481 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
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
Fixed-parameter tractable canonization and isomorphism test for graphs of bounded treewidth
We give a fixed-parameter tractable algorithm that, given a parameter and
two graphs , either concludes that one of these graphs has treewidth
at least , or determines whether and are isomorphic. The running
time of the algorithm on an -vertex graph is ,
and this is the first fixed-parameter algorithm for Graph Isomorphism
parameterized by treewidth.
Our algorithm in fact solves the more general canonization problem. We namely
design a procedure working in time that, for a
given graph on vertices, either concludes that the treewidth of is
at least , or: * finds in an isomorphic-invariant way a graph
that is isomorphic to ; * finds an isomorphism-invariant
construction term --- an algebraic expression that encodes together with a
tree decomposition of of width .
Hence, the isomorphism test reduces to verifying whether the computed
isomorphic copies or the construction terms for and are equal.Comment: Full version of a paper presented at FOCS 201
Induced Minor Free Graphs: Isomorphism and Clique-width
Given two graphs and , we say that contains as an induced
minor if a graph isomorphic to can be obtained from by a sequence of
vertex deletions and edge contractions. We study the complexity of Graph
Isomorphism on graphs that exclude a fixed graph as an induced minor. More
precisely, we determine for every graph that Graph Isomorphism is
polynomial-time solvable on -induced-minor-free graphs or that it is
GI-complete. Additionally, we classify those graphs for which
-induced-minor-free graphs have bounded clique-width. These two results
complement similar dichotomies for graphs that exclude a fixed graph as an
induced subgraph, minor, or subgraph.Comment: 16 pages, 5 figures. An extended abstract of this paper previously
appeared in the proceedings of the 41st International Workshop on
Graph-Theoretic Concepts in Computer Science (WG 2015
Graph Isomorphism in Quasipolynomial Time Parameterized by Treewidth
We extend Babai's quasipolynomial-time graph isomorphism test (STOC 2016) and
develop a quasipolynomial-time algorithm for the multiple-coset isomorphism
problem. The algorithm for the multiple-coset isomorphism problem allows to
exploit graph decompositions of the given input graphs within Babai's
group-theoretic framework.
We use it to develop a graph isomorphism test that runs in time
where is the number of vertices and is
the minimum treewidth of the given graphs and is
some polynomial in . Our result generalizes Babai's
quasipolynomial-time graph isomorphism test.Comment: 52 pages, 1 figur
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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