Movable dominating sensor sets in networks

In this paper we consider 1-movable dominating sets, motivated by the use of sensors employed to detect certain events in networks, where the sensors have a limited ability to react under changing conditions in the network. A 1-movable dominating set is a dominating set S ⊆ V(G) such that for every v ∈ S, either S - {v} is a dominating set, or there exists a vertex u ∈ (V(G) - S) ∩ N(v) such that (S - {v}) ∪ {u} is a dominating set. We present computational complexity results and bounds on the size of 1-movable dominating sets in arbitrary graphs. We also give a polynomial time algorithm to find minimum 1-movable dominating sets for trees. We conclude by extending this idea to k-movable dominating sets.