| Summary: | For any integer , a set of vertices of a graph is -cost-effective if for every . In this paper we study the minimum cardinality of a maximal -cost-effective set and the maximum cardinality of a -cost-effective set. We obtain Gallai-type results involving the -cost-effective and global -offensive alliance parameters, and we provide bounds on the maximum -cost-effective number. Finally, we consider -cost-effective sets that are also dominating. We show that computing the -cost-effective domination number is NP-complete for bipartite graphs. Moreover, we note that not all trees have a -cost-effective dominating set and give a constructive characterization of those that do.
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