An application of the Turán theorem to domination in graphs

A function f:V(G)→{+1,−1}f:V(G)→{+1,−1} defined on the vertices of a graph GG is a signed dominating function   if for any vertex vv the sum of function values over its closed neighborhood is at least 1. The signed domination number  γs(G)γs(G) of GG is the minimum weight of a signed dominating function on GG. By simply changing “{+1,−1}{+1,−1}” in the above definition to “{+1,0,−1}{+1,0,−1}”, we can define the minus dominating function and the minus domination number   of GG. In this note, by applying the Turán theorem, we present sharp lower bounds on the signed domination number for a graph containing no (k+1)(k+1)-cliques. As a result, we generalize a previous result due to Kang et al. on the minus domination number of kk-partite graphs to graphs containing no (k+1)(k+1)-cliques and characterize the extremal graphs.