Surjective H-Colouring over Reflexive Digraphs

TL;DR

This paper classifies the computational complexity of the Surjective H-Colouring problem for reflexive digraphs, introducing endo-triviality to distinguish cases where the problem is tractable or NP-complete.

Contribution

It introduces the concept of endo-triviality for reflexive digraphs and applies it to establish a complexity dichotomy for Surjective H-Colouring on reflexive tournaments.

Findings

Surjective H-Colouring is NL-complete for transitive reflexive tournaments.

Surjective H-Colouring is NP-complete for non-transitive reflexive tournaments.

The classification extends to partially reflexive digraphs of size up to 3.

Abstract

The Surjective H-Colouring problem is to test if a given graph allows a vertex-surjective homomorphism to a fixed graph H. The complexity of this problem has been well studied for undirected (partially) reflexive graphs. We introduce endo-triviality, the property of a structure that all of its endomorphisms that do not have range of size 1 are automorphisms, as a means to obtain complexity-theoretic classifications of Surjective H-Colouring in the case of reflexive digraphs. Chen [2014] proved, in the setting of constraint satisfaction problems, that Surjective H-Colouring is NP-complete if H has the property that all of its polymorphisms are essentially unary. We give the first concrete application of his result by showing that every endo-trivial reflexive digraph H has this property. We then use the concept of endo-triviality to prove, as our main result, a dichotomy for Surjective H-Colouring when H is a reflexive tournament: if H is transitive, then Surjective H-Colouring is in NL, otherwise it is NP-complete. By combining this result with some known and new results we obtain a complexity classification for Surjective H-Colouring when H is a partially reflexive digraph of size at most 3.