Surjective H-Colouring over Reflexive Digraphs

Abstract

The S urjective H-C olouring 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 S urjective H-C olouring in the case of reflexive digraphs . Chen (2014) proved, in the setting of constraint satisfaction problems, that S urjective H-C olouring 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 S urjective H-C olouring when H is a reflexive tournament: if H is transitive, then S urjective H-C olouring is in NL; otherwise, it is NP-complete. By combining this result with some known and new results, we obtain a complexity classification for S urjective H-C olouring when H is a partially reflexive digraph of size at most 3.