385 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
Isomorphism test for digraphs with weighted edges
Colour refinement is at the heart of all the most efficient graph isomorphism software packages. In this paper we present a method for extending the applicability of refinement algorithms to directed graphs with weighted edges. We use Traces as a reference software, but the proposed solution is easily transferrable to any other refinement-based graph isomorphism tool in the literature. We substantiate the claim that the performances of the original algorithm remain substantially unchanged by showing experiments for some classes of benchmark graphs
Out-degree reducing partitions of digraphs
Let be a fixed integer. We determine the complexity of finding a
-partition of the vertex set of a given digraph such
that the maximum out-degree of each of the digraphs induced by , () is at least smaller than the maximum out-degree of . We show
that this problem is polynomial-time solvable when and -complete otherwise. The result for and answers a question
posed in \cite{bangTCS636}. We also determine, for all fixed non-negative
integers , the complexity of deciding whether a given digraph of
maximum out-degree has a -partition such that the digraph
induced by has maximum out-degree at most for . It
follows from this characterization that the problem of deciding whether a
digraph has a 2-partition such that each vertex has at
least as many neighbours in the set as in , for is
-complete. This solves a problem from \cite{kreutzerEJC24} on
majority colourings.Comment: 11 pages, 1 figur
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