A set S ⊆ V of vertices in a graph G = (V,E) is called a dominating set if every vertex in V − S is adjacent to at least one vertex in S. Domination in graphs is a well-studied branch of graph theory, and is the subject of two books by Haynes, Hedetniemi and Slater [8, 9]. However, about 90% of the papers on domination have considered only undirected graphs. Thus, relatively little is known abo...

This paper establishes a necessary and sufficient condition for a CO-irredundant set of vertices of a graph to be maximal and shows that the smallest cardinality of a maximal CO-irredundant set in an n vertex graph with maximum degree is bounded below by n/2 for = 2, 4n/13 for = 3 and 2n/(3 − 3) for 4. This result is best possible and extremal graphs are characterised for 3. © 2005 Elsevier B.V...

A dominating set of a graph G=(V(G),E(G)) is vertex D such that every in V(G)∖D adjacent to D. The cardinality smallest G called the domination number and denoted by γ(G). k-isolating if G−NG[D] contains no k-cliques. minimum k-isolation ιk(G). Clearly, γ(G)=ι1(G). I irredundant if, for non-isolated v G[I], there exists u V(G)∖I NG(u)∩I={v}. An maximal I∪{u} longer any u∈V(G)∖I. irredundance ir...