Nearly perfect sets in graphs

In a graph G = (V; E), a set of vertices S is nearly perfect if every vertex in V ? S is adjacent to at most one vertex in S. Nearly perfect sets are closely related to 2-packings of graphs, strongly stable sets, dominating sets and eecient dominating sets. We say a nearly perfect set S is 1-minimal if for every vertex u in S, the set S ? fug is not nearly perfect. Similarly, a nearly perfect set S is 1-maximal if for every vertex u in V ? S, S fug is not a nearly perfect set. Lastly, we deene n p (G) to be the minimum cardinality of a 1-maximal nearly perfect set, and N p (G) to be the maximum cardinality of a 1-minimal nearly perfect set. In this paper we calculate these parameters for some classes of graphs. We show that the decision problem for n p (G) is NP-complete; we give a linear algorithm for determining n p (T) for any tree T ; and we show that N p (G) can be calculated for any graph G in polynomial time.