eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-02-27
49:1
49:14
10.4230/LIPIcs.STACS.2018.49
article
Surjective H-Colouring over Reflexive Digraphs
Larose, Benoit
Martin, Barnaby
Paulusma, Daniel
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol096-stacs2018/LIPIcs.STACS.2018.49/LIPIcs.STACS.2018.49.pdf
Surjective H-Coloring
Computational Complexity
Algorithmic Graph Theory
Universal Algebra
Constraint Satisfaction