TY - JOUR

T1 - Convex drawings of internally triconnected plane graphs on O(n2) grids

AU - Zhou, Xiao

AU - Nishizeki, Takao

N1 - 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).
Publisher Copyright:
© 2010 World Scientific Publishing Company.

PY - 2010/9/1

Y1 - 2010/9/1

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(n3) area if the triconnected component decomposition tree of G has at most four leaves. In this paper, we improve the area bound O(n3) to O(n2), 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(n3) area if the triconnected component decomposition tree of G has at most four leaves. In this paper, we improve the area bound O(n3) to O(n2), 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.

KW - Convex drawing

KW - plane graph

KW - triconnected component decomposition

UR - http://www.scopus.com/inward/record.url?scp=85027204824&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85027204824&partnerID=8YFLogxK

U2 - 10.1142/S179383091000070X

DO - 10.1142/S179383091000070X

M3 - Article

AN - SCOPUS:85027204824

VL - 2

SP - 347

EP - 362

JO - Discrete Mathematics, Algorithms and Applications

JF - Discrete Mathematics, Algorithms and Applications

SN - 1793-8309

IS - 3

ER -