Small grid drawings of planar graphs with balanced partition

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 is known that every planar graph G of n vertices has a grid drawing on an (n?2)×(n?2) or (4n/3)×(2n/3) integer grid. In this paper we show that if a planar graph G has a balanced partition then G has a grid drawing with small grid area. More precisely, if a separation pair bipartitions G into two edge-disjoint subgraphs G 1 and G 2, then G has a max?{n 1,n 2}×max?{n 1,n 2} grid drawing, where n 1 and n 2 are the numbers of vertices in G 1 and G 2, respectively. In particular, we show that every series-parallel graph G has a (2n/3)×(2n/3) grid drawing and a grid drawing with area smaller than 0.3941n 2 (<(2/3)2 n 2).