Convex Grid Drawings of Plane Graphs with Rectangular Contours

In a convex drawing of a plane graph, all edges are drawn as straightline 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. In this paper, we show that an internally triconnected plane graph G has a convex grid drawing on a 2n × n grid if T (G) has exactly four leaves. We also present an algorithm to find such a drawing in linear time. Our convex grid drawing has a rectangular contour, while most of the known algorithms produce grid drawings having triangular contours. Submitted: August 2007 Reviewed: June 2008 Revised: July 2008 Accepted: July 2008 Final: July 2008 Published: October 2008 Article type: Regular Paper Communicated by: X. He E-mail addresses: [email protected] (Kazuyuki Miura) [email protected] (Akira Kamada) [email protected] (Takao Nishizeki) 198 Miura, Kamada, Nishizeki Convex Grid Drawings of Plane Graphs...