152 research outputs found
Small H-coloring problems for bounded degree digraphs
An NP-complete coloring or homomorphism problem may become polynomial time
solvable when restricted to graphs with degrees bounded by a small number, but
remain NP-complete if the bound is higher. For instance, 3-colorability of
graphs with degrees bounded by 3 can be decided by Brooks' theorem, while for
graphs with degrees bounded by 4, the 3-colorability problem is NP-complete. We
investigate an analogous phenomenon for digraphs, focusing on the three
smallest digraphs H with NP-complete H-colorability problems. It turns out that
in all three cases the H-coloring problem is polynomial time solvable for
digraphs with degree bounds , (or
, ). On the other hand with degree bounds
, , all three problems are again
NP-complete. A conjecture proposed for graphs H by Feder, Hell and Huang states
that any variant of the -coloring problem which is NP-complete without
degree constraints is also NP-complete with degree constraints, provided the
degree bounds are high enough. Our study is the first confirmation that the
conjecture may also apply to digraphs.Comment: 10 page
The Dichotomy of List Homomorphisms for Digraphs
The Dichotomy Conjecture for constraint satisfaction problems has been
verified for conservative problems (or, equivalently, for list homomorphism
problems) by Andrei Bulatov. An earlier case of this dichotomy, for list
homomorphisms to undirected graphs, came with an elegant structural distinction
between the tractable and intractable cases. Such structural characterization
is absent in Bulatov's classification, and Bulatov asked whether one can be
found. We provide an answer in the case of digraphs; the technique will apply
in a broader context. The key concept we introduce is that of a digraph
asteroidal triple (DAT). The dichotomy then takes the following form. If a
digraph H has a DAT, then the list homomorphism problem for H is NP-complete;
and a DAT-free digraph H has a polynomial time solvable list homomorphism
problem. DAT-free graphs can be recognized in polynomial time
Complexity of Correspondence Homomorphisms
Correspondence homomorphisms are both a generalization of standard
homomorphisms and a generalization of correspondence colourings. For a fixed
target graph , the problem is to decide whether an input graph , with
each edge labeled by a pair of permutations of , admits a homomorphism to
`corresponding' to the labels, in a sense explained below.
We classify the complexity of this problem as a function of the fixed graph
. It turns out that there is dichotomy -- each of the problems is
polynomial-time solvable or NP-complete. While most graphs yield
NP-complete problems, there are interesting cases of graphs for which the
problem is solved by Gaussian elimination.
We also classify the complexity of the analogous correspondence {\em list
homomorphism} problems, and also the complexity of a {\em bipartite version} of
both problems. We emphasize the proofs for the case when is reflexive, but,
for the record, we include a rough sketch of the remaining proofs in an
Appendix.Comment: 12 pages, 5 figure
Duality for Min-Max Orderings and Dichotomy for Min Cost Homomorphisms
Min-Max orderings correspond to conservative lattice polymorphisms. Digraphs
with Min-Max orderings have polynomial time solvable minimum cost homomorphism
problems. They can also be viewed as digraph analogues of proper interval
graphs and bigraphs.
We give a forbidden structure characterization of digraphs with a Min-Max
ordering which implies a polynomial time recognition algorithm. We also
similarly characterize digraphs with an extended Min-Max ordering, and we apply
this characterization to prove a conjectured form of dichotomy for minimum cost
homomorphism problems
Point determining digraphs, -matrix partitions, and dualities in full homomorphisms
We prove that every point-determining digraph contains a vertex such
that is also point determining. We apply this result to show that for any
-matrix , with diagonal zeros and diagonal ones, the
size of a minimal -obstruction is at most . This extends the
results of Sumner, and of Feder and Hell, from undirected graphs and symmetric
matrices to digraphs and general matrices.Comment: 12 page
Minimal digraph obstructions for small matrices
Given a -matrix , a minimal -obstruction is a digraph
such that is not -partitionable, but every proper induced subdigraph
of is. In this note we present a list of all the -obstructions for every
matrix .
