A set $Wsubset V (G)$ is called a resolving set, if for every two distinct vertices $u, v in V (G)$ there exists $win W$ such that $d(u,w) not = d(v,w)$, 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.