The Metric-Location-Domination Number of k-Paths

Abstract

Let G be a connected graph with vertex set V (G) and edge set E(G). For ordered subset W = {w 1, w 2 …, wk } of V (G) and v ∈ V (G), the representation of v with respect to W is the k-vector r(v|W ) = (d(v, w 1), d(v, w 2), …, d(v, wk )), where d(x, y) is the distance between vertices x and y. The set W is called a resolving set of G if all vertices of G have distinct representations with respect to W . A subset S ⊆ V (G) is called a dominating set of G if every vertex of G\\S is adjacent to some vertex of S. The cardinality of a minimum dominating set is the domination number γ(G) of G. An ordered W ⊆ V (G) is called a metric-locating-dominating set of G if W is a dominating as well as resolving set. The metric-location-domination number of G, denoted by γM (G), is the cardinality of a minimum dominating resolving set. A k-path G is the graph with vertex set V (G) = {v 1, v 2, …, vn } and edge set E(G) = {vivj : |j −i| ≤ k}. In this paper, we determine the metric-location-domination number of k-path, and in particular, we show that If n > k(2k + 1) then γM (G) = γ(G).