On a Class of Planar Graphs with Straight-Line Grid Drawings on Linear Area

Abstract

A straight-line grid drawing of a planar graph G is a drawing of G on an integer grid such that each vertex is drawn as a grid point and each edge is drawn as a straight-line segment without edge crossings. It is well known that a planar graph of n vertices admits a straight-line grid drawing on a grid of area O(n²). A lower bound of Ω(n²) on the area-requirement for straight-line grid drawings of certain planar graphs are also known. In this paper, we introduce a fairly large class of planar graphs which admits a straight-line grid drawing on a grid of area O(n). We give a linear-time algorithm to find such a drawing. Our new class of planar graphs, which we call "doughnut graphs," is a subclass of 5-connected planar graphs. We show several interesting properties of "doughnut graphs" in this paper. One can easily observe that any spanning subgraph of a "doughnut graph" also admits a straight-line grid drawing with linear area. But the recognition of a spanning subgraph of a "doughnut graph" seems to be a non-trivial problem, since the recognition of a spanning subgraph of a given graph is an NP-complete problem in general. We establish a necessary and sufficient condition for a 4-connected planar graph G to be a spanning subgraph of a "doughnut graph." We also give a linear-time algorithm to augment a 4-connected planar graph G to a "doughnut graph" if G satisfies the necessary and sufficient condition.