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