On distance-s locating and distance-t dominating sets in graphs

Let s and t be positive integers, and let G be a simple and connected graph with vertex set V. For x,y is an element of V, let d(x,y) denote the length of an x - y geodesic in G and let ds(x,y) =min{d(x,y),s + 1}. Let W subset of V. A set W is a distance-s locating set of G if, for any distinct x,y is an element of V, there exists a vertex z is an element of W such that ds(x,z)not equal ds(y,z), and the distance-s location number, dims(G), of G is the minimum cardinality among all distance-s locating sets of G. A set W is a distance-t dominating set of G if, for each vertex u is an element of V, there exists a vertex v is an element of W such that d(u,v) <= t, and the distance-t domination number, ?t(G), of G is the minimum cardinality among all distance-t dominating sets of G. The (s,t)-location-domination number of G, denoted by ?L(s,t)(G), is the minimum cardinality among all sets W subset of V such that W is both a distance-s locating set and a distance-t dominating set of G. For any connected graph G of order at least two, we show that 1 <= max{dims(G),?t(G)}<= ?L(s,t)(G) <= min{dim a(G) + 1,|V |- 1}, where a =min{s,t}. We characterize connected graphs G satisfying ?L(s,t)(G) equals 1 and |V |- 1, respectively. We examine the relationship among dims(G), ?t(G) and ?L(s,t)(G); along the way, we show that ?L(s,t)(G) =dim s(G) if s < t. We also show that there exist graphs H and G with H subset of G such that ?L(s,t)(H)/ ?L(s,t)(G) can be arbitrarily large. Moreover, we examine some graph classes.