Plane Spanning Trees in Edge-Colored Simple Drawings of Kn

Oswin Aichholzer; Michael Hoffmann; Johannes Obenaus; Rosna Paul; Daniel Perz; Nadja Seiferth; Birgit Vogtenhuber; Alexandra Weinberger

doi:10.1007/978-3-030-68766-3_37

Plane Spanning Trees in Edge-Colored Simple Drawings of Kn

Oswin Aichholzer, Michael Hoffmann, Johannes Obenaus, Rosna Paul, Daniel Perz, Nadja Seiferth, Birgit Vogtenhuber, Alexandra Weinberger

Institute of Software Technology (7160)

Research output: Chapter in Book/Report/Conference proceeding › Conference paper › peer-review

Abstract

Károlyi, Pach, and Tóth proved that every 2-edge-colored straight-line drawing of the complete graph contains a monochromatic plane spanning tree. It is open if this statement generalizes to other classes of drawings, specifically, to simple drawings of the complete graph. These are drawings where edges are represented by Jordan arcs, any two of which intersect at most once. We present two partial results towards such a generalization. First, we show that the statement holds for cylindrical simple drawings. (In a cylindrical drawing, all vertices are placed on two concentric circles and no edge crosses either circle.) Second, we introduce a relaxation of the problem in which the graph is k-edge-colored, and the target structure must be hypochromatic, that is, avoid (at least) one color class. In this setting, we show that every ⌈(n+5)/6⌉-edge-colored monotone simple drawing of Kn contains a hypochromatic plane spanning tree. (In a monotone drawing, every edge is represented as an x-monotone curve.)

Original language	English
Title of host publication	Graph Drawing and Network Visualization - 28th International Symposium, GD 2020, Revised Selected Papers
Editors	David Auber, Pavel Valtr
Pages	482-489
Number of pages	8
ISBN (Electronic)	978-3-030-68766-3
DOIs	https://doi.org/10.1007/978-3-030-68766-3_37
Publication status	Published - Feb 2021
Event	28th International Symposium on Graph Drawing and Network Visualization: Graph Drawing 2020 - Virtuell, Canada Duration: 16 Sept 2020 → 18 Sept 2020 https://gd2020.cs.ubc.ca/

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	12590 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	28th International Symposium on Graph Drawing and Network Visualization
Country/Territory	Canada
City	Virtuell
Period	16/09/20 → 18/09/20
Internet address	https://gd2020.cs.ubc.ca/

Keywords

Cylindrical drawing
Monotone drawing
Plane subdrawing
Simple drawing

ASJC Scopus subject areas

Theoretical Computer Science
Computer Science(all)

Fields of Expertise

Information, Communication & Computing

Access to Document

10.1007/978-3-030-68766-3_37

Cite this

Aichholzer, O., Hoffmann, M., Obenaus, J., Paul, R., Perz, D., Seiferth, N., Vogtenhuber, B., & Weinberger, A. (2021). Plane Spanning Trees in Edge-Colored Simple Drawings of Kn. In D. Auber, & P. Valtr (Eds.), Graph Drawing and Network Visualization - 28th International Symposium, GD 2020, Revised Selected Papers (pp. 482-489). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 12590 LNCS). https://doi.org/10.1007/978-3-030-68766-3_37

Plane Spanning Trees in Edge-Colored Simple Drawings of Kn. / Aichholzer, Oswin; Hoffmann, Michael; Obenaus, Johannes et al.
Graph Drawing and Network Visualization - 28th International Symposium, GD 2020, Revised Selected Papers. ed. / David Auber; Pavel Valtr. 2021. p. 482-489 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 12590 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference paper › peer-review

Aichholzer, O, Hoffmann, M, Obenaus, J, Paul, R, Perz, D, Seiferth, N, Vogtenhuber, B & Weinberger, A 2021, Plane Spanning Trees in Edge-Colored Simple Drawings of Kn. in D Auber & P Valtr (eds), Graph Drawing and Network Visualization - 28th International Symposium, GD 2020, Revised Selected Papers. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 12590 LNCS, pp. 482-489, 28th International Symposium on Graph Drawing and Network Visualization, Virtuell, Canada, 16/09/20. https://doi.org/10.1007/978-3-030-68766-3_37

Aichholzer O, Hoffmann M, Obenaus J, Paul R, Perz D, Seiferth N et al. Plane Spanning Trees in Edge-Colored Simple Drawings of Kn. In Auber D, Valtr P, editors, Graph Drawing and Network Visualization - 28th International Symposium, GD 2020, Revised Selected Papers. 2021. p. 482-489. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). Epub 2020 Sept. doi: 10.1007/978-3-030-68766-3_37

Aichholzer, Oswin ; Hoffmann, Michael ; Obenaus, Johannes et al. / Plane Spanning Trees in Edge-Colored Simple Drawings of Kn. Graph Drawing and Network Visualization - 28th International Symposium, GD 2020, Revised Selected Papers. editor / David Auber ; Pavel Valtr. 2021. pp. 482-489 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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AB - Károlyi, Pach, and Tóth proved that every 2-edge-colored straight-line drawing of the complete graph contains a monochromatic plane spanning tree. It is open if this statement generalizes to other classes of drawings, specifically, to simple drawings of the complete graph. These are drawings where edges are represented by Jordan arcs, any two of which intersect at most once. We present two partial results towards such a generalization. First, we show that the statement holds for cylindrical simple drawings. (In a cylindrical drawing, all vertices are placed on two concentric circles and no edge crosses either circle.) Second, we introduce a relaxation of the problem in which the graph is k-edge-colored, and the target structure must be hypochromatic, that is, avoid (at least) one color class. In this setting, we show that every ⌈(n+5)/6⌉-edge-colored monotone simple drawing of Kn contains a hypochromatic plane spanning tree. (In a monotone drawing, every edge is represented as an x-monotone curve.)

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Plane Spanning Trees in Edge-Colored Simple Drawings of Kn

Abstract

Publication series

Conference

Keywords

ASJC Scopus subject areas

Fields of Expertise

Access to Document

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Doctoral Program: Discrete Mathematics

4th DACH Workshop on Arrangements and Drawings

28th International Symposium on Graph Drawing and Network Visualization