Convex drawings of internally triconnected plane graphs on O(n 2) grids

Abstract

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(n 3) area if the triconnected component decomposition tree of G has at most four leaves. In this paper, we improve the area bound O(n 3) to O(n 2), 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. © 2009 Springer-Verlag Berlin Heidelberg.