Minimum monopoly in regular and tree graphs

In this paper we consider a graph optimization problem called minimum monopoly problem, in which it is required to find a minimum cardinality set S ⊆ V , such that, for each u ∈ V , |N [u] ∩ S| |N [u]|/2 in a given graph G = (V ,E). We show that this optimization problem does not have a polynomial-time approximation scheme for k-regular graphs (k 5), unless P=NP. We show this by establishing two L-reductions (an approximation preserving reduction) from minimum dominating set problem for k-regular graphs to minimum monopoly problem for 2k-regular graphs and to minimum monopoly problem for (2k − 1)-regular graphs, where k 3. We also show that, for tree graphs, a minimum monopoly set can be computed in linear time. © 2006 Elsevier B.V. All rights reserved.