A special graph for the connected metric dimension of graphs

Given a connected graph G = (V, E), let d(x, y) represent the separation between x and y at its vertices. If each vertex in a collection B is uniquely identified by its vector of distances to the vertices in B, then that set of vertices resolves a graph G. A metric dimension of G is represented by dim(G) and is the smallest cardinality of a resolving set of G. If the subgraph (Formula Presented) induced by B is a nontrivial connected subgraph of G, then a resolving set B of G is connected. The metric dimension of G is the cardinality of the minimal resolving set, while the connected metric dimension of G is the cardinality of the smallest connected resolving set. The connected metric dimension of the knots graph, whitehead link graph and jewel graph are determined in this study. Finally, we derive the explicit formulas for the triangular book graph, quadrilateral book graph and crystal planar map.