Metric-Locating-Dominating Sets in Graphs

If u and v are vertices of a graph, then d(u, v) denotes the distance from u to v. Let S = {v1, v2, . . . , vk} be a set of vertices in a connected graph G. For each v ∈ V (G), the k-vector cS(v) is defined by cS(v) = (d(v, v1), d(v, v2), · · · , d(v, vk)). A dominating set S = {v1, v2, . . . , vk} in a connected graph G is a metric-locatingdominating set, or an MLD-set, if the k-vectors cS(v) for v ∈ V (G) are distinct. The metric-location-domination number γM (G) of G is the minimum cardinality of an MLD-set in G. We determine the metric-location-domination number of a tree in terms of its domination number. In particular, we show that γ(T ) = γM (T ) if and only if T contains no vertex that is adjacent to two or more end-vertices. We show that for a tree T the ratio γL(T )/γM (T ) is bounded above by 2, where γL(G) is the locationdomination number defined by Slater (Dominating and reference sets in graphs, J. Math. Phys. Sci. 22 (1988), 445–455). We establish that if G is a connected graph of order n ≥ 2, then γM (T ) = n−1 if and only if G = K1,n−1 or G = Kn. The connected graphs G of order n ≥ 4 for which γM (T ) = n− 2 are characterized in terms of seven families of graphs.