Convex drawings of internally triconnected plane graphs on O(n 2) grids

Xiao Zhou; Takao Nishizeki

doi:10.1007/978-3-642-10631-6_77

Convex drawings of internally triconnected plane graphs on O(n ²) grids

Xiao Zhou, Takao Nishizeki

INF - System Information Sciences

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

1 Citation (Scopus)

Abstract

In a convex grid drawing of a plane graph, every edge is drawn as a straight-line segment without any edge-intersection, every vertex is located at a grid point, and every facial cycle is drawn as a convex polygon. A plane graph G has a convex drawing if and only if G is internally triconnected. It has been known that an internally triconnected plane graph G of n vertices has a convex grid drawing on a grid of O(n ³) area if the triconnected component decomposition tree of G has at most four leaves. In this paper, we improve the area bound O(n ³) to O(n ²), which is optimal up to a constant factor. More precisely, we show that G has a convex grid drawing on a 2n×4n grid. We also present an algorithm to find such a drawing in linear time.

Original language	English
Title of host publication	Algorithms and Computation - 20th International Symposium, ISAAC 2009, Proceedings
Pages	760-770
Number of pages	11
DOIs	https://doi.org/10.1007/978-3-642-10631-6_77
Publication status	Published - 2009
Event	20th International Symposium on Algorithms and Computation, ISAAC 2009 - Honolulu, HI, United States Duration: 2009 Dec 16 → 2009 Dec 18

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	5878 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Other

Other	20th International Symposium on Algorithms and Computation, ISAAC 2009
Country/Territory	United States
City	Honolulu, HI
Period	09/12/16 → 09/12/18

ASJC Scopus subject areas

Theoretical Computer Science
Computer Science(all)

Access to Document

10.1007/978-3-642-10631-6_77

Cite this

Zhou, X., & Nishizeki, T. (2009). Convex drawings of internally triconnected plane graphs on O(n ²) grids. In Algorithms and Computation - 20th International Symposium, ISAAC 2009, Proceedings (pp. 760-770). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5878 LNCS). https://doi.org/10.1007/978-3-642-10631-6_77

Convex drawings of internally triconnected plane graphs on O(n ²) grids. / Zhou, Xiao; Nishizeki, Takao.

Algorithms and Computation - 20th International Symposium, ISAAC 2009, Proceedings. 2009. p. 760-770 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5878 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Zhou, X & Nishizeki, T 2009, Convex drawings of internally triconnected plane graphs on O(n ²) grids. in Algorithms and Computation - 20th International Symposium, ISAAC 2009, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 5878 LNCS, pp. 760-770, 20th International Symposium on Algorithms and Computation, ISAAC 2009, Honolulu, HI, United States, 09/12/16. https://doi.org/10.1007/978-3-642-10631-6_77

Zhou X, Nishizeki T. Convex drawings of internally triconnected plane graphs on O(n ²) grids. In Algorithms and Computation - 20th International Symposium, ISAAC 2009, Proceedings. 2009. p. 760-770. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-642-10631-6_77

@inproceedings{4dde9e0e7436456396b1778e3abead8a,

title = "Convex drawings of internally triconnected plane graphs on O(n 2) grids",

abstract = "In a convex grid drawing of a plane graph, every edge is drawn as a straight-line segment without any edge-intersection, every vertex is located at a grid point, and every facial cycle is drawn as a convex polygon. A plane graph G has a convex drawing if and only if G is internally triconnected. It has been known that an internally triconnected plane graph G of n vertices has a convex grid drawing on a grid of O(n 3) area if the triconnected component decomposition tree of G has at most four leaves. In this paper, we improve the area bound O(n 3) to O(n 2), which is optimal up to a constant factor. More precisely, we show that G has a convex grid drawing on a 2n×4n grid. We also present an algorithm to find such a drawing in linear time.",

author = "Xiao Zhou and Takao Nishizeki",

note = "Funding Information: This work is supported in part by a Grant-in-Aid for Scientific Research (C) 19500001 from Japan Society for the Promotion of Science (JSPS).; 20th International Symposium on Algorithms and Computation, ISAAC 2009 ; Conference date: 16-12-2009 Through 18-12-2009",

year = "2009",

doi = "10.1007/978-3-642-10631-6_77",

language = "English",

isbn = "3642106307",

series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

pages = "760--770",

booktitle = "Algorithms and Computation - 20th International Symposium, ISAAC 2009, Proceedings",

}

TY - GEN

T1 - Convex drawings of internally triconnected plane graphs on O(n 2) grids

AU - Zhou, Xiao

AU - Nishizeki, Takao

PY - 2009

Y1 - 2009

N2 - In a convex grid drawing of a plane graph, every edge is drawn as a straight-line segment without any edge-intersection, every vertex is located at a grid point, and every facial cycle is drawn as a convex polygon. A plane graph G has a convex drawing if and only if G is internally triconnected. It has been known that an internally triconnected plane graph G of n vertices has a convex grid drawing on a grid of O(n 3) area if the triconnected component decomposition tree of G has at most four leaves. In this paper, we improve the area bound O(n 3) to O(n 2), which is optimal up to a constant factor. More precisely, we show that G has a convex grid drawing on a 2n×4n grid. We also present an algorithm to find such a drawing in linear time.

AB - In a convex grid drawing of a plane graph, every edge is drawn as a straight-line segment without any edge-intersection, every vertex is located at a grid point, and every facial cycle is drawn as a convex polygon. A plane graph G has a convex drawing if and only if G is internally triconnected. It has been known that an internally triconnected plane graph G of n vertices has a convex grid drawing on a grid of O(n 3) area if the triconnected component decomposition tree of G has at most four leaves. In this paper, we improve the area bound O(n 3) to O(n 2), which is optimal up to a constant factor. More precisely, we show that G has a convex grid drawing on a 2n×4n grid. We also present an algorithm to find such a drawing in linear time.

UR - http://www.scopus.com/inward/record.url?scp=75649145000&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=75649145000&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-10631-6_77

DO - 10.1007/978-3-642-10631-6_77

M3 - Conference contribution

AN - SCOPUS:75649145000

SN - 3642106307

SN - 9783642106309

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 760

EP - 770

BT - Algorithms and Computation - 20th International Symposium, ISAAC 2009, Proceedings

T2 - 20th International Symposium on Algorithms and Computation, ISAAC 2009

Y2 - 16 December 2009 through 18 December 2009

ER -