Rough sets in graphs using similarity relations

In this paper, we investigate the theory of rough set to study graphs using the concept of orbits. Rough sets are based on a clustering criterion and we use the idea of similarity of vertices under automorphism as a criterion. We introduce indiscernibility relation in terms of orbits and prove necessary and sufficient conditions under which the indiscernibility partitions remain the same when associated with different attribute sets. We show that automorphisms of the graph $\mathcal{G}$ preserve the indiscernibility partitions. Further, we prove that for any graph $\mathcal{G}$ with $k$ orbits, any reduct $\mathcal{R}$ consists of one element from $k-1$ orbits of the graph. We also study the rough membership functions for paths, cycles, complete and complete bipartite graphs. Moreover, we introduce essential sets and discernibility matrices induced by orbits of graphs and study their relationship. We also prove that every essential set consists of union of any two orbits of the graph.