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**Published in:** LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

We consider a hierarchy of graph invariants that naturally extends the spectral invariants defined by Fürer (Lin. Alg. Appl. 2010) based on the angles formed by the set of standard basis vectors and their projections onto eigenspaces of the adjacency matrix. We provide a purely combinatorial characterization of this hierarchy in terms of the walk counts. This allows us to give a complete answer to Fürer’s question about the strength of his invariants in distinguishing non-isomorphic graphs in comparison to the 2-dimensional Weisfeiler-Leman algorithm, extending the recent work of Rattan and Seppelt (SODA 2023). As another application of the characterization, we prove that almost all graphs are determined up to isomorphism in terms of the spectrum and the angles, which is of interest in view of the long-standing open problem whether almost all graphs are determined by their eigenvalues alone. Finally, we describe the exact relationship between the hierarchy and the Weisfeiler-Leman algorithms for small dimensions, as also some other important spectral characteristics of a graph such as the generalized and the main spectra.

V. Arvind, Frank Fuhlbrück, Johannes Köbler, and Oleg Verbitsky. On a Hierarchy of Spectral Invariants for Graphs. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{arvind_et_al:LIPIcs.STACS.2024.6, author = {Arvind, V. and Fuhlbr\"{u}ck, Frank and K\"{o}bler, Johannes and Verbitsky, Oleg}, title = {{On a Hierarchy of Spectral Invariants for Graphs}}, booktitle = {41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)}, pages = {6:1--6:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-311-9}, ISSN = {1868-8969}, year = {2024}, volume = {289}, editor = {Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.6}, URN = {urn:nbn:de:0030-drops-197166}, doi = {10.4230/LIPIcs.STACS.2024.6}, annote = {Keywords: Graph Isomorphism, spectra of graphs, combinatorial refinement, strongly regular graphs} }

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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

We show that a canonical labeling of a random n-vertex graph can be obtained by assigning to each vertex x the triple (w₁(x),w₂(x),w₃(x)), where w_k(x) is the number of walks of length k starting from x. This takes time 𝒪(n²), where n² is the input size, by using just two matrix-vector multiplications. The linear-time canonization of a random graph is the classical result of Babai, Erdős, and Selkow. For this purpose they use the well-known combinatorial color refinement procedure, and we make a comparative analysis of the two algorithmic approaches.

Oleg Verbitsky and Maksim Zhukovskii. Canonization of a Random Graph by Two Matrix-Vector Multiplications. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 100:1-100:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{verbitsky_et_al:LIPIcs.ESA.2023.100, author = {Verbitsky, Oleg and Zhukovskii, Maksim}, title = {{Canonization of a Random Graph by Two Matrix-Vector Multiplications}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {100:1--100:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.100}, URN = {urn:nbn:de:0030-drops-187536}, doi = {10.4230/LIPIcs.ESA.2023.100}, annote = {Keywords: Graph Isomorphism, canonical labeling, random graphs, walk matrix, color refinement, linear time} }

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**Published in:** LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

It is well known that the isomorphism problem for vertex-colored graphs with color multiplicity at most 3 is solvable by the classical 2-dimensional Weisfeiler-Leman algorithm (2-WL). On the other hand, the prominent Cai-Fürer-Immerman construction shows that even the multidimensional version of the algorithm does not suffice for graphs with color multiplicity 4. We give an efficient decision procedure that, given a graph G of color multiplicity 4, recognizes whether or not G is identifiable by 2-WL, that is, whether or not 2-WL distinguishes G from any non-isomorphic graph. In fact, we solve the more general problem of recognizing whether or not a given coherent configuration of maximum fiber size 4 is separable. This extends our recognition algorithm to directed graphs of color multiplicity 4 with colored edges.
Our decision procedure is based on an explicit description of the class of graphs with color multiplicity 4 that are not identifiable by 2-WL. The Cai-Fürer-Immerman graphs of color multiplicity 4 distinctly appear here as a natural subclass, which demonstrates that the Cai-Fürer-Immerman construction is not ad hoc. Our classification reveals also other types of graphs that are hard for 2-WL. One of them arises from patterns known as (n₃)-configurations in incidence geometry.

Frank Fuhlbrück, Johannes Köbler, and Oleg Verbitsky. Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 43:1-43:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fuhlbruck_et_al:LIPIcs.STACS.2020.43, author = {Fuhlbr\"{u}ck, Frank and K\"{o}bler, Johannes and Verbitsky, Oleg}, title = {{Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {43:1--43:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-140-5}, ISSN = {1868-8969}, year = {2020}, volume = {154}, editor = {Paul, Christophe and Bl\"{a}ser, Markus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.43}, URN = {urn:nbn:de:0030-drops-119046}, doi = {10.4230/LIPIcs.STACS.2020.43}, annote = {Keywords: Graph Isomorphism, Weisfeiler-Leman Algorithm, Cai-F\"{u}rer-Immerman Graphs, coherent Configurations} }

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**Published in:** LIPIcs, Volume 82, 26th EACSL Annual Conference on Computer Science Logic (CSL 2017)

Given a graph F, let I(F) be the class of graphs containing F
as an induced subgraph. Let W[F] denote the minimum k such that
I(F) is definable in k-variable first-order logic. The recognition
problem of I(F), known as Induced Subgraph Isomorphism (for the pattern graph F), is solvable in time O(n^{W[F]}). Motivated by this fact, we are interested in determining or estimating the value of W[F]. Using Olariu's characterization of paw-free graphs, we show that I(K_3+e) is definable by a first-order sentence of quantifier depth 3, where K_3+e denotes the paw graph. This provides an example of a graph F with W[F] strictly less than the number of vertices in F.
On the other hand, we prove that W[F]=4 for all F on 4 vertices
except the paw graph and its complement. If F is a graph on t vertices, we prove a general lower bound W[F]>(1/2-o(1))t, where the function in the little-o notation approaches 0 as t increases. This bound holds true even for a related parameter W^*[F], which is
defined as the minimum k such that I(F) is definable in the k-variable
infinitary logic. We show that W^*[F] can be strictly less than W[F]. Specifically, W^*[P_4]=3 for P_4 being the path graph on 4 vertices.

