Relations Between Graphs

TL;DR

This paper explores conditions for relations between graphs that generate one graph's edges from another, generalizing homomorphisms and introducing concepts like R-cores and R-cocores, which are unique and computable efficiently.

Contribution

It introduces a generalized framework for graph relations, extending homomorphism concepts, and establishes the uniqueness and polynomial-time computability of R-cores and R-cocores.

Findings

R-cores and R-cocores are unique up to isomorphism

They can be computed in polynomial time

The framework generalizes existing graph homomorphism concepts

Abstract

Given two graphs G and H, we ask under which conditions there is a relation R that generates the edges of H given the structure of graph G. This construction can be seen as a form of multihomomorphism. It generalizes surjective homomorphisms of graphs and naturally leads to notions of R-retractions, R-cores, and R-cocores of graphs. Both R-cores and R-cocores of graphs are unique up to isomorphism and can be computed in polynomial time.