# The complexity of digraph homomorphisms: Local tournaments, injective homomorphisms and polymorphisms

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## Date

2008-12-19T21:24:07Z

## Authors

Swarts, Jacobus Stephanus

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## Abstract

In this thesis we examine the computational complexity of certain digraph homomorphism problems. A homomorphism between digraphs, denoted by $f: G \to H$, is a mapping from the vertices of $G$ to the vertices of $H$ such that the arcs of $G$ are preserved. The problem of deciding whether a homomorphism to a fixed digraph $H$ exists is known as the $H$-colouring problem.
We prove a generalization of a theorem due to Bang-Jensen, Hell and MacGillivray. Their theorem shows that for every semi-complete digraph $H$, $H$-colouring exhibits a dichotomy: $H$-colouring is either polynomial time solvable or it is NP-complete. We show that the class of local tournaments also exhibit a dichotomy. The NP-completeness results are found using direct NP-completeness reductions, indicator and vertex (and arc) sub-indicator constructions. The polynomial cases are handled by appealing to a result of Gutjhar, Woeginger and Welzl: the \underbar{$X$}-graft extension. We also provide a new proof of their result that follows directly from the consistency check. An unexpected result is the existence of unicyclic local tournaments with NP-complete homomorphism problems.
During the last decade a new approach to studying the complexity of digraph homomorphism problems has emerged. This approach focuses attention on so-called polymorphisms as a measure of the complexity of a digraph homomorphism problem. For a digraph $H$, a polymorphism of arity $k$ is a homomorphism $f: H^k \to H$.
Certain special polymorphisms are conjectured to be the key to understanding $H$-colouring problems. These polymorphisms are known as weak near unanimity functions (WNUFs). A WNUF of arity $k$ is a polymorphism $f: H^k \to H$ such that $f$ is idempotent an $f(y,x,x,\ldots,x)=f(x,y,x,\ldots,x)=f(x,x,y,\ldots,x) = \cdots = f(x,x,x,\ldots,y)$. We prove that a large class of polynomial time $H$-colouring problems all have a $\WNUF$. Furthermore we also prove some non-existence results for $\WNUF$s on certain digraphs. In proving these results, we develop a vertex (and arc) sub-indicator construction as well as an indicator construction in analogy with the ones developed by Hell and Ne{\v{s}}et{\v{r}}il. This is then used to show that all tournaments with at least two cycles do not admit a $\WNUF_k$ for $k>1$. This furnishes a new proof (in the case of tournaments) of the result by Bang-Jensen, Hell and MacGillivray referred to at the start. These results lend some support to the conjecture that $\WNUF$s are the ``right'' functions for measuring the complexity of $H$-colouring problems.
We also study a related notion, namely that of an injective homomorphism. A homomorphism $f: G \to H$ is injective if the restriction of $f$ to the in-neighbours of every vertex in $G$ is an injective mapping. In order to classify the complexity of these problems we develop an indicator construction that is suited to injective homomorphism problems.
For this type of digraph homomorphism problem we consider two cases: reflexive and irreflexive targets. In the case of reflexive targets we are able to classify all injective homomorphism problems as either belonging to the class of polynomial time solvable problems or as being NP-complete. Irreflexive targets pose more of a problem. The problem lies with targets of maximum in-degree equal to two. Targets with maximum in-degree one are polynomial, while targets with in-degree at least three are NP-complete. There is a transformation from (ordinary) graph homomorphism problems to injective, in-degree two, homomorphism problems (a reverse transformation also exists). This transformation provides some explanation as to the difficulty of the in-degree two case. We nonetheless classify all injective homomorphisms to irreflexive tournaments as either being a problem in P or a problem in the class of NP-complete problems. We also discuss some upper bounds on the injective oriented irreflexive (reflexive) chromatic number.

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## Keywords

digraph homomorphisms, computational complexity