Convex Grid Drawings of Plane Graphs with Pentagonal Contours on O(n²) grids

抄録

In a convex grid drawing of a plane graph, all edges are drawn as straight-line segments without any edge-intersection, all vertices are put on grid points and all facial cycles are drawn as convex polygons. 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. Furthermore, an internally triconnected plane graph G has a convex grid drawing on a 6n×n² grid if T (G) has exactly five leaves. In this paper, we show that an internally triconnected plane graph G has a convex grid drawing on a 20n × 16n 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 O(n2) size.