Abstract
The S urjective H-C olouring 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 S urjective H-C olouring in the case of reflexive digraphs . Chen (2014) proved, in the setting of constraint satisfaction problems, that S urjective H-C olouring 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 S urjective H-C olouring when H is a reflexive tournament: if H is transitive, then S urjective H-C olouring is in NL; otherwise, it is NP-complete. By combining this result with some known and new results, we obtain a complexity classification for S urjective H-C olouring when H is a partially reflexive digraph of size at most 3.
Highlights
The classical homomorphism problem, known as H-Colouring, involves a fixed structure H, with input another structure G, of the same signature, invoking the question as to whether there is a function from the domain of G to the domain of H that is a homomorphism from G to H
This paper concerns the computational complexity of the surjective homomorphism problem, known in the literature as Surjective H-Colouring [15, 16] and H-VertexCompaction [30]
This problem requires the homomorphism to be surjective. It is a cousin of the list homomorphism problem and is even more closely related to the retraction and compaction problems
Summary
The classical homomorphism problem, known as H-Colouring, involves a fixed structure H, with input another structure G, of the same signature, invoking the question as to whether there is a function from the domain of G to the domain of H that is a homomorphism from G to H. The algebraic method is not so far advanced for surjective homomorphism problems It only exists in the work of Chen [7], who proved that Surjective H-Colouring is NP-complete if H has the property that all of its polymorphisms depend only on one variable, that is, are essentially unary. Combining this result with the aforementioned result of Chen [7] immediately yields that Surjective H-Colouring is NP-complete for any such digraph H. As the case k ≤ 2 is trivial, this gives a classification of Surjective H-Colouring for reflexive directed cycles, which we believe form a natural class of digraphs to consider given the results in [24, 31]. We are not aware of an existing classification for H-Retraction on this class, but we do build on one existing for List H-Colouring from [13]
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