Fractional hypergraph isomorphism and fractional invariants

TL;DR

This paper extends fractional graph isomorphism to hypergraphs, establishing invariance of several fractional hypergraph parameters and classifying their behavior under this equivalence.

Contribution

It introduces fractional hypergraph isomorphism, providing an alternative characterization and analyzing which fractional invariants are preserved under this new concept.

Findings

Fractional packing, covering, matching, and transversal numbers are invariant.

Fractional matching, vertex and edge cover, independence, and domination numbers are invariant.

Fractional chromatic, clique, and clique cover numbers are not invariant.

Abstract

Fractional graph isomorphism is the linear relaxation of an integer programming formulation of graph isomorphism. It preserves some invariants of graphs, like degree sequences and equitable partitions, but it does not preserve others like connectivity, clique and independence numbers, chromatic number, vertex and edge cover numbers, matching number, domination and total domination numbers. In this work, we extend the concept of fractional graph isomorphism to hypergraphs, and give an alternative characterization, analogous to one of those that are known for graphs. With this new concept we prove that the fractional packing, covering, matching and transversal numbers on hypergraphs are invariant under fractional hypergraph isomorphism. As a consequence, fractional matching, vertex and edge cover, independence, domination and total domination numbers are invariant under fractional graph isomorphism. This is not the case of fractional chromatic, clique, and clique cover numbers. In this way, most of the classical fractional parameters are classified with respect to their invariance under fractional graph isomorphism.