9 Search Results for "Adamaszek, Michal"

Document
DOI: 10.4230/LIPIcs.SoCG.2026.82
A Persistent Version of Latschev’s Theorem

Authors: Steve Oudot and Lukas Waas

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
Latschev’s theorem provides sufficient conditions on a metric space M and δ > 0 for the homotopy type of M to agree with that of the Vietoris-Rips complex ℛ^δ(𝕄) of any nearby space 𝕄 in the Gromov-Hausdorff distance. We prove a persistent version of this theorem, providing sufficient conditions on a pair (M, f:M → ℝ^N) and δ > 0 for the persistent homotopy type of the sublevel set filtration of (M, f) to be interleaved with that of the function-Rips complex ℛ^δ(𝕄^•) of any nearby pair (𝕄, 𝕗). In particular, our result answers a longstanding question on the related topic of estimating sublevel set persistent homology from finite point samples.
Cite as

Steve Oudot and Lukas Waas. A Persistent Version of Latschev’s Theorem. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 82:1-82:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{oudot_et_al:LIPIcs.SoCG.2026.82,
  author =	{Oudot, Steve and Waas, Lukas},
  title =	{{A Persistent Version of Latschev’s Theorem}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{82:1--82:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.82},
  URN =		{urn:nbn:de:0030-drops-258891},
  doi =		{10.4230/LIPIcs.SoCG.2026.82},
  annote =	{Keywords: Topological data analysis (TDA), metric geometry, Vietoris-Rips complex, homotopy theory, multi-parameter persistent homology}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.93
Simplicial Approximation to CW Complexes with Spherical Delaunay Triangulations

Authors: Raphaël Tinarrage

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
Simplicial approximation provides a framework for constructing simplicial complexes that are homotopy equivalent to a given manifold, provided a CW structure is explicitly known. However, its conventional implementation quickly becomes intractable on a computer: barycentric subdivision produces poorly shaped simplices, and the star condition introduces many vertices. To address these limitations, this article develops a subdivision scheme based on spherical Delaunay triangulations, which attains better refinement properties than barycentric subdivisions. Moreover, the star condition is reframed as two independent problems, one geometric and the other combinatorial, respectively tackled in the language of locally equiconnected spaces and the list homomorphism problem, allowing an exponential reduction in the number of vertices. Via a prototype implementation, we obtain simplicial complexes homotopy equivalent to Grassmannians and Stiefel manifolds up to dimension 5.
Cite as

Raphaël Tinarrage. Simplicial Approximation to CW Complexes with Spherical Delaunay Triangulations. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 93:1-93:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{tinarrage:LIPIcs.SoCG.2026.93,
  author =	{Tinarrage, Rapha\"{e}l},
  title =	{{Simplicial Approximation to CW Complexes with Spherical Delaunay Triangulations}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{93:1--93:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.93},
  URN =		{urn:nbn:de:0030-drops-258991},
  doi =		{10.4230/LIPIcs.SoCG.2026.93},
  annote =	{Keywords: Triangulation of manifolds, Simplicial approximation, CW complexes, Delaunay complexes, List homomorphism problem, Topological Data Analysis}
}
Document
DOI: 10.4230/LIPIcs.WADS.2025.17
Dynamic Streaming Algorithms for Geometric Independent Set

Authors: Timothy M. Chan and Yuancheng Yu

Published in: LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025)

Abstract
We present the first space-efficient, fully dynamic streaming algorithm for computing a constant-factor approximation of the maximum independent set size of n axis-aligned rectangles in two dimensions. For an arbitrarily small constant δ > 0, our algorithm obtains an O((1/δ)²) approximation and requires O(U^δ polylog n) space and update time with high probability, assuming that coordinates are integers bounded by U. We also obtain a similar result for fat objects in any constant dimension. This extends recent non-streaming algorithms by Bhore and Chan from SODA'25, and also greatly extends previous streaming results, which were limited to special types of geometric objects such as one-dimensional intervals and unit disks.
Cite as

