Pushing the Boundaries of Combinatorial Graph Isomorphism Algorithms

A computational problem well studied in the field of Complexity Theory is Graph Isomorphism. Color Refinement is a classical technique for testing the non-existence of isomorphism between two given graphs. Tinhofer's algorithm is an extension of the Color Refinement algorithm and the same technique does not succeed for all graphs. We characterize Tinhofer graphs algebraically. We propose and study in detail, a new graph hierarchy based on Tinhofer's algorithm which also prove for all the new graph classes in the hierarchy a sufficient condition. We look at the relations of the new graph classes with each other and also with the already known classes. We also facilitate an effective graph isomorphism algorithm for the lower classes in the hierarchy. Getting an algebraic characterization for the new hierarchy classes, understanding the exact location of other graphs in this hierarchy and extending the efficient algorithm for higher graph classes are identified as potential areas of future interest.