399 research outputs found
Structure Theorem and Isomorphism Test for Graphs with Excluded Topological Subgraphs
We generalize the structure theorem of Robertson and Seymour for graphs
excluding a fixed graph as a minor to graphs excluding as a topological
subgraph. We prove that for a fixed , every graph excluding as a
topological subgraph has a tree decomposition where each part is either "almost
embeddable" to a fixed surface or has bounded degree with the exception of a
bounded number of vertices. Furthermore, we prove that such a decomposition is
computable by an algorithm that is fixed-parameter tractable with parameter
.
We present two algorithmic applications of our structure theorem. To
illustrate the mechanics of a "typical" application of the structure theorem,
we show that on graphs excluding as a topological subgraph, Partial
Dominating Set (find vertices whose closed neighborhood has maximum size)
can be solved in time time. More significantly, we show
that on graphs excluding as a topological subgraph, Graph Isomorphism can
be solved in time . This result unifies and generalizes two
previously known important polynomial-time solvable cases of Graph Isomorphism:
bounded-degree graphs and -minor free graphs. The proof of this result needs
a generalization of our structure theorem to the context of invariant treelike
decomposition
Deciding first-order properties of nowhere dense graphs
Nowhere dense graph classes, introduced by Nesetril and Ossona de Mendez,
form a large variety of classes of "sparse graphs" including the class of
planar graphs, actually all classes with excluded minors, and also bounded
degree graphs and graph classes of bounded expansion.
We show that deciding properties of graphs definable in first-order logic is
fixed-parameter tractable on nowhere dense graph classes. At least for graph
classes closed under taking subgraphs, this result is optimal: it was known
before that for all classes C of graphs closed under taking subgraphs, if
deciding first-order properties of graphs in C is fixed-parameter tractable,
then C must be nowhere dense (under a reasonable complexity theoretic
assumption).
As a by-product, we give an algorithmic construction of sparse neighbourhood
covers for nowhere dense graphs. This extends and improves previous
constructions of neighbourhood covers for graph classes with excluded minors.
At the same time, our construction is considerably simpler than those. Our
proofs are based on a new game-theoretic characterisation of nowhere dense
graphs that allows for a recursive version of locality-based algorithms on
these classes. On the logical side, we prove a "rank-preserving" version of
Gaifman's locality theorem.Comment: 30 page
Parameterized Directed -Chinese Postman Problem and Arc-Disjoint Cycles Problem on Euler Digraphs
In the Directed -Chinese Postman Problem (-DCPP), we are given a
connected weighted digraph and asked to find non-empty closed directed
walks covering all arcs of such that the total weight of the walks is
minimum. Gutin, Muciaccia and Yeo (Theor. Comput. Sci. 513 (2013) 124--128)
asked for the parameterized complexity of -DCPP when is the parameter.
We prove that the -DCPP is fixed-parameter tractable.
We also consider a related problem of finding arc-disjoint directed
cycles in an Euler digraph, parameterized by . Slivkins (ESA 2003) showed
that this problem is W[1]-hard for general digraphs. Generalizing another
result by Slivkins, we prove that the problem is fixed-parameter tractable for
Euler digraphs. The corresponding problem on vertex-disjoint cycles in Euler
digraphs remains W[1]-hard even for Euler digraphs
Compact Labelings For Efficient First-Order Model-Checking
We consider graph properties that can be checked from labels, i.e., bit
sequences, of logarithmic length attached to vertices. We prove that there
exists such a labeling for checking a first-order formula with free set
variables in the graphs of every class that is \emph{nicely locally
cwd-decomposable}. This notion generalizes that of a \emph{nicely locally
tree-decomposable} class. The graphs of such classes can be covered by graphs
of bounded \emph{clique-width} with limited overlaps. We also consider such
labelings for \emph{bounded} first-order formulas on graph classes of
\emph{bounded expansion}. Some of these results are extended to counting
queries
Logics for Unranked Trees: An Overview
Labeled unranked trees are used as a model of XML documents, and logical
languages for them have been studied actively over the past several years. Such
logics have different purposes: some are better suited for extracting data,
some for expressing navigational properties, and some make it easy to relate
complex properties of trees to the existence of tree automata for those
properties. Furthermore, logics differ significantly in their model-checking
properties, their automata models, and their behavior on ordered and unordered
trees. In this paper we present a survey of logics for unranked trees
Does Treewidth Help in Modal Satisfiability?
Many tractable algorithms for solving the Constraint Satisfaction Problem
(CSP) have been developed using the notion of the treewidth of some graph
derived from the input CSP instance. In particular, the incidence graph of the
CSP instance is one such graph. We introduce the notion of an incidence graph
for modal logic formulae in a certain normal form. We investigate the
parameterized complexity of modal satisfiability with the modal depth of the
formula and the treewidth of the incidence graph as parameters. For various
combinations of Euclidean, reflexive, symmetric and transitive models, we show
either that modal satisfiability is FPT, or that it is W[1]-hard. In
particular, modal satisfiability in general models is FPT, while it is
W[1]-hard in transitive models. As might be expected, modal satisfiability in
transitive and Euclidean models is FPT.Comment: Full version of the paper appearing in MFCS 2010. Change from v1:
improved section 5 to avoid exponential blow-up in formula siz
Visual Ontology Cleaning: Cognitive Principles and Applicability
In this paper we connect two research areas, the Qualitative
Spatial Reasoning and visual reasoning on ontologies. We discuss the logical
limitations of the mereotopological approach to the visual ontology
cleaning, from the point of view of its formal support. The analysis is
based on three different spatial interpretations wich are based in turn on
three different spatial interpretations of the concepts of an ontology.Ministerio de Educación y Ciencia TIN2004-0388
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