Timothy M. Chan and Yuancheng Yu. Dynamic Streaming Algorithms for Geometric Independent Set. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 17:1-17:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chan_et_al:LIPIcs.WADS.2025.17,
  author =	{Chan, Timothy M. and Yu, Yuancheng},
  title =	{{Dynamic Streaming Algorithms for Geometric Independent Set}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{17:1--17:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.17},
  URN =		{urn:nbn:de:0030-drops-242481},
  doi =		{10.4230/LIPIcs.WADS.2025.17},
  annote =	{Keywords: Geometric Independent Set, Dynamic Streaming Algorithms}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.14
Extremal Betti Numbers and Persistence in Flag Complexes

Authors: Lies Beers and Magnus Bakke Botnan

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
We investigate several problems concerning extremal Betti numbers and persistence in filtrations of flag complexes. For graphs on n vertices, we show that β_k(X(G)) is maximal when G = 𝒯_{n,k+1}, the Turán graph on k+1 partition classes, where X(G) denotes the flag complex of G. Building on this, we construct an edgewise (one edge at a time) filtration 𝒢 = G₁ ⊆ ⋯ ⊆ 𝒯_{n,k+1} for which β_k(X(G_i)) is maximal for all graphs on n vertices and i edges. Moreover, the persistence barcode ℬ_k(X(G)) achieves a maximal number of intervals, and total persistence, among all edgewise filtrations with |E(𝒯_{n,k+1})| edges. For k = 1, we consider edgewise filtrations of the complete graph K_n. We show that the maximal number of intervals in the persistence barcode is obtained precisely when G_{⌈n/2⌉ ⋅ ⌊n/2⌋} = 𝒯_{n,2}. Among such filtrations, we characterize those achieving maximal total persistence. We further show that no filtration can optimize β₁(X(G_i)) for all i, and conjecture that our filtrations maximize the total persistence over all edgewise filtrations of K_n.
Cite as

Lies Beers and Magnus Bakke Botnan. Extremal Betti Numbers and Persistence in Flag Complexes. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{beers_et_al:LIPIcs.SoCG.2025.14,
  author =	{Beers, Lies and Bakke Botnan, Magnus},
  title =	{{Extremal Betti Numbers and Persistence in Flag Complexes}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{14:1--14:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.14},
  URN =		{urn:nbn:de:0030-drops-231668},
  doi =		{10.4230/LIPIcs.SoCG.2025.14},
  annote =	{Keywords: Topological data analysis, Extremal graph theory}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.24
An 11/6-Approximation Algorithm for Vertex Cover on String Graphs

Authors: Édouard Bonnet and Paweł Rzążewski

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
We present a 1.8334-approximation algorithm for Vertex Cover on string graphs given with a representation, which takes polynomial time in the size of the representation; the exact approximation factor is 11/6. Recently, the barrier of 2 was broken by Lokshtanov, Panolan, Saurabh, Xue, and Zehavi [SoGC '24] with a 1.9999-approximation algorithm. Thus we increase by three orders of magnitude the distance of the approximation ratio to the trivial bound of 2. Our algorithm is very simple. The intricacies reside in its analysis, where we mainly establish that string graphs without odd cycles of length at most 11 are 8-colorable. Previously, Chudnovsky, Scott, and Seymour [JCTB '21] showed that string graphs without odd cycles of length at most 7 are 80-colorable, and string graphs without odd cycles of length at most 5 have bounded chromatic number.
Cite as

Édouard Bonnet and Paweł Rzążewski. An 11/6-Approximation Algorithm for Vertex Cover on String Graphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bonnet_et_al:LIPIcs.SoCG.2025.24,
  author =	{Bonnet, \'{E}douard and Rz\k{a}\.{z}ewski, Pawe{\l}},
  title =	{{An 11/6-Approximation Algorithm for Vertex Cover on String Graphs}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{24:1--24:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.24},
  URN =		{urn:nbn:de:0030-drops-231764},
  doi =		{10.4230/LIPIcs.SoCG.2025.24},
  annote =	{Keywords: Approximation algorithms, string graphs, Vertex Cover, Coloring, odd girth}
}
Document
DOI: 10.4230/LIPIcs.ESA.2018.65
Quasi-Polynomial Time Approximation Schemes for Packing and Covering Problems in Planar Graphs

