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Logarithmic Weisfeiler-Leman Identifies All Planar Graphs [article]

Martin Grohe, Sandra Kiefer
2021 arXiv   pre-print
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.  ...  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  ...  Introduction The Weisfeiler-Leman (WL) algorithm is a well-known combinatorial procedure for detecting symmetries in graphs.  ... 
arXiv:2106.16218v1 fatcat:pewgcuhr3je27ph5s3yaudckt4

Logarithmic Weisfeiler-Leman Identifies All Planar Graphs

Martin Grohe, Sandra Kiefer
2021
It has been shown that for suitable constants k, the algorithm WL k identifies all planar graphs [13] , all graphs of bounded tree width [18] , and all graphs in many other natural graph classes [12, 14  ...  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.  ...  Introduction The Weisfeiler-Leman (WL) algorithm is a combinatorial procedure for detecting symmetries in graphs.  ... 
doi:10.18154/rwth-2021-06468 fatcat:4i4ilkygxjfibopkyoojm7olte

Logarithmic Weisfeiler-Leman Identifies All Planar Graphs

Martin Grohe, Sandra Kiefer, Nikhil Bansal, Emanuela Merelli, James Worrell
2021
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.  ...  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  ...  Introduction The Weisfeiler-Leman (WL) algorithm is a combinatorial procedure for detecting symmetries in graphs.  ... 
doi:10.4230/lipics.icalp.2021.134 fatcat:grvwrx2xtrhc7cijciyjzq4kry

Weisfeiler and Leman go Machine Learning: The Story so far [article]

Christopher Morris, Yaron Lipman, Haggai Maron, Bastian Rieck, Nils M. Kriege, Martin Grohe, Matthias Fey, Karsten Borgwardt
2021 arXiv   pre-print
In recent years, algorithms and neural architectures based on the Weisfeiler-Leman algorithm, a well-known heuristic for the graph isomorphism problem, emerged as a powerful tool for machine learning with  ...  graphs and relational data.  ...  Since planar graphs exclude the complete 5-node graph 𝐾 5 as a minor, planar graphs have a bounded WL dimension.  ... 
arXiv:2112.09992v1 fatcat:r5ahhxsvhrbotfi6grerkzxuui

The Graph Isomorphism Problem (Dagstuhl Seminar 15511)

László Babai, Anuj Dawar, Pascal Schweitzer, Jacobo Torán, Marc Herbstritt
2016 Dagstuhl Reports  
This report documents the program and the outcomes of Dagstuhl Seminar 15511 "The Graph Isomorphism Problem".  ...  A highlight of the conference was the presentation of a new quasi-polynomial time algorithm for the Graph Isomorphism Problem, providing the first improvement since 1983.  ...  An efficient characterization of this class of graphs has been obtained recently in [4] and [5]. Using the last result, we prove that all CR-definable graphs are compact.  ... 
doi:10.4230/dagrep.5.12.1 dblp:journals/dagstuhl-reports/BabaiDST15 fatcat:meig4kvcgffnjiq5wpa6wjs6am

The Weisfeiler-Leman dimension of planar graphs is at most 3

Sandra Kiefer, Ilia Ponomarenko, Pascal Schweitzer
2017 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)  
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.  ...  Weisfeiler-Leman algorithm distinguishes all non-isomorphic graphs in G.  ... 
doi:10.1109/lics.2017.8005107 dblp:conf/lics/KieferPS17 fatcat:ubfsasbpxnba5e3imfqhgfli4i

On the Weisfeiler-Leman Dimension of Finite Groups [article]

Jendrik Brachter, Pascal Schweitzer
2021 arXiv   pre-print
Using graphs of high Weisfeiler-Leman dimension, we construct highly similar but non-isomorphic groups with equal Θ(√(log n))-subgroup-profiles, which nevertheless have Weisfeiler-Leman dimension 3.  ...  The results indicate that the Weisfeiler-Leman algorithm can be more effective in distinguishing groups than in distinguishing graphs based on similar combinatorial constructions.  ...  These groups are nevertheless identified by the 3-dimensional Weisfeiler Leman algorithm.  ... 
arXiv:2003.13745v2 fatcat:x67yohs4cnbsjk2e2kkwjvwe5m

word2vec, node2vec, graph2vec, X2vec: Towards a Theory of Vector Embeddings of Structured Data [article]

Martin Grohe
2020 arXiv   pre-print
Vector representations of graphs and relational structures, whether hand-crafted feature vectors or learned representations, enable us to apply standard data analysis and machine learning techniques to  ...  [94] under the name Weisfeiler-Leman subtree kernel. They also introduce variants such as a Weisfeiler-Leman shortest path kernel.  ...  In Section 3, we introduce the Weisfeiler-Leman algorithm.  ... 
arXiv:2003.12590v1 fatcat:ak4ue5wuqzdzbm3bu5tfziy6wy

