A Generalisation of Isomorphisms with Applications

TL;DR

This paper introduces TF-isomorphisms, a broad generalization of graph isomorphisms, and explores their invariance properties, providing new tools for analyzing graph structures and related problems.

Contribution

It generalizes isomorphisms to TF-isomorphisms, studies their invariants, and connects these to graph properties like incidence double covers and TF-orbitals.

Findings

TF-isomorphisms preserve alternating trails and incidence double covers.

An equivalence relation related to incidence double covers is established.

A new invariant under TF-isomorphisms is introduced, aiding in graph analysis.

Abstract

In this paper, we study the behaviour of TF-isomorphisms, a natural generalisation of isomorphisms. TF-isomorphisms allow us to simplify the approach to seemingly unrelated problems. In particular, we mention the Neighbourhood Reconstruction problem, the Matrix Symmetrization problem and Stability of Graphs. We start with a study of invariance under TF-isomorphisms. In particular, we show that alternating trails and incidence double covers are conserved by TF-isomorphisms, irrespective of whether they are TF-isomorphisms between graphs or digraphs. We then define an equivalence relation and subsequently relate its equivalence classes to the incidence double cover of a graph. By directing the edges of an incidence double cover from one colour class to the other and discarding isolated vertices we obtain an invariant under TF-isomorphisms which gathers a number of invariants. This can be used to study TF-orbitals, an analogous generalisation of the orbitals of a permutation group.