168 research outputs found
On the Complexity of Digraph Colourings and Vertex Arboricity
It has been shown by Bokal et al. that deciding 2-colourability of digraphs
is an NP-complete problem. This result was later on extended by Feder et al. to
prove that deciding whether a digraph has a circular -colouring is
NP-complete for all rational . In this paper, we consider the complexity
of corresponding decision problems for related notions of fractional colourings
for digraphs and graphs, including the star dichromatic number, the fractional
dichromatic number and the circular vertex arboricity. We prove the following
results:
Deciding if the star dichromatic number of a digraph is at most is
NP-complete for every rational .
Deciding if the fractional dichromatic number of a digraph is at most is
NP-complete for every .
Deciding if the circular vertex arboricity of a graph is at most is
NP-complete for every rational .
To show these results, different techniques are required in each case. In
order to prove the first result, we relate the star dichromatic number to a new
notion of homomorphisms between digraphs, called circular homomorphisms, which
might be of independent interest. We provide a classification of the
computational complexities of the corresponding homomorphism colouring problems
similar to the one derived by Feder et al. for acyclic homomorphisms.Comment: 21 pages, 1 figur
Functors on relational structures which admit both left and right adjoints
This paper describes several cases of adjunction in the homomorphism preorder
of relational structures. We say that two functors and
between thin categories of relational structures are adjoint if for all
structures and , we have that maps
homomorphically to if and only if maps homomorphically
to . If this is the case is called the left
adjoint to and the right adjoint to . In 2015,
Foniok and Tardif described some functors on the category of digraphs that
allow both left and right adjoints. The main contribution of Foniok and Tardif
is a construction of right adjoints to some of the functors identified as right
adjoints by Pultr in 1970. We generalise results of Foniok and Tardif to
arbitrary relational structures, and coincidently, we also provide more right
adjoints on digraphs, and since these constructions are connected to finite
duality, we also provide a new construction of duals to trees. Our results are
inspired by an application in promise constraint satisfaction -- it has been
shown that such functors can be used as efficient reductions between these
problems
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