Authors: Michal Pilipczuk, Erik Jan van Leeuwen, and Andreas Wiese

Published in: LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

Abstract
We consider two optimization problems in planar graphs. In {Maximum Weight Independent Set of Objects} we are given a graph G and a family D of {objects}, each being a connected subgraph of G with a prescribed weight, and the task is to find a maximum-weight subfamily of D consisting of pairwise disjoint objects. In {Minimum Weight Distance Set Cover} we are given an edge-weighted graph G, two sets D,C of vertices of G, where vertices of D have prescribed weights, and a nonnegative radius r. The task is to find a minimum-weight subset of D such that every vertex of C is at distance at most r from some selected vertex. Via simple reductions, these two problems generalize a number of geometric optimization tasks, notably {Maximum Weight Independent Set} for polygons in the plane and {Weighted Geometric Set Cover} for unit disks and unit squares. We present {quasi-polynomial time approximation schemes} (QPTASs) for both of the above problems in planar graphs: given an accuracy parameter epsilon>0 we can compute a solution whose weight is within multiplicative factor of (1+epsilon) from the optimum in time 2^{poly(1/epsilon,log |D|)}* n^{O(1)}, where n is the number of vertices of the input graph. Our main technical contribution is to transfer the techniques used for recursive approximation schemes for geometric problems due to Adamaszek, Har-Peled, and Wiese [Adamaszek and Wiese, 2013; Adamaszek and Wiese, 2014; Sariel Har-Peled, 2014] to the setting of planar graphs. In particular, this yields a purely combinatorial viewpoint on these methods.
Cite as

Michal Pilipczuk, Erik Jan van Leeuwen, and Andreas Wiese. Quasi-Polynomial Time Approximation Schemes for Packing and Covering Problems in Planar Graphs. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 65:1-65:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{pilipczuk_et_al:LIPIcs.ESA.2018.65,
  author =	{Pilipczuk, Michal and van Leeuwen, Erik Jan and Wiese, Andreas},
  title =	{{Quasi-Polynomial Time Approximation Schemes for Packing and Covering Problems in Planar Graphs}},
  booktitle =	{26th Annual European Symposium on Algorithms (ESA 2018)},
  pages =	{65:1--65:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-081-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{112},
  editor =	{Azar, Yossi and Bast, Hannah and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.65},
  URN =		{urn:nbn:de:0030-drops-95282},
  doi =		{10.4230/LIPIcs.ESA.2018.65},
  annote =	{Keywords: QPTAS, planar graphs, Voronoi diagram}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2018.3
Vietoris-Rips and Cech Complexes of Metric Gluings

Authors: Michal Adamaszek, Henry Adams, Ellen Gasparovic, Maria Gommel, Emilie Purvine, Radmila Sazdanovic, Bei Wang, Yusu Wang, and Lori Ziegelmeier

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract
We study Vietoris-Rips and Cech complexes of metric wedge sums and metric gluings. We show that the Vietoris-Rips (resp. Cech) complex of a wedge sum, equipped with a natural metric, is homotopy equivalent to the wedge sum of the Vietoris-Rips (resp. Cech) complexes. We also provide generalizations for certain metric gluings, i.e. when two metric spaces are glued together along a common isometric subset. As our main example, we deduce the homotopy type of the Vietoris-Rips complex of two metric graphs glued together along a sufficiently short path. As a result, we can describe the persistent homology, in all homological dimensions, of the Vietoris-Rips complexes of a wide class of metric graphs.
Cite as

Michal Adamaszek, Henry Adams, Ellen Gasparovic, Maria Gommel, Emilie Purvine, Radmila Sazdanovic, Bei Wang, Yusu Wang, and Lori Ziegelmeier. Vietoris-Rips and Cech Complexes of Metric Gluings. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{adamaszek_et_al:LIPIcs.SoCG.2018.3,
  author =	{Adamaszek, Michal and Adams, Henry and Gasparovic, Ellen and Gommel, Maria and Purvine, Emilie and Sazdanovic, Radmila and Wang, Bei and Wang, Yusu and Ziegelmeier, Lori},
  title =	{{Vietoris-Rips and Cech Complexes of Metric Gluings}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{3:1--3:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.3},
  URN =		{urn:nbn:de:0030-drops-87162},
  doi =		{10.4230/LIPIcs.SoCG.2018.3},
  annote =	{Keywords: Vietoris-Rips and Cech complexes, metric space gluings and wedge sums, metric graphs, persistent homology}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2017.42
Approximation and Parameterized Algorithms for Geometric Independent Set with Shrinking

