A set of vertices \(S\subseteq V\) in a graph \(G=(V,E)\) is called an internal minority set if for every vertex \(v\in S\) , a minority of the neighbors of v are in S, or equivalently, every vertex \(v\in S\) has strictly more neighbors in \(V-S\) than it has in S. As we will show, minority sets in graphs are closely related to, but different than, a variety of sets that have been studied, such as defensive and offensive alliances, cost effective and very cost effective sets, unfriendly partitions in graphs, and independent and dominating sets in graphs. Sets similar to minority sets can also be defined by specifying that similar conditions apply to every vertex \(w\in V-S\) , giving rise to external minority sets, and to all vertices \(u\in V\) , giving rise to total minority sets in graphs. In this paper we introduce the study of these types of sets. Various properties and results are obtained, a corollary of which is a new lower bound for the chromatic number of a graph. Moreover, the complexity issues of two minority related problems are addressed.
