. As an example, we prove that the signed domination number of a tree of order with ( leaves and s support vertices is at least (n + 1 + 2(l - s))/3.