An application of the Turn theorem to domination in graphs ?
A function f:V(G)?{+1,?1} f : V ( G ) ? { + 1 , ? 1 } defined on the vertices of a graph G G is a signed dominating function  if for any vertex v v the sum of function values over its closed neighborhood is at least 1. The signed domination number ?s(G) ? s ( G ) of G G is the minimum weight of a signed dominating function on G G . 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 G G . In this note, by applying the Turn 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 etal. on the minus domination number of k k -partite graphs to graphs containing no (k+1) ( k + 1 ) -cliques and characterize the extremal graphs.
turn theorem signed domination minus domination clique