102,078 research outputs found
The Power of the Weisfeiler-Leman Algorithm to Decompose Graphs
The Weisfeiler-Leman procedure is a widely-used approach for graph
isomorphism testing that works by iteratively computing an
isomorphism-invariant coloring of vertex tuples. Meanwhile, a fundamental tool
in structural graph theory, which is often exploited in approaches to tackle
the graph isomorphism problem, is the decomposition into 2- and 3-connected
components.
We prove that the 2-dimensional Weisfeiler-Leman algorithm implicitly
computes the decomposition of a graph into its 3-connected components. Thus,
the dimension of the algorithm needed to distinguish two given graphs is at
most the dimension required to distinguish the corresponding decompositions
into 3-connected components (assuming it is at least 2).
This result implies that for k >= 2, the k-dimensional algorithm
distinguishes k-separators, i.e., k-tuples of vertices that separate the graph,
from other vertex k-tuples. As a byproduct, we also obtain insights about the
connectivity of constituent graphs of association schemes.
In an application of the results, we show the new upper bound of k on the
Weisfeiler-Leman dimension of graphs of treewidth at most k. Using a
construction by Cai, F\"urer, and Immerman, we also provide a new lower bound
that is asymptotically tight up to a factor of 2.Comment: 30 pages, 4 figures, full version of a paper accepted at MFCS 201
Computing with Tangles
Tangles of graphs have been introduced by Robertson and Seymour in the
context of their graph minor theory. Tangles may be viewed as describing
"k-connected components" of a graph (though in a twisted way). They play an
important role in graph minor theory. An interesting aspect of tangles is that
they cannot only be defined for graphs, but more generally for arbitrary
connectivity functions (that is, integer-valued submodular and symmetric set
functions).
However, tangles are difficult to deal with algorithmically. To start with,
it is unclear how to represent them, because they are families of separations
and as such may be exponentially large. Our first contribution is a data
structure for representing and accessing all tangles of a graph up to some
fixed order.
Using this data structure, we can prove an algorithmic version of a very
general structure theorem due to Carmesin, Diestel, Harman and Hundertmark (for
graphs) and Hundertmark (for arbitrary connectivity functions) that yields a
canonical tree decomposition whose parts correspond to the maximal tangles.
(This may be viewed as a generalisation of the decomposition of a graph into
its 3-connected components.
Balanced Crown Decomposition for Connectivity Constraints
We introduce the balanced crown decomposition that captures the structure imposed on graphs by their connected induced subgraphs of a given size. Such subgraphs are a popular modeling tool in various application areas, where the non-local nature of the connectivity condition usually results in very challenging algorithmic tasks. The balanced crown decomposition is a combination of a crown decomposition and a balanced partition which makes it applicable to graph editing as well as graph packing and partitioning problems. We illustrate this by deriving improved approximation algorithms and kernelization for a variety of such problems.
In particular, through this structure, we obtain the first constant-factor approximation for the Balanced Connected Partition (BCP) problem, where the task is to partition a vertex-weighted graph into k connected components of approximately equal weight. We derive a 3-approximation for the two most commonly used objectives of maximizing the weight of the lightest component or minimizing the weight of the heaviest component
Complexity results on the decomposition of a digraph into directed linear forests and out-stars
We consider two decomposition problems in directed graphs. We say that a
digraph is -bounded for some if each of its
connected components contains at most arcs.
For the first problem, a directed linear forest is a collection of
vertex-disjoint directed paths and we consider the problem of decomposing a
given digraph into a -bounded and an -bounded directed linear forest
for some fixed . We give a full
dichotomy for this problem by showing that it can be solved in polynomial time
if and is NP-complete otherwise. This answers a question of
Campbell, H\"orsch, and Moore.
For the second problem, we say that an out-galaxy is a vertex-disjoint
collection of out-stars. Again, we give a full dichotomy of when a given
digraph can be edge-decomposed into a -bounded and an -bounded
out-galaxy for fixed . More
precisely, we show that the problem can be solved in polynomial time if
and is NP-complete otherwise
A Complete Grammar for Decomposing a Family of Graphs into 3-connected Components
Tutte has described in the book "Connectivity in graphs" a canonical
decomposition of any graph into 3-connected components. In this article we
translate (using the language of symbolic combinatorics)
Tutte's decomposition into a general grammar expressing any family of graphs
(with some stability conditions) in terms of the 3-connected subfamily. A key
ingredient we use is an extension of the so-called dissymmetry theorem, which
yields negative signs in the grammar.
As a main application we recover in a purely combinatorial way the analytic
expression found by Gim\'enez and Noy for the series counting labelled planar
graphs (such an expression is crucial to do asymptotic enumeration and to
obtain limit laws of various parameters on random planar graphs). Besides the
grammar, an important ingredient of our method is a recent bijective
construction of planar maps by Bouttier, Di Francesco and Guitter.Comment: 39 page
Dynamic Programming for Graphs on Surfaces
We provide a framework for the design and analysis of dynamic programming
algorithms for surface-embedded graphs on n vertices and branchwidth at most k.
Our technique applies to general families of problems where standard dynamic
programming runs in 2^{O(k log k)} n steps. Our approach combines tools from
topological graph theory and analytic combinatorics. In particular, we
introduce a new type of branch decomposition called "surface cut
decomposition", generalizing sphere cut decompositions of planar graphs
introduced by Seymour and Thomas, which has nice combinatorial properties.
Namely, the number of partial solutions that can be arranged on a surface cut
decomposition can be upper-bounded by the number of non-crossing partitions on
surfaces with boundary. It follows that partial solutions can be represented by
a single-exponential (in the branchwidth k) number of configurations. This
proves that, when applied on surface cut decompositions, dynamic programming
runs in 2^{O(k)} n steps. That way, we considerably extend the class of
problems that can be solved in running times with a single-exponential
dependence on branchwidth and unify/improve most previous results in this
direction.Comment: 28 pages, 3 figure
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