SpeqNets: Sparsity-aware Permutation-equivariant Graph Networks [article]

Christopher Morris, Gaurav Rattan, Sandra Kiefer, Siamak Ravanbakhsh
2022 arXiv   pre-print
They either operate on k-order tensors or consider all k-node subgraphs, implying an exponential dependence on k in memory requirements, and do not adapt to the sparsity of the graph.  ...  networks do not scale to large graphs.  ...  Here, graph kernels based on the 1-dimensional Weisfeiler-Leman algorithm (1-WL) [Weisfeiler and Leman, 1968 ], a simple heuristic for the graph isomorphism problem, and corresponding GNNs [Morris et  ... 
arXiv:2203.13913v3 fatcat:bazb6yobm5cfdmiicex7r43qba

A Study of Weisfeiler-Leman Colorings on Planar Graphs [article]

Sandra Kiefer, Daniel Neuen
2022 arXiv   pre-print
The Weisfeiler-Leman (WL) algorithm is a combinatorial procedure that computes colorings on graphs, which can often be used to detect their (non-)isomorphism.  ...  In particular, the graphs from case (a) are identified by 2-WL.  ...  WL algorithm suffices to identify all planar graphs in a logarithmic number of refinement rounds [25] , extending previous results for 3-connected planar graphs [44] .  ... 
arXiv:2206.10557v1 fatcat:zmgfbrowczak3msvfe5vvpjqgy

Weisfeiler and Leman Go Infinite: Spectral and Combinatorial Pre-Colorings [article]

Or Feldman, Amit Boyarski, Shai Feldman, Dani Kogan, Avi Mendelson, Chaim Baskin
2022 arXiv   pre-print
., obtained via the Weisfeiler-Leman (WL) test) and spectral invariants.  ...  Graph isomorphism testing is usually approached via the comparison of graph invariants.  ...  The weisfeiler-leman dimension of planar graphs is at most . Journal of the ACM (JACM), ( ): -, .  ... 
arXiv:2201.13410v2 fatcat:nqvzk544szd4zhpdtikavrkj3a

Near-Optimal Lower Bounds on Quantifier Depth and Weisfeiler-Leman Refinement Steps [article]

Christoph Berkholz, Jakob Nordström
2016 arXiv   pre-print
Our trade-offs also apply to first-order counting logic, and by the known connection to the k-dimensional Weisfeiler--Leman algorithm imply near-optimal lower bounds on the number of refinement iterations  ...  The first author wishes to acknowledge useful feedback from the participants of Dagstuhl Seminar 15511 The Graph Isomorphism Problem.  ...  Weisfeiler-Leman has also been used as a subroutine in algorithms that solve graph isomorphism on all graphs.  ... 
arXiv:1608.08704v2 fatcat:zwcbr43z2vglviqttnobu2jkyu

Graph Isomorphism for unit square graphs [article]

Daniel Neuen
2016 arXiv   pre-print
An interesting family of graph classes arises from intersection graphs of geometric objects.  ...  In this work we show that the Graph Isomorphism Problem for unit square graphs, intersection graphs of axis-parallel unit squares in the plane, can be solved in polynomial time.  ...  Since k-dimensional Weisfeiler-Leman identifies all interval graphs this is true for all v ∈ X.  ... 
arXiv:1602.08371v2 fatcat:elacbqko5vfnnb3z7tdiwj2wni

Recent Advances on the Graph Isomorphism Problem [article]

Martin Grohe, Daniel Neuen
2021 arXiv   pre-print
A second focus will be the combinatorial Weisfeiler-Leman algorithm.  ...  We give an overview of recent advances on the graph isomorphism problem.  ...  An early breakthrough was Hopcroft and Tarjan's O(n log n) isomorphism test for planar graphs [45] .  ... 
arXiv:2011.01366v2 fatcat:drrbl2cx3zabtkxo75f6nc5jlq

A Study of Weisfeiler-Leman Colorings on Planar Graphs

Sandra Kiefer, Daniel Neuen, Mikołaj Bojańczyk, Emanuela Merelli, David P. Woodruff
2022
The Weisfeiler-Leman (WL) algorithm is a combinatorial procedure that computes colorings on graphs, which can often be used to detect their (non-)isomorphism.  ...  In particular, the graphs from case (a) are identified by 2-WL.  ...  Moreover, it was recently shown that a constant dimension of the WL algorithm suffices to identify all planar graphs in a logarithmic number of refinement rounds [23] , extending previous results for  ... 
doi:10.4230/lipics.icalp.2022.81 fatcat:awelidk5nng5bbeyj7oyzv2aji
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