71 research outputs found
The Weisfeiler-Leman Dimension of Planar Graphs is at most 3
We prove that the Weisfeiler-Leman (WL) dimension of the class of all finite
planar graphs is at most 3. In particular, every finite planar graph is
definable in first-order logic with counting using at most 4 variables. The
previously best known upper bounds for the dimension and number of variables
were 14 and 15, respectively.
First we show that, for dimension 3 and higher, the WL-algorithm correctly
tests isomorphism of graphs in a minor-closed class whenever it determines the
orbits of the automorphism group of any arc-colored 3-connected graph belonging
to this class.
Then we prove that, apart from several exceptional graphs (which have
WL-dimension at most 2), the individualization of two correctly chosen vertices
of a colored 3-connected planar graph followed by the 1-dimensional
WL-algorithm produces the discrete vertex partition. This implies that the
3-dimensional WL-algorithm determines the orbits of a colored 3-connected
planar graph.
As a byproduct of the proof, we get a classification of the 3-connected
planar graphs with fixing number 3.Comment: 34 pages, 3 figures, extended version of LICS 2017 pape
A Linear Upper Bound on the Weisfeiler-Leman Dimension of Graphs of Bounded Genus
The Weisfeiler-Leman (WL) dimension of a graph is a measure for the inherent descriptive complexity of the graph. While originally derived from a combinatorial graph isomorphism test called the Weisfeiler-Leman algorithm, the WL dimension can also be characterised in terms of the number of variables that is required to describe the graph up to isomorphism in first-order logic with counting quantifiers.
It is known that the WL dimension is upper-bounded for all graphs that exclude some fixed graph as a minor [M. Grohe, 2017]. However, the bounds that can be derived from this general result are astronomic. Only recently, it was proved that the WL dimension of planar graphs is at most 3 [S. Kiefer et al., 2017].
In this paper, we prove that the WL dimension of graphs embeddable in a surface of Euler genus g is at most 4g+3. For the WL dimension of graphs embeddable in an orientable surface of Euler genus g, our approach yields an upper bound of 2g + 3
Canonisation and Definability for Graphs of Bounded Rank Width
We prove that the combinatorial Weisfeiler-Leman algorithm of dimension
is a complete isomorphism test for the class of all graphs of rank
width at most . Rank width is a graph invariant that, similarly to tree
width, measures the width of a certain style of hierarchical decomposition of
graphs; it is equivalent to clique width. It was known that isomorphism of
graphs of rank width is decidable in polynomial time (Grohe and Schweitzer,
FOCS 2015), but the best previously known algorithm has a running time
for a non-elementary function . Our result yields an isomorphism
test for graphs of rank width running in time . Another
consequence of our result is the first polynomial time canonisation algorithm
for graphs of bounded rank width. Our second main result is that fixed-point
logic with counting captures polynomial time on all graph classes of bounded
rank width.Comment: 32 page
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
On the Weisfeiler-Leman dimension of some polyhedral graphs
Let be a positive integer, a graph with vertex set , and
the coloring of the Cartesian -power , obtained by
the -dimensional Weisfeiler-Leman algorithm. The -dimension of the
graph is defined to be the smallest for which the coloring determines up to isomorphism. It is known that the -dimension of any planar graph is or , but no planar graph of -dimension is known. We prove that the -dimension of a
polyhedral (i.e., -connected planar) graph is at most if the color
classes of the coloring are the orbits of the componentwise
action of the group on
Logarithmic Weisfeiler-Leman Identifies All Planar Graphs
The Weisfeiler-Leman (WL) algorithm is a well-known combinatorial procedure for detecting symmetries in graphs and it is widely used in graph-isomorphism tests. It proceeds by iteratively refining a colouring of vertex tuples. The number of iterations needed to obtain the final output is crucial for the parallelisability of the algorithm.
We show that there is a constant k such that every planar graph can be identified (that is, distinguished from every non-isomorphic graph) by the k-dimensional WL algorithm within a logarithmic number of iterations. This generalises a result due to Verbitsky (STACS 2007), who proved the same for 3-connected planar graphs.
The number of iterations needed by the k-dimensional WL algorithm to identify a graph corresponds to the quantifier depth of a sentence that defines the graph in the (k+1)-variable fragment C^{k+1} of first-order logic with counting quantifiers. Thus, our result implies that every planar graph is definable with a C^{k+1}-sentence of logarithmic quantifier depth
Logarithmic Weisfeiler--Leman and Treewidth
In this paper, we show that the -dimensional Weisfeiler--Leman
algorithm can identify graphs of treewidth in rounds. This
improves the result of Grohe & Verbitsky (ICALP 2006), who previously
established the analogous result for -dimensional Weisfeiler--Leman. In
light of the equivalence between Weisfeiler--Leman and the logic (Cai, F\"urer, & Immerman, Combinatorica 1992), we obtain an
improvement in the descriptive complexity for graphs of treewidth .
Precisely, if is a graph of treewidth , then there exists a
-variable formula in with
quantifier depth that identifies up to isomorphism
On the Expressivity of Persistent Homology in Graph Learning
Persistent homology, a technique from computational topology, has recently
shown strong empirical performance in the context of graph classification.
Being able to capture long range graph properties via higher-order topological
features, such as cycles of arbitrary length, in combination with multi-scale
topological descriptors, has improved predictive performance for data sets with
prominent topological structures, such as molecules. At the same time, the
theoretical properties of persistent homology have not been formally assessed
in this context. This paper intends to bridge the gap between computational
topology and graph machine learning by providing a brief introduction to
persistent homology in the context of graphs, as well as a theoretical
discussion and empirical analysis of its expressivity for graph learning tasks
Weisfeiler--Leman and Graph Spectra
We devise a hierarchy of spectral graph invariants, generalising the
adjacency spectra and Laplacian spectra, which are commensurate in power with
the hierarchy of combinatorial graph invariants generated by the
Weisfeiler--Leman (WL) algorithm. More precisely, we provide a spectral
characterisation of -WL indistinguishability after iterations, for .
Most of the well-known spectral graph invariants such as adjacency or
Laplacian spectra lie in the regime between 1-WL and 2-WL. We show that
individualising one vertex plus running 1-WL is already more powerful than all
such spectral invariants in terms of their ability to distinguish
non-isomorphic graphs. Building on this result, we resolve an open problem of
F\"urer (2010) about spectral invariants and strengthen a result due to Godsil
(1981) about commute distances
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