Convex Grid Drawings of Plane Graphs with Pentagonal Contours

Kazuyuki MIURA  

Publication
IEICE TRANSACTIONS on Information and Systems   Vol.E97-D   No.3   pp.413-420
Publication Date: 2014/03/01
Online ISSN: 1745-1361
DOI: 10.1587/transinf.E97.D.413
Print ISSN: 0916-8532
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,  

Full Text: PDF(612.8KB)>>
Buy this Article




Summary: 
In a convex drawing of a plane graph, all edges are drawn as straight-line segments without any edge-intersection 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 (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 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×n2 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.