Solving Graph Isomorphism Problem for a Special case

TL;DR

This paper introduces a polynomial-time algorithm for solving the graph isomorphism problem for a specific class of graphs with unique neighborhood degree properties, improving efficiency over existing methods.

Contribution

It proposes a novel graph representation and an $O(n^4)$ time algorithm tailored for a special class of graphs with unique neighborhood degree lists.

Findings

Algorithm runs in $O(n^4)$ time.

Faster than quasi-polynomial algorithms for the studied graph class.

Successfully determines isomorphism for the special graph type.

Abstract

Graph isomorphism is an important computer science problem. The problem for the general case is unknown to be in polynomial time. The base algorithm for the general case works in quasi-polynomial time. The solutions in polynomial time for some special type of classes are known. In this work, we have worked with a special type of graphs. We have proposed a method to represent these graphs and finding isomorphism between these graphs. The method uses a modified version of the degree list of a graph and neighbourhood degree list. These special type of graphs have a property that neighbourhood degree list of any two immediate neighbours is different for every vertex.The representation becomes invariant to the order in which the node was selected for giving the representation making the isomorphism problem trivial for this case. The algorithm works in $O(n^4)$ time, where n is the number of vertices present in the graph. The proposed algorithm runs faster than quasi-polynomial time for the graphs used in the study.