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Durham University

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Publication details for Dr Barnaby Martin

Larose, B., Martin, B. & Paulusma, D. (2018), Surjective H-Colouring over Reflexive Digraphs, in Niedermeier, Rolf & Vallée, Brigitte eds, Leibniz International Proceedings in Informatics (LIPIcs) 96: 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Caen, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 49:1-49:14.

Author(s) from Durham


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.