Notice that this note will be part of a larger paper, but we are archiving it
now so we can cite the results.Comment: 9 pages, 2 figure
Obstructions to chordal circular-arc graphs of small independence number
A blocking quadruple (BQ) is a quadruple of vertices of a graph such that any
two vertices of the quadruple either miss (have no neighbours on) some path
connecting the remaining two vertices of the quadruple, or are connected by
some path missed by the remaining two vertices. This is akin to the notion of
asteroidal triple used in the classical characterization of interval graphs by
Lekkerkerker and Boland. We show that a circular-arc graph cannot have a
blocking quadruple. We also observe that the absence of blocking quadruples is
not in general sufficient to guarantee that a graph is a circular-arc graph.
Nonetheless, it can be shown to be sufficient for some special classes of
graphs, such as those investigated by Bonomo et al. In this note, we focus on
chordal graphs, and study the relationship between the structure of chordal
graphs and the presence/absence of blocking quadruples. Our contribution is
two-fold. Firstly, we provide a forbidden induced subgraph characterization of
chordal graphs without blocking quadruples. In particular, we observe that all
the forbidden subgraphs are variants of the subgraphs forbidden for interval
graphs. Secondly, we show that the absence of blocking quadruples is sufficient
to guarantee that a chordal graph with no independent set of size five is a
circular-arc graph. In our proof we use a novel geometric approach,
constructing a circular-arc representation by traversing around a carefully
chosen clique tree
Forbidden structure characterization of circular-arc graphs and a certifying recognition algorithm
A circular-arc graph is the intersection graph of arcs of a circle. It is a
well-studied graph model with numerous natural applications. A certifying
algorithm is an algorithm that outputs a certificate, along with its answer (be
it positive or negative), where the certificate can be used to easily justify
the given answer. While the recognition of circular-arc graphs has been known
to be polynomial since the 1980s, no polynomial-time certifying recognition
algorithm is known to date, despite such algorithms being found for many
subclasses of circular-arc graphs. This is largely due to the fact that a
forbidden structure characterization of circular-arc graphs is not known, even
though the problem has been intensely studied since the seminal work of Klee in
the 1960s.
In this contribution, we settle this problem. We present the first forbidden
structure characterization of circular-arc graphs. Our obstruction has the form
of mutually avoiding walks in the graph. It naturally extends a similar
obstruction that characterizes interval graphs. As a consequence, we give the
first polynomial-time certifying algorithm for the recognition of circular-arc
graphs.Comment: 26 pages, 3 figure
On edge-sets of bicliques in graphs
A biclique is a maximal induced complete bipartite subgraph of a graph. We
investigate the intersection structure of edge-sets of bicliques in a graph.
Specifically, we study the associated edge-biclique hypergraph whose hyperedges
are precisely the edge-sets of all bicliques. We characterize graphs whose
edge-biclique hypergraph is conformal (i.e., it is the clique hypergraph of its
2-section) by means of a single forbidden induced obstruction, the triangular
prism. Using this result, we characterize graphs whose edge-biclique hypergraph
is Helly and provide a polynomial time recognition algorithm. We further study
a hereditary version of this property and show that it also admits polynomial
time recognition, and, in fact, is characterized by a finite set of forbidden
induced subgraphs. We conclude by describing some interesting properties of the
2-section graph of the edge-biclique hypergraph.Comment: This version corrects an error in Theorem 11 found after the paper
went into prin
Colourings, Homomorphisms, and Partitions of Transitive Digraphs
We investigate the complexity of generalizations of colourings (acyclic
colourings, -colourings, homomorphisms, and matrix partitions), for
the class of transitive digraphs. Even though transitive digraphs are nicely
structured, many problems are intractable, and their complexity turns out to be
difficult to classify. We present some motivational results and several open
problems.Comment: 13 pages, 3 figure
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