License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2018.49
URN: urn:nbn:de:0030-drops-84882
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8488/
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Larose, Benoit ; Martin, Barnaby ; Paulusma, Daniel

Surjective H-Colouring over Reflexive Digraphs

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LIPIcs-STACS-2018-49.pdf (0.5 MB)

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 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.

BibTeX - Entry

@InProceedings{larose_et_al:LIPIcs:2018:8488,
  author =	{Benoit Larose and Barnaby Martin and Daniel Paulusma},
  title =	{{Surjective H-Colouring over Reflexive Digraphs}},
  booktitle =	{35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
  pages =	{49:1--49:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-062-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{96},
  editor =	{Rolf Niedermeier and Brigitte Vall{\'e}e},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8488},
  URN =		{urn:nbn:de:0030-drops-84882},
  doi =		{10.4230/LIPIcs.STACS.2018.49},
  annote =	{Keywords: Surjective H-Coloring, Computational Complexity, Algorithmic Graph Theory, Universal Algebra, Constraint Satisfaction}
}

Keywords: Surjective H-Coloring, Computational Complexity, Algorithmic Graph Theory, Universal Algebra, Constraint Satisfaction
Collection: 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)
Issue Date: 2018
Date of publication: 27.02.2018

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