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(n). A lower bound of Ω(n) 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 lineartime 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. Submitted: April 2008 Reviewed: January 2009 Revised: February 2009 Accepted: April 2009 Final: April 2009 Published: June 2009 Article type: Regular paper Communicated by: Petra Mutzel E-mail addresses: rkarim@univdhaka.edu (Md. Rezaul Karim) saidurrahman@cse.buet.ac.bd (Md. Saidur Rahman) 154 Karim and Rahman Straight-Line Grid Drawings on Linear Area