A Polynomial Time Graph Isomorphism Algorithm For Graphs That Are Not Locally Triangle-Free

TL;DR

This paper presents a polynomial time algorithm for graph isomorphism applicable to graphs that are not locally triangle-free, leveraging neighborhood sub-graph permutations and automorphisms, with potential implications for group isomorphism.

Contribution

It introduces a polynomial time algorithm for a specific class of graphs, expanding the scope of efficient graph isomorphism testing.

Findings

Algorithm works in polynomial time for non-locally triangle-free graphs

Method constructs automorphisms based on neighborhood permutations

Potential to improve general graph isomorphism algorithms

Abstract

In this paper, we show the existence of a polynomial time graph isomorphism algorithm for all graphs excluding graphs that are locally trianglefree. This particular class of graphs allows to divide the graph into neighbourhood sub-graph where each of induced sub-graph (neighbourhood) has at least 2 vertices. We construct all possible permutations for each induced sub-graph using a search tree. We construct automorphisms of subgraphs based on these permutations. Finally, we decide isomorphism through automorphisms . The author expects that the solution, present in this paper, may lead to a faster algorithm for the general case of graph isomorphism (using " barycentric subdivision" ). The paper might affect group isomorphism also as we may construct graphs (corresponds to a particular group) in way so we can avoid it to be a triangle free graph. Since,for a given group G , each choice of a generating set will give a different Cayley graph.