On the Metric-Location-Domination Number of Some Exponential Graphs

Abstract

Abstract-If u and v are vertices in a simple and undirected graph. A distance d(u; v) is the shortest path between u and v in G. For an ordered set= fw₁; w₂; :::; w_kg of vertices and a vertex v 2 G, the representation of v with respect to W is the ordered k-tuple r(vjW ) = fd(v; w₁); d(v; w₂); :::; d(v; w_k)g. A dominating set W in a connected graph G is a metric-locating-dominating set (MLD), if r(vjW ) for v 2 V (G) are distinct. The metric-location-domination number of G denotes _M (G) is the minimum cardinality of an MLD-set in G. The purpose of this paper is to find the metric-location-domination number of some exponential graphs. By an exponential graph, we mean a graph formed by combining two graphs G and H, where each edge of graph G is replaced by the graph H, denote by G^H . We have determined the metric-location-domination number of some exponential graph, namely P_n^Btm and P_n^Wm. 2010 Mathematics Subject Classification: 05C78 Index Terms: metric-locating-dominating, metric-location-domination number, exponential graph.