Oleg Verbitsky and Maksim Zhukovskii. On the First-Order Complexity of Induced Subgraph Isomorphism. In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 82, pp. 40:1-40:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{verbitsky_et_al:LIPIcs.CSL.2017.40, author = {Verbitsky, Oleg and Zhukovskii, Maksim}, title = {{On the First-Order Complexity of Induced Subgraph Isomorphism}}, booktitle = {26th EACSL Annual Conference on Computer Science Logic (CSL 2017)}, pages = {40:1--40:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-045-3}, ISSN = {1868-8969}, year = {2017}, volume = {82}, editor = {Goranko, Valentin and Dam, Mads}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2017.40}, URN = {urn:nbn:de:0030-drops-76841}, doi = {10.4230/LIPIcs.CSL.2017.40}, annote = {Keywords: the induced subgraph isomorphism problem, descriptive and computational complexity, finite-variable first-order logic, quantifier depth and variable w} }

Document

**Published in:** LIPIcs, Volume 23, Computer Science Logic 2013 (CSL 2013)

Given two structures G and H distinguishable in FO^k (first-order logic with k variables), let A^k(G,H) denote the minimum alternation depth of a FO^k formula distinguishing G from H. Let A^k(n) be the maximum value of A^k(G,H) over n-element structures. We prove the strictness of the quantifier alternation hierarchy of FO^2 in a strong quantitative form, namely A^2(n) >= n/8-2, which is tight up to a constant factor. For each k >= 2, it holds that A^k(n) > log_(k+1) n-2 even over colored trees, which is also tight up to a constant factor if k >= 3. For k >= 3 the last lower bound holds also over uncolored trees, while the alternation hierarchy of FO^2 collapses even over all uncolored graphs.
We also show examples of colored graphs G and H on n vertices that can be distinguished in FO^2 much more succinctly if the alternation number is increased just by one: while in Sigma_i it is possible to distinguish G from H with bounded quantifier depth, in Pi_i this requires quantifier depth Omega(n2). The quadratic lower bound is best possible here because, if G and H can be distinguished in FO^k with i quantifier alternations, this can be done with quantifier depth n^(2k-2).

Christoph Berkholz, Andreas Krebs, and Oleg Verbitsky. Bounds for the quantifier depth in finite-variable logics: Alternation hierarchy. In Computer Science Logic 2013 (CSL 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 23, pp. 61-80, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{berkholz_et_al:LIPIcs.CSL.2013.61, author = {Berkholz, Christoph and Krebs, Andreas and Verbitsky, Oleg}, title = {{Bounds for the quantifier depth in finite-variable logics: Alternation hierarchy}}, booktitle = {Computer Science Logic 2013 (CSL 2013)}, pages = {61--80}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-60-6}, ISSN = {1868-8969}, year = {2013}, volume = {23}, editor = {Ronchi Della Rocca, Simona}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2013.61}, URN = {urn:nbn:de:0030-drops-41907}, doi = {10.4230/LIPIcs.CSL.2013.61}, annote = {Keywords: Alternation hierarchy, finite-variable logic} }

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**Published in:** LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

We present a logspace algorithm that constructs a canonical intersection model for a given proper circular-arc graph, where canonical means that isomorphic graphs receive identical models. This implies that the recognition and the isomorphism problems for these graphs are solvable in logspace. For the broader class of concave-round graphs, which still possess (not necessarily proper) circular-arc models, we show that a canonical circular-arc model can also be constructed in logspace. As a building block for these results, we design a logspace algorithm for computing canonical circular-arc models of circular-arc hypergraphs; this important class of hypergraphs corresponds to matrices with the circular ones property. Furthermore, we consider the Star System Problem that consists in reconstructing a graph from its closed neighborhood hypergraph. We show that this problem is solvable in logarithmic space for the classes of proper circular-arc, concave-round, and co-convex graphs.

Johannes Köbler, Sebastian Kuhnert, and Oleg Verbitsky. Solving the Canonical Representation and Star System Problems for Proper Circular-Arc Graphs in Logspace. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 387-399, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{kobler_et_al:LIPIcs.FSTTCS.2012.387, author = {K\"{o}bler, Johannes and Kuhnert, Sebastian and Verbitsky, Oleg}, title = {{Solving the Canonical Representation and Star System Problems for Proper Circular-Arc Graphs in Logspace}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {387--399}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-47-7}, ISSN = {1868-8969}, year = {2012}, volume = {18}, editor = {D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.387}, URN = {urn:nbn:de:0030-drops-38757}, doi = {10.4230/LIPIcs.FSTTCS.2012.387}, annote = {Keywords: Proper circular-arc graphs, graph isomorphism, canonization, circular ones property, logspace complexity} }

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