A Survey on Alliances and Related Parameters in Graphs

In this paper, we show that several graph parameters are known in different areas under completely different names.More specifically, our observations connect signed domination, monopolies, $\alpha$-domination, $\alpha$-independence,positive influence domination,and a parameter associated to fast information propagationin networks to parameters related to various notions of global $r$-alliances in graphs.We also propose a new framework, called (global) $(D,O)$-alliances, not only in order to characterizevarious known variants of alliance and domination parameters, but also to suggest a unifying framework for the study of alliances and domination.Finally, we also give a survey on the mentioned graph parameters, indicating how results transfer due to our observations

On the approximability and exact algorithms for vector domination and related problems in graphs

We consider two graph optimization problems called vector domination and total vector domination. In vector domination one seeks a small subset S of vertices of a graph such that any vertex outside S has a prescribed number of neighbors in S. In total vector domination, the requirement is extended to all vertices of the graph. We prove that these problems (and several variants thereof) cannot be approximated to within a factor of clnn, where c is a suitable constant and n is the number of the vertices, unless P = NP. We also show that two natural greedy strategies have approximation factors ln D+O(1), where D is the maximum degree of the input graph. We also provide exact polynomial time algorithms for several classes of graphs. Our results extend, improve, and unify several results previously known in the literature.Comment: In the version published in DAM, weaker lower bounds for vector domination and total vector domination were stated. Being these problems generalization of domination and total domination, the lower bounds of 0.2267 ln n and (1-epsilon) ln n clearly hold for both problems, unless P = NP or NP \subseteq DTIME(n^{O(log log n)}), respectively. The claims are corrected in the present versio