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Surjective H-Colouring over Reflexive Digraphs Larose, Benoit Martin, Barnaby Paulusma, Daniel Surjective H-Coloring Computational Complexity Algorithmic Graph Theory Universal Algebra Constraint Satisfaction 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Benoit Larose and Barnaby Martin and Daniel Paulusma 2018 Is Part Of LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.STACS.2018.49 urn:nbn:de:0030-drops-84882 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.49 eng https://creativecommons.org/licenses/by/3.0/legalcode