Surjective H-Colouring over Reflexive Digraphs
Benoit Larose, Barnaby Martin, Daniel Paulusma

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…
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