4,211 research outputs found
Exact Algorithm for Graph Homomorphism and Locally Injective Graph Homomorphism
For graphs and , a homomorphism from to is a function , which maps vertices adjacent in to adjacent vertices
of . A homomorphism is locally injective if no two vertices with a common
neighbor are mapped to a single vertex in . Many cases of graph homomorphism
and locally injective graph homomorphism are NP-complete, so there is little
hope to design polynomial-time algorithms for them. In this paper we present an
algorithm for graph homomorphism and locally injective homomorphism working in
time , where is the bandwidth of the
complement of
The Complexity of Surjective Homomorphism Problems -- a Survey
We survey known results about the complexity of surjective homomorphism
problems, studied in the context of related problems in the literature such as
list homomorphism, retraction and compaction. In comparison with these
problems, surjective homomorphism problems seem to be harder to classify and we
examine especially three concrete problems that have arisen from the
literature, two of which remain of open complexity
On retracts, absolute retracts, and folds in cographs
Let G and H be two cographs. We show that the problem to determine whether H
is a retract of G is NP-complete. We show that this problem is fixed-parameter
tractable when parameterized by the size of H. When restricted to the class of
threshold graphs or to the class of trivially perfect graphs, the problem
becomes tractable in polynomial time. The problem is also soluble when one
cograph is given as an induced subgraph of the other. We characterize absolute
retracts of cographs.Comment: 15 page
Constraint Satisfaction with Counting Quantifiers
We initiate the study of constraint satisfaction problems (CSPs) in the
presence of counting quantifiers, which may be seen as variants of CSPs in the
mould of quantified CSPs (QCSPs). We show that a single counting quantifier
strictly between exists^1:=exists and exists^n:=forall (the domain being of
size n) already affords the maximal possible complexity of QCSPs (which have
both exists and forall), being Pspace-complete for a suitably chosen template.
Next, we focus on the complexity of subsets of counting quantifiers on clique
and cycle templates. For cycles we give a full trichotomy -- all such problems
are in L, NP-complete or Pspace-complete. For cliques we come close to a
similar trichotomy, but one case remains outstanding. Afterwards, we consider
the generalisation of CSPs in which we augment the extant quantifier
exists^1:=exists with the quantifier exists^j (j not 1). Such a CSP is already
NP-hard on non-bipartite graph templates. We explore the situation of this
generalised CSP on bipartite templates, giving various conditions for both
tractability and hardness -- culminating in a classification theorem for
general graphs. Finally, we use counting quantifiers to solve the complexity of
a concrete QCSP whose complexity was previously open
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