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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 20*n* × 16*n* 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 10*n* × 5*n* grid if *T*(*G*) has exactly five leaves. We also present a linear-time algorithm to find such a drawing.

- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E106-A No.9 pp.1092-1099

- Publication Date
- 2023/09/01

- Publicized
- 2023/03/06

- Online ISSN
- 1745-1337

- DOI
- 10.1587/transfun.2022DMP0009

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

- Category
- Algorithms and Data Structures

Kazuyuki MIURA

Fukushima University

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Kazuyuki MIURA, "Convex Grid Drawings of Internally Triconnected Plane Graphs with Pentagonal Contours" in IEICE TRANSACTIONS on Fundamentals,
vol. E106-A, no. 9, pp. 1092-1099, September 2023, doi: 10.1587/transfun.2022DMP0009.

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 20*n* × 16*n* 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 10*n* × 5*n* grid if *T*(*G*) has exactly five leaves. We also present a linear-time algorithm to find such a drawing.

URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.2022DMP0009/_p

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@ARTICLE{e106-a_9_1092,

author={Kazuyuki MIURA, },

journal={IEICE TRANSACTIONS on Fundamentals},

title={Convex Grid Drawings of Internally Triconnected Plane Graphs with Pentagonal Contours},

year={2023},

volume={E106-A},

number={9},

pages={1092-1099},

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 20*n* × 16*n* 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 10*n* × 5*n* grid if *T*(*G*) has exactly five leaves. We also present a linear-time algorithm to find such a drawing.},

keywords={},

doi={10.1587/transfun.2022DMP0009},

ISSN={1745-1337},

month={September},}

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TY - JOUR

TI - Convex Grid Drawings of Internally Triconnected Plane Graphs with Pentagonal Contours

T2 - IEICE TRANSACTIONS on Fundamentals

SP - 1092

EP - 1099

AU - Kazuyuki MIURA

PY - 2023

DO - 10.1587/transfun.2022DMP0009

JO - IEICE TRANSACTIONS on Fundamentals

SN - 1745-1337

VL - E106-A

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JA - IEICE TRANSACTIONS on Fundamentals

Y1 - September 2023

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 20*n* × 16*n* 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 10*n* × 5*n* grid if *T*(*G*) has exactly five leaves. We also present a linear-time algorithm to find such a drawing.

ER -