On the Metric Dimension of Joins of Two Graphs

For an ordered set { } k w w w W , , , 2 1  = of vertices and a vertex v in a connected graph G, the (metric) representation of v with respect to W is the k-vector r(v/W) = (d(v, w1), d(v, w2), ..., d(v, wk)), where d(x, y)representsthe distance between the vertices x and y. A resolving set of minimum cardinality is called a minimum resolving set or abasis and the cardinality of a basis for G is its dimension dimG. For the graph G1 = (V1, E1) and G2 = (V2, E2) their joinis denoted by G1 +G2 is the graph whose vertex set is V1∪V2 and the edge set is { } V V E E v u uv E 2 1 2 1 , : ∈ ∈ =   . In this paper, we determine the metric dimension of join of paths, paths and cycles, path and stars, complete graphs,complete graphs and paths. Keywords—Join, Metric dimension, Metric basis, Resolving set. ————————————————————