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Documents authored by Tóth, Géza

Document
DOI: 10.4230/LIPIcs.ESA.2026.28
Rerouting Curves on Surfaces

Authors: Timo Brand, Stefan Felsner, Henry Förster, Stephen Kobourov, Anna Lubiw, Yoshio Okamoto, János Pach, Csaba D. Tóth, Géza Tóth, Torsten Ueckerdt, and Pavel Valtr

Published in: LIPIcs, Volume 388, 34th Annual European Symposium on Algorithms (ESA 2026)

Abstract
We study the problem of reconfiguring a crossing-free embedding of a graph on a surface, with edges represented as curves, into another crossing-free embedding of the same graph on the same surface with the same fixed vertex positions. In this process, we reroute one edge at a time while maintaining crossing-free intermediate embeddings. This problem was introduced by Ito et al. [TALG 2025], who showed that even if the graph is a matching of two edges, reconfiguration is not always possible in the plane, but is always possible on the torus. For matchings of two or more edges, they gave a necessary and sufficient condition for reconfigurable embeddings in the plane, but not on the torus. Our main result is that for matchings, trees and forests, reconfiguration is always possible on the torus, and consequently, on any orientable surface of genus at least one. In addition, we provide sufficient conditions for reconfiguration on orientable surfaces of genus at least one and in the projective plane. For more general graphs, we show that reconfiguration is not always possible.
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Timo Brand, Stefan Felsner, Henry Förster, Stephen Kobourov, Anna Lubiw, Yoshio Okamoto, János Pach, Csaba D. Tóth, Géza Tóth, Torsten Ueckerdt, and Pavel Valtr. Rerouting Curves on Surfaces. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 28:1-28:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{brand_et_al:LIPIcs.ESA.2026.28,
  author =	{Brand, Timo and Felsner, Stefan and F\"{o}rster, Henry and Kobourov, Stephen and Lubiw, Anna and Okamoto, Yoshio and Pach, J\'{a}nos and T\'{o}th, Csaba D. and T\'{o}th, G\'{e}za and Ueckerdt, Torsten and Valtr, Pavel},
  title =	{{Rerouting Curves on Surfaces}},
  booktitle =	{34th Annual European Symposium on Algorithms (ESA 2026)},
  pages =	{28:1--28:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-445-1},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{388},
  editor =	{Bille, Philip and Pettie, Seth and Storandt, Sabine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.28},
  URN =		{urn:nbn:de:0030-drops-271641},
  doi =		{10.4230/LIPIcs.ESA.2026.28},
  annote =	{Keywords: Combinatorial reconfiguration, orientable surface, non-orientable surface, rerouting}
}
Document
DOI: 10.4230/LIPIcs.GD.2024.6
1-Planar Unit Distance Graphs

Authors: Panna Gehér and Géza Tóth

Published in: LIPIcs, Volume 320, 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

Abstract
A matchstick graph is a plane graph with edges drawn as unit distance line segments. This class of graphs was introduced by Harborth who conjectured that a matchstick graph on n vertices can have at most ⌊3n-√{12n-3}⌋ edges. Recently his conjecture was settled by Lavollée and Swanepoel. In this paper we consider 1-planar unit distance graphs. We say that a graph is a 1-planar unit distance graph if it can be drawn in the plane such that all edges are drawn as unit distance line segments while each of them are involved in at most one crossing. We show that such graphs on n vertices can have at most 3n-∜{n}/10 edges.
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Panna Gehér and Géza Tóth. 1-Planar Unit Distance Graphs. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 6:1-6:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{geher_et_al:LIPIcs.GD.2024.6,
  author =	{Geh\'{e}r, Panna and T\'{o}th, G\'{e}za},
  title =	{{1-Planar Unit Distance Graphs}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{6:1--6:9},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-343-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{320},
  editor =	{Felsner, Stefan and Klein, Karsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2024.6},
  URN =		{urn:nbn:de:0030-drops-212900},
  doi =		{10.4230/LIPIcs.GD.2024.6},
  annote =	{Keywords: unit distance graph, 1-planar, matchstick graph}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.56
Disjointness Graphs of Short Polygonal Chains

