On Isomorphism of Graphs and the k-clique Problem

Abstract

In this paper we develop three characterizations for isomorphism of graphs. The first characterization is obtained by associating certain bitableaux with the graphs. We order these bitableaux by suitably defined lexicographic order and denote the bitableau that is least in this order as the standard representation for the associated graph. The standard representation characterizes graphs uniquely. The second characterization is obtained in terms of associated rooted, unordered, pseudo trees. We show that the isomorphism of two given graphs is implied by the isomorphism of their associated pseudo trees. The third characterization is obtained in terms of ordered adjacency lists to be associated with two given labeled graphs. We show the two given labeled graphs are isomorphic if and only if their associated ordered adjacency lists can be made identical by the action of suitable transpositions on any one of these lists. We discuss in brief the complexity of these characterizations described in this paper for deciding isomorphism of graphs. Finally, we discuss the k-clique problem in the light of these characterizations towards the end of the paper.