Emergency Response Sets in Graphs

We introduce a k-response set as a set of vertices where responders can be placed so that given any set of k emergencies, these responders can respond, one per emergency, where each responder covers its own vertex and its neighbors. A weak k-response set does not have to worry about emergencies at the vertices of the set. We define Rk and rk as the minimum cardinality of such sets. We provide bounds on these parameters and discuss connections with domination invariants. For example, for a graph G of order n and minimum degree at least 2, R2(G) ≤ 2n/3, while r2(G) ≤ n/2 provided G is also connected and notK3. We also provide bounds for trees T of order n. We observe that there are for each k trees for which rk(T ) ≤ n/2, but that the minimum Rk(T ) appears to grows with k; a novel computer algorithm is used to show that R3(T ) > n/2. As expected, these parameters are NP-hard to compute, and we provide a linear-time algorithm for trees for fixed k.