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In a convex grid drawing of a plane graph, all edges are drawn as straight-line segments without any edge-intersection, all vertices are put on grid points and all facial cycles are drawn as convex polygons. A plane graph *G* has a convex drawing if and only if *G* is internally triconnected, and an internally triconnected plane graph *G* has a convex grid drawing on an (*n*-1)×(*n*-1) grid if either *G* is triconnected or the triconnected component decomposition tree *T*(*G*) of *G* has two or three leaves, where *n* is the number of vertices in *G*. An internally triconnected plane graph *G* has a convex grid drawing on a 2*n*×2*n* grid if *T*(*G*) has exactly four leaves. Furthermore, an internally triconnected plane graph *G* has a convex grid drawing on a 6*n*×*n*^{2} grid if *T*(*G*) has exactly five leaves. In this paper, we show that an internally triconnected plane graph *G* has a convex grid drawing on a 20*n*×16*n* grid if *T*(*G*) has exactly five leaves. We also present an algorithm to find such a drawing in linear time. This is the first algorithm that finds a convex grid drawing of such a plane graph *G* in a grid of *O*(*n*^{2}) size.

- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E104-A No.9 pp.1142-1149

- Publication Date
- 2021/09/01

- Publicized
- 2021/03/10

- Online ISSN
- 1745-1337

- DOI
- 10.1587/transfun.2020DMP0011

- Type of Manuscript
- Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)

- Category
- Graphs and Networks

Kei SATO

Fukushima University

Kazuyuki MIURA

Fukushima University

The copyright of the original papers published on this site belongs to IEICE. Unauthorized use of the original or translated papers is prohibited. See IEICE Provisions on Copyright for details.

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Kei SATO, Kazuyuki MIURA, "Convex Grid Drawings of Plane Graphs with Pentagonal Contours on O(n2) Grids" in IEICE TRANSACTIONS on Fundamentals,
vol. E104-A, no. 9, pp. 1142-1149, September 2021, doi: 10.1587/transfun.2020DMP0011.

Abstract: In a convex grid drawing of a plane graph, all edges are drawn as straight-line segments without any edge-intersection, all vertices are put on grid points and all facial cycles are drawn as convex polygons. A plane graph *G* has a convex drawing if and only if *G* is internally triconnected, and an internally triconnected plane graph *G* has a convex grid drawing on an (*n*-1)×(*n*-1) grid if either *G* is triconnected or the triconnected component decomposition tree *T*(*G*) of *G* has two or three leaves, where *n* is the number of vertices in *G*. An internally triconnected plane graph *G* has a convex grid drawing on a 2*n*×2*n* grid if *T*(*G*) has exactly four leaves. Furthermore, an internally triconnected plane graph *G* has a convex grid drawing on a 6*n*×*n*^{2} grid if *T*(*G*) has exactly five leaves. In this paper, we show that an internally triconnected plane graph *G* has a convex grid drawing on a 20*n*×16*n* grid if *T*(*G*) has exactly five leaves. We also present an algorithm to find such a drawing in linear time. This is the first algorithm that finds a convex grid drawing of such a plane graph *G* in a grid of *O*(*n*^{2}) size.

URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.2020DMP0011/_p

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@ARTICLE{e104-a_9_1142,

author={Kei SATO, Kazuyuki MIURA, },

journal={IEICE TRANSACTIONS on Fundamentals},

title={Convex Grid Drawings of Plane Graphs with Pentagonal Contours on O(n2) Grids},

year={2021},

volume={E104-A},

number={9},

pages={1142-1149},

abstract={In a convex grid drawing of a plane graph, all edges are drawn as straight-line segments without any edge-intersection, all vertices are put on grid points and all facial cycles are drawn as convex polygons. A plane graph *G* has a convex drawing if and only if *G* is internally triconnected, and an internally triconnected plane graph *G* has a convex grid drawing on an (*n*-1)×(*n*-1) grid if either *G* is triconnected or the triconnected component decomposition tree *T*(*G*) of *G* has two or three leaves, where *n* is the number of vertices in *G*. An internally triconnected plane graph *G* has a convex grid drawing on a 2*n*×2*n* grid if *T*(*G*) has exactly four leaves. Furthermore, an internally triconnected plane graph *G* has a convex grid drawing on a 6*n*×*n*^{2} grid if *T*(*G*) has exactly five leaves. In this paper, we show that an internally triconnected plane graph *G* has a convex grid drawing on a 20*n*×16*n* grid if *T*(*G*) has exactly five leaves. We also present an algorithm to find such a drawing in linear time. This is the first algorithm that finds a convex grid drawing of such a plane graph *G* in a grid of *O*(*n*^{2}) size.},

keywords={},

doi={10.1587/transfun.2020DMP0011},

ISSN={1745-1337},

month={September},}

Copy

TY - JOUR

TI - Convex Grid Drawings of Plane Graphs with Pentagonal Contours on O(n2) Grids

T2 - IEICE TRANSACTIONS on Fundamentals

SP - 1142

EP - 1149

AU - Kei SATO

AU - Kazuyuki MIURA

PY - 2021

DO - 10.1587/transfun.2020DMP0011

JO - IEICE TRANSACTIONS on Fundamentals

SN - 1745-1337

VL - E104-A

IS - 9

JA - IEICE TRANSACTIONS on Fundamentals

Y1 - September 2021

AB - In a convex grid drawing of a plane graph, all edges are drawn as straight-line segments without any edge-intersection, all vertices are put on grid points and all facial cycles are drawn as convex polygons. A plane graph *G* has a convex drawing if and only if *G* is internally triconnected, and an internally triconnected plane graph *G* has a convex grid drawing on an (*n*-1)×(*n*-1) grid if either *G* is triconnected or the triconnected component decomposition tree *T*(*G*) of *G* has two or three leaves, where *n* is the number of vertices in *G*. An internally triconnected plane graph *G* has a convex grid drawing on a 2*n*×2*n* grid if *T*(*G*) has exactly four leaves. Furthermore, an internally triconnected plane graph *G* has a convex grid drawing on a 6*n*×*n*^{2} grid if *T*(*G*) has exactly five leaves. In this paper, we show that an internally triconnected plane graph *G* has a convex grid drawing on a 20*n*×16*n* grid if *T*(*G*) has exactly five leaves. We also present an algorithm to find such a drawing in linear time. This is the first algorithm that finds a convex grid drawing of such a plane graph *G* in a grid of *O*(*n*^{2}) size.

ER -