A set S of vertices in a graph G=(V,E) is a dominating set if every vertex in V\S is adjacent to at least one vertex in S, and the minimum cardinality of a dominating set of G is the domination number of G. A vertex v in a dominating set S is said to be very cost effective if it is adjacent to more vertices in V\S than to vertices in S. A dominating set S is very cost effective if every vertex in S is very cost effective. The minimum cardinality of a very cost effective dominating set of G is the very cost effective domination number of G. We first give necessary conditions for a graph to have equal domination and very cost effective domination numbers. Then we determine an upper bound on the very cost effective domination number for trees in terms of their domination number, and characterize the trees which attain this bound. lastly, we show that no such bound exists for graphs in general, even when restricted to bipartite graphs.

A set S of vertices in a graph G = (V,E) is a dominating set if every vertex in V \ S is adjacent to at least one vertex in S. A vertex v in a dominating set S is said to be it cost effective if it is adjacent to at least as many vertices in V \ S as it is in S. A dominating set S is cost effective if every vertex in S is cost effective. The minimum cardinality of a cost effective dominating set of G is the cost effective domination number of G. In addition to some preliminary results for general graphs, we give lower and upper bounds on the cost effective domination number of trees in terms of their domination number and characterize the trees that achieve the upper bound. We show that every value of the cost effective domination number between these bounds is realizable.

For any integer k≥0, a set of vertices S of a graph G=(V,E) is k-cost-effective if for every v∈S,|N(v)∩(V∖S)|≥|N(v)∩S|+k. In this paper we study the minimum cardinality of a maximal k-cost-effective set and the maximum cardinality of a k-cost-effective set. We obtain Gallai-type results involving the k-cost-effective and global k-offensive alliance parameters, and we provide bounds on the maximum k-cost-effective number. Finally, we consider k-cost-effective sets that are also dominating. We show that computing the k-cost-effective domination number is NP-complete for bipartite graphs. Moreover, we note that not all trees have a k-cost-effective dominating set and give a constructive characterization of those that do.