Solis Graphs and Uniquely Metric Basis Graphs

A set W subset of V(G) is called a resolving set, if for every two distinct vertices u, v is an element of V(C) there exists w is an element of W such that d(u, w) not equal d(v, w), where d(x, y) is the distance between the vertices x and y. A resolving set for G with minimum cardinality is called a metric basis. A graph with a unique metric basis is called a uniquely dimensional graph. In this paper, we establish a family of graph called Solis graph, and we prove that if G is a minimal edge unique base graph with the base of size two, then G belongs to the Solis graphs family. Finally, an algorithm is given for finding the metric dimension of a Solis graph.