Adjoint functors and tree duality

TL;DR

This paper proves that if a digraph has tree duality, then its arc graph also has tree duality, and extends this result to right adjoint functors on relational structures, preserving key properties.

Contribution

It demonstrates that tree duality is preserved under arc graphs and right adjoint functors, generalizing previous results in graph homomorphism theory.

Findings

Arc graphs of digraphs with tree duality also have tree duality.

Right adjoint functors preserve tree duality, polynomial CSPs, and near-unanimity functions.

Abstract

A family T of digraphs is a complete set of obstructions for a digraph H if for an arbitrary digraph G the existence of a homomorphism from G to H is equivalent to the non-existence of a homomorphism from any member of T to G. A digraph H is said to have tree duality if there exists a complete set of obstructions T consisting of orientations of trees. We show that if H has tree duality, then its arc graph delta H also has tree duality, and we derive a family of tree obstructions for delta H from the obstructions for H. Furthermore we generalise our result to right adjoint functors on categories of relational structures. We show that these functors always preserve tree duality, as well as polynomial CSPs and the existence of near-unanimity functions.