An application of the Turán theorem to domination in graphs

Abstract

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