A polynomial graph extension procedure for improving graph isomorphism algorithms

TL;DR

This paper introduces a polynomial graph extension method that enhances graph isomorphism algorithms by propagating constraints through weighted edges, enabling more effective identification of incompatible vertex mappings.

Contribution

The novel polynomial extension procedure encodes isomorphism constraints into extended graphs, improving the detection of incompatible vertex pairs in graph isomorphism testing.

Findings

The extension propagates isomorphism constraints via weighted edges.

The method produces a forbidding matrix that filters incompatible mappings.

Tests show the matrix often leaves only one compatible candidate per vertex.

Abstract

We present in this short note a polynomial graph extension procedure that can be used to improve any graph isomorphism algorithm. This construction propagates new constraints from the isomorphism constraints of the input graphs (denoted by $G(V,E)$ and $G'(V',E')$). Thus, information from the edge structures of $G$ and $G'$ is "hashed" into the weighted edges of the extended graphs. A bijective mapping is an isomorphism of the initial graphs if and only if it is an isomorphism of the extended graphs. As such, the construction enables the identification of pair of vertices $i\in V$ and $i'\in V'$ that can not be mapped by any isomorphism $h^*:V \to V'$ (e.g. if the extended edges of $i$ and $i'$ are different). A forbidding matrix $F$, that encodes all pairs of incompatible mappings $(i,i')$, is constructed in order to be used by a different algorithm. Moreover, tests on numerous graph classes show that the matrix $F$ might leave only one compatible element for each $i \in V$.