Small grid drawings of planar graphs with balanced bipartition

Authors:
Xiao Zhou;Takashi Hikino;Takao Nishizeki
Affiliations:
Graduate School of Information Sciences, Tohoku University, Sendai, Japan;Graduate School of Information Sciences, Tohoku University, Sendai, Japan;Graduate School of Information Sciences, Tohoku University, Sendai, Japan
Venue:
WALCOM'10 Proceedings of the 4th international conference on Algorithms and Computation
Year:
2010

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Abstract

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.