A dominating set in a graph G is a set of vertices D such that every vertex of G is either in D or is adjacent some vertex of D. The domination number y(G) of G is the minimum cardinality of any dominating set. A graph is vertex domination-critical if the removal of any vertex decreases its domination number. This paper gives examples and properties of vertex domination-critical graphs, presents a method of constructing them, and poses some open questions. In the process several results for arbitrary graphs are presented.

No abstract

No abstract

No abstract

a b s t r a c tFor a graph G = (V, E), a non-empty set S ⊆ V is a defensive alliance if for every vertex v in S, v has at most one more neighbor in V − S than it has in S, and S is an offensive alliance if for every v ∈ V − S that has a neighbor in S, v has more neighbors in S than in V − S. A powerful alliance is both defensive and offensive. We initiate the study of powerful alliances in graphs.