Small Grid Drawings of Planar Graphs with Balanced Bipartition

• Published in: WALCOM: Algorithms and Computation pp. 47 - 57
• Main Authors: Zhou, Xiao, Hikino, Takashi, Nishizeki, Takao
• Format: Book Chapter
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg 2010
• Series: Lecture Notes in Computer Science
• ISBN: 9783642114397; 3642114393
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-642-11440-3_5

In a grid drawing of a planar graph, every vertex is located at a grid point, and every edge is drawn as a straight-line segment without any edge-intersection. It has been known that every planar graph G of n vertices has a grid drawing on an (n − 2)×(n − 2) integer grid and such a drawing can be found in linear time. In this paper we show that if a planar graph G has a balanced bipartition then G has a grid drawing with small grid area. More precisely, if a separation pair bipartitions G into two edge-disjoint subgraphs G1 and G2, then G has a grid drawing on a W×H grid such that both the width W and height H are smaller than the larger number of vertices in G1 and in G2. In particular, we show that every series-parallel graph G has a grid drawing on a (2n/3)×(2n/3) grid and such a drawing can be found in linear time.