The main aim of work is presenting application on a j-pre-topology generated by the shortest path problem and j-dependency of a vertices of a subgraph Hi in step i on graph G with respect to j-pre-topology; j = 1, 2. Furthermore, we discuss the restriction and separators in pre-topology generated by the shortest path problem. Finally, we study the rate of liaison in pre-topology between two subgraphs. It is formally shown that the new distance measure is a metric.