The Fundamental Morphism Theorem in the Categories of Graphs & Graph Reconstruction

TL;DR

This paper extends the Fundamental Morphism Theorem to categories of graphs with loops and multiple edges, and uses it to reformulate the graph reconstruction conjectures in categorical terms.

Contribution

It establishes the Fundamental Morphism Theorem in two graph categories and reformulates the reconstruction conjectures as existence problems for specific graph homomorphisms.

Findings

Fundamental Morphism Theorem holds in these graph categories.

Reconstruction conjectures are equivalent to certain homomorphism existence.

Provides a new categorical perspective on graph reconstruction.

Abstract

The Fundamental Morphism Theorem is a categorical version of the First Noether Isomorphism Theorem for categories that do not have kernels or cokernels. We consider two categories of graphs. Both categories will admit graphs with multiple edges and loops, and are distinguished by allowing two different types of homomorphisms, standard graph homomorphisms and more general graph homomorphisms where the contraction of an edge is allowed. We establish the Fundamental Morphism Theorem in these two categories of graphs. We then use the result to provide an equivalent reformulation of the vertex and edge reconstruction conjectures. This reformulation shows that reconstructability is equivalent to the existence of a graph homomorphism satisfying an equation.