On graph equivalences preserved under extensions

TL;DR

This paper investigates how certain graph equivalence relations, especially those based on properties like connectivity, colorability, Hamiltonicity, and planarity, are affected when extended through unions with arbitrary graphs.

Contribution

It introduces the concept of strengthening of graph equivalence relations and analyzes their behavior for various graph properties.

Findings

Strengthening relations preserve certain properties under graph unions.

Results characterize when properties like k-connectivity, k-colorability, Hamiltonicity, and planarity are maintained.

Provides a framework for understanding graph property invariance under extensions.

Abstract

Let R be an equivalence relation on graphs. By the strengthening of R we mean the relation R' such that graphs G and H are in the relation R' if for every graph F, the union of the graphs G and F is in the relation R with the union of the graphs H and F. We study strengthenings of equivalence relations on graphs. The most important case that we consider concerns equivalence relations defined by graph properties. We obtain results on the strengthening of equivalence relations determined by the properties such as being a k-connected graph, k-colorable, hamiltonian and planar.