Constant Time Calculation of the Metric Dimension of the Join of Path Graphs

The distance between two vertices of a simple connected graph G, denoted as d(u,v), is the length of the shortest path from u to v and is always symmetrical. An ordered subset W=w1,w2,w3,⋯,wk of V(G) is a resolving set for G, if for ∀u,v∈V(G), there exists wi∈W ∋ d(u,wi)≠d(v,wi). A resolving set with minimal cardinality is called the metric basis. The metric dimension of G is the cardinality of metric basis of G and is denoted as dim(G). For the graph G1=(V1,E1,) and G2=(V2,E2), their join is denoted by G1+G2. The vertex set of G1+G2 is V1∪V2 and the edge set is E=E1∪E2∪uv,u∈V1,v∈V2. In this article, we show that the metric dimension of the join of two path graphs is unbounded because of its dependence on the size of the paths. We also provide a general formula to determine this metric dimension. We also develop algorithms to obtain metric dimensions and a metric basis for the join of path graphs, with respect to its symmetries.