Solis Graphs and Uniquely Metric Basis Graphs

A set 𝑊 ⊂ 𝑉 (𝐺) is called a resolving set, if for every two distinct vertices u; 𝑣 ∈ 𝑉 (𝐺) there exists w ∈ W such that d(𝑢, 𝑤) 6= 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 nding the metric dimension of a Solis graph.