Authors: Michal Pilipczuk, Erik Jan van Leeuwen, and Andreas Wiese

Published in: LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Abstract
Consider the Maximum Weight Independent Set problem for rectangles: given a family of weighted axis-parallel rectangles in the plane, find a maximum-weight subset of non-overlapping rectangles. The problem is notoriously hard both in the approximation and in the parameterized setting. The best known polynomial-time approximation algorithms achieve super-constant approximation ratios [Chalermsook & Chuzhoy, Proc. SODA 2009; Chan & Har-Peled, Discrete & Comp. Geometry, 2012], even though there is a (1+epsilon)-approximation running in quasi-polynomial time [Adamaszek & Wiese, Proc. FOCS 2013; Chuzhoy & Ene, Proc. FOCS 2016]. When parameterized by the target size of the solution, the problem is W[1]-hard even in the unweighted setting [Marx, ESA 2005]. To achieve tractability, we study the following shrinking model: one is allowed to shrink each input rectangle by a multiplicative factor 1-delta for some fixed delta > 0, but the performance is still compared against the optimal solution for the original, non-shrunk instance. We prove that in this regime, the problem admits an EPTAS with running time f(epsilon,delta) n^{O(1)}, and an FPT algorithm with running time f(k,delta) n^{O(1)}, in the setting where a maximum-weight solution of size at most k is to be computed. This improves and significantly simplifies a PTAS given earlier for this problem [Adamaszek, Chalermsook & Wiese, Proc. APPROX/RANDOM 2015], and provides the first parameterized results for the shrinking model. Furthermore, we explore kernelization in the shrinking model, by giving efficient kernelization procedures for several variants of the problem when the input rectangles are squares.
Cite as

Michal Pilipczuk, Erik Jan van Leeuwen, and Andreas Wiese. Approximation and Parameterized Algorithms for Geometric Independent Set with Shrinking. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 42:1-42:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{pilipczuk_et_al:LIPIcs.MFCS.2017.42,
  author =	{Pilipczuk, Michal and van Leeuwen, Erik Jan and Wiese, Andreas},
  title =	{{Approximation and Parameterized Algorithms for Geometric Independent Set with Shrinking}},
  booktitle =	{42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)},
  pages =	{42:1--42:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-046-0},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{83},
  editor =	{Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.42},
  URN =		{urn:nbn:de:0030-drops-80917},
  doi =		{10.4230/LIPIcs.MFCS.2017.42},
  annote =	{Keywords: Combinatorial optimization, Approximation algorithms, Fixed-parameter algorithms}
}
Document
DOI: 10.4230/LIPIcs.STACS.2010.2442
Large-Girth Roots of Graphs

Authors: Anna Adamaszek and Michal Adamaszek

Published in: LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

Abstract
We study the problem of recognizing graph powers and computing roots of graphs. We provide a polynomial time recognition algorithm for $r$-th powers of graphs of girth at least $2r+3$, thus improving a bound conjectured by Farzad et al. (STACS 2009). Our algorithm also finds all $r$-th roots of a given graph that have girth at least $2r+3$ and no degree one vertices, which is a step towards a recent conjecture of Levenshtein that such root should be unique. On the negative side, we prove that recognition becomes an NP-complete problem when the bound on girth is about twice smaller. Similar results have so far only been attempted for $r=2,3$.
Cite as

Anna Adamaszek and Michal Adamaszek. Large-Girth Roots of Graphs. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 35-46, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{adamaszek_et_al:LIPIcs.STACS.2010.2442,
  author =	{Adamaszek, Anna and Adamaszek, Michal},
  title =	{{Large-Girth Roots of Graphs}},
  booktitle =	{27th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{35--46},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-16-3},
  ISSN =	{1868-8969},
  year =	{2010},
  volume =	{5},
  editor =	{Marion, Jean-Yves and Schwentick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2442},
  URN =		{urn:nbn:de:0030-drops-24429},
  doi =		{10.4230/LIPIcs.STACS.2010.2442},
  annote =	{Keywords: Graph roots, Graph powers, NP-completeness, Recognition algorithms}
}
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