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Convex Grid Drawings of Plane Graphs with Pentagonal Contours
Kazuyuki MIURA
Publication
IEICE TRANSACTIONS on Information and Systems
Vol.E97D
No.3
pp.413420 Publication Date: 2014/03/01
Online ISSN: 17451361
DOI: 10.1587/transinf.E97.D.413
Print ISSN: 09168532 Type of Manuscript: Special Section PAPER (Special Section on Foundations of Computer Science —New Trends in Theory of Computation and Algorithm—) Category: Graph Algorithms Keyword: algorithm, convex grid drawing, graph drawing, plane graph, triconnected,
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Summary:
In a convex drawing of a plane graph, all edges are drawn as straightline segments without any edgeintersection and all facial cycles are drawn as convex polygons. In a convex grid drawing, all vertices are put on grid points. 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 (n1)×(n1) 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 2n×2n grid if T(G) has exactly four leaves. In this paper, we show that an internally triconnected plane graph G has a convex grid drawing on a 6n×n^{2} 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 polynomial size.