Authors: János Pach, Gábor Tardos, and Géza Tóth

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
The disjointness graph of a set system is a graph whose vertices are the sets, two being connected by an edge if and only if they are disjoint. It is known that the disjointness graph G of any system of segments in the plane is χ-bounded, that is, its chromatic number χ(G) is upper bounded by a function of its clique number ω(G). Here we show that this statement does not remain true for systems of polygonal chains of length 2. We also construct systems of polygonal chains of length 3 such that their disjointness graphs have arbitrarily large girth and chromatic number. In the opposite direction, we show that the class of disjointness graphs of (possibly self-intersecting) 2-way infinite polygonal chains of length 3 is χ-bounded: for every such graph G, we have χ(G) ≤ (ω(G))³+ω(G).
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János Pach, Gábor Tardos, and Géza Tóth. Disjointness Graphs of Short Polygonal Chains. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 56:1-56:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{pach_et_al:LIPIcs.SoCG.2022.56,
  author =	{Pach, J\'{a}nos and Tardos, G\'{a}bor and T\'{o}th, G\'{e}za},
  title =	{{Disjointness Graphs of Short Polygonal Chains}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{56:1--56:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.56},
  URN =		{urn:nbn:de:0030-drops-160645},
  doi =		{10.4230/LIPIcs.SoCG.2022.56},
  annote =	{Keywords: chi-bounded, disjointness graph}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2018.65
A Crossing Lemma for Multigraphs

Authors: János Pach and Géza Tóth

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

Abstract
Let G be a drawing of a graph with n vertices and e>4n edges, in which no two adjacent edges cross and any pair of independent edges cross at most once. According to the celebrated Crossing Lemma of Ajtai, Chvátal, Newborn, Szemerédi and Leighton, the number of crossings in G is at least c{e^3 over n^2}, for a suitable constant c>0. In a seminal paper, Székely generalized this result to multigraphs, establishing the lower bound c{e^3 over mn^2}, where m denotes the maximum multiplicity of an edge in G. We get rid of the dependence on m by showing that, as in the original Crossing Lemma, the number of crossings is at least c'{e^3 over n^2} for some c'>0, provided that the "lens" enclosed by every pair of parallel edges in G contains at least one vertex. This settles a conjecture of Bekos, Kaufmann, and Raftopoulou.
Cite as

János Pach and Géza Tóth. A Crossing Lemma for Multigraphs. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 65:1-65:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{pach_et_al:LIPIcs.SoCG.2018.65,
  author =	{Pach, J\'{a}nos and T\'{o}th, G\'{e}za},
  title =	{{A Crossing Lemma for Multigraphs}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{65:1--65:13},
  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.2018.65},
  URN =		{urn:nbn:de:0030-drops-87781},
  doi =		{10.4230/LIPIcs.SoCG.2018.65},
  annote =	{Keywords: crossing number, Crossing Lemma, multigraph, separator theorem}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2017.59
Disjointness Graphs of Segments

Authors: János Pach, Gábor Tardos, and Géza Tóth

Published in: LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Abstract
The disjointness graph G=G(S) of a set of segments S in R^d, d>1 is a graph whose vertex set is S and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We prove that the chromatic number of G satisfies chi(G)<=omega(G)^4+omega(G)^3 where omega(G) denotes the clique number of G. It follows, that S has at least cn^{1/5} pairwise intersecting or pairwise disjoint elements. Stronger bounds are established for lines in space, instead of segments. We show that computing omega(G) and chi(G) for disjointness graphs of lines in space are NP-hard tasks. However, we can design efficient algorithms to compute proper colorings of G in which the number of colors satisfies the above upper bounds. One cannot expect similar results for sets of continuous arcs, instead of segments, even in the plane. We construct families of arcs whose disjointness graphs are triangle-free (omega(G)=2), but whose chromatic numbers are arbitrarily large.
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János Pach, Gábor Tardos, and Géza Tóth. Disjointness Graphs of Segments. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 59:1-59:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{pach_et_al:LIPIcs.SoCG.2017.59,
  author =	{Pach, J\'{a}nos and Tardos, G\'{a}bor and T\'{o}th, G\'{e}za},
  title =	{{Disjointness Graphs of Segments}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{59:1--59:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Aronov, Boris and Katz, Matthew J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.59},
  URN =		{urn:nbn:de:0030-drops-71960},
  doi =		{10.4230/LIPIcs.SoCG.2017.59},
  annote =	{Keywords: disjointness graph, chromatic number, clique number, chi-bounded}
}
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