The Homomorphism Domination Exponent

TL;DR

This paper introduces the homomorphism domination exponent, a measure comparing homomorphism counts between graphs, and explores its properties, bounds, and computability, especially for specific graph classes like chordal and series-parallel graphs.

Contribution

It defines the homomorphism domination exponent, analyzes its properties, and provides a linear program for computing it in certain graph classes, advancing understanding in graph homomorphism theory.

Findings

Established bounds for the homomorphism domination exponent.

Identified classes of graphs where HDE is computable.

Presented a linear program for calculating HDE in special cases.

Abstract

We initiate a study of the homomorphism domination exponent of a pair of graphs F and G, defined as the maximum real number c such that |Hom(F,T)| \geq |Hom(G,T)|^c for every graph T. The problem of determining whether HDE(F,G) \geq 1 is known as the homomorphism domination problem and its decidability is an important open question arising in the theory of relational databases. We investigate the combinatorial and computational properties of the homomorphism domination exponent, proving upper and lower bounds and isolating classes of graphs F and G for which HDE(F,G) is computable. In particular, we present a linear program computing HDE(F,G) in the special case where F is chordal and G is series-parallel.