Secondary domination in graphs

Given a dominating set S ⊆ V in a graph G =( V,E), place one guard at each vertex in S. Should there be a problem at a vertex v ∈ V − S, we can send a guard at a vertex u ∈ S adjacent to v to handle the problem. If for some reason this guard needs assistance, as econd guard can be sent fromS to v, but the question is: how long will it take for a second guard to arrive? This is the issue of what we call secondary domination. We will focus primarily on dominating sets in which a second guard can arrive in at most two time steps. A (1,2) -dominating set in a graph G =( V,E )i s a setS having the property that for every vertex v ∈ V − S there is at least one vertex in S at distance 1 from v and a second vertex in S at distance at most 2 from v. We present a variety of results about secondary domination, relating this to several other well-studied types of domination.