432 research outputs found
An Upper Bound on the Size of Obstructions for Bounded Linear Rank-Width
We provide a doubly exponential upper bound in on the size of forbidden
pivot-minors for symmetric or skew-symmetric matrices over a fixed finite field
of linear rank-width at most . As a corollary, we obtain a
doubly exponential upper bound in on the size of forbidden vertex-minors
for graphs of linear rank-width at most . This solves an open question
raised by Jeong, Kwon, and Oum [Excluded vertex-minors for graphs of linear
rank-width at most . European J. Combin., 41:242--257, 2014]. We also give a
doubly exponential upper bound in on the size of forbidden minors for
matroids representable over a fixed finite field of path-width at most .
Our basic tool is the pseudo-minor order used by Lagergren [Upper Bounds on
the Size of Obstructions and Interwines, Journal of Combinatorial Theory Series
B, 73:7--40, 1998] to bound the size of forbidden graph minors for bounded
path-width. To adapt this notion into linear rank-width, it is necessary to
well define partial pieces of graphs and merging operations that fit to
pivot-minors. Using the algebraic operations introduced by Courcelle and
Kant\'e, and then extended to (skew-)symmetric matrices by Kant\'e and Rao, we
define boundaried -labelled graphs and prove similar structure theorems for
pivot-minor and linear rank-width.Comment: 28 pages, 1 figur
Linear rank-width and linear clique-width of trees
We show that for every forest T the linear rank-width of T is equal to the path-width of T, and the linear clique-width of T equals the path-width of T plus two, provided that T contains a path of length three. It follows that both linear rank-width and linear clique-width of forests can be computed in linear time. Using our characterization of linear rank-width of forests, we determine the set of minimal excluded acyclic vertex-minors for the class of graphs of linear rank-width at most k
Linear Rank-Width of Distance-Hereditary Graphs
We present a characterization of the linear rank-width of distance-hereditary graphs. Using the characterization, we show that the linear rank-width of every n-vertex distance-hereditary graph can be computed in time O(n²⋅log(n)), and a linear layout witnessing the linear rank-width can be computed with the same time complexity. For our characterization, we combine modifications of canonical split decompositions with an idea of [Megiddo, Hakimi, Garey, Johnson, Papadimitriou: The complexity of searching a graph. JACM 1988], used for computing the path-width of trees. We also provide a set of distance-hereditary graphs which contains the set of distance-hereditary vertex-minor obstructions for linear rank-width. The set given in [Jeong, Kwon, Oum: Excluded vertex-minors for graphs of linear rank-width at most k. STACS 2013: 221–232] is a subset of our obstruction set
On the Monadic Second-Order Transduction Hierarchy
We compare classes of finite relational structures via monadic second-order
transductions. More precisely, we study the preorder where we set C \subseteq K
if, and only if, there exists a transduction {\tau} such that
C\subseteq{\tau}(K). If we only consider classes of incidence structures we can
completely describe the resulting hierarchy. It is linear of order type
{\omega}+3. Each level can be characterised in terms of a suitable variant of
tree-width. Canonical representatives of the various levels are: the class of
all trees of height n, for each n \in N, of all paths, of all trees, and of all
grids
Successor-Invariant First-Order Logic on Graphs with Excluded Topological Subgraphs
We show that the model-checking problem for successor-invariant first-order
logic is fixed-parameter tractable on graphs with excluded topological
subgraphs when parameterised by both the size of the input formula and the size
of the exluded topological subgraph. Furthermore, we show that model-checking
for order-invariant first-order logic is tractable on coloured posets of
bounded width, parameterised by both the size of the input formula and the
width of the poset.
Our result for successor-invariant FO extends previous results for this logic
on planar graphs (Engelmann et al., LICS 2012) and graphs with excluded minors
(Eickmeyer et al., LICS 2013), further narrowing the gap between what is known
for FO and what is known for successor-invariant FO. The proof uses Grohe and
Marx's structure theorem for graphs with excluded topological subgraphs. For
order-invariant FO we show that Gajarsk\'y et al.'s recent result for FO
carries over to order-invariant FO
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
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