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).
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References
Biedl T (2002) Drawing outer-planar graphs in o(nlog n) area. In: Goodrich MT, Kobourov SG (eds) Proc of GD 2002. LNCS, vol 2528, pp 54–65
Biedl T (2010) On small drawings of series-parallel graphs and other subclasses of planar graphs. In: Proc of GD 2009. LNCS, vol 5849, pp 280–291
Brandenburg FJ (2008) Drawing planar graphs on 8n 2/9 area. Electron Notes Discrete Math 31:37–40
Chrobak M, Kant G (1997) Convex grid drawings of 3-connected planar graphs. Int J Comput Geom Appl 7:211–223
Chrobak M, Payne TH (1995) A linear-time algorithm for drawing a planar graph on a grid. Inf Process Lett 54:241–246
de Fraysseix H, Pach J, Pollack R (1990) How to draw a planar graph on a grid. Combinatorica 10:41–51
Di Battista G, Frati F (2009) Small area drawings of outerplanar graphs. Algorithmica 54:25–53
Dolev D, Leighton FT, Trickey H (1984) Planar embedding of planar graphs. Adv Comput Res 2:147–161
Frati F (2008) A lower bound on the area requirements of series-parallel graphs. In: Broersma H, Erlebach T, Friedetzky T, Paulusma D (eds) Proc of WG 2008. LNCS, vol 5344, pp 159–170
Frati F, Patrignami M (2007) A note on minimum-area straight-line drawings of planar graphs. In: Hong SH, Nishizeki T, Quan W (eds) Proc of GD 2007. LNCS, vol 4875, pp 339–344
Miura K, Nakano S, Nishizeki T (2001) Grid drawings of 4-connected plane graphs. Discrete Comput Geom 26:73–87
Nishizeki T, Chiba N (2008) Planar graphs: theory and algorithms. Dover, New York
Nishizeki T, Rahman MS (2004) Planar graph drawing. Singapore, World Scientific
Schnyder W (1990) Embedding planar graphs on the grid. In: Proc of first ACM-SIAM symposium on discrete algorithms, pp 138–148
Shiloach Y (1976) Arrangements of planar graphs on the planar lattice. PhD thesis, Weizmann Institute of Science
Takamizawa K, Nishizeki T, Saito N (1982) Linear-time computability of combinatorial problems on series-parallel graphs. J ACM 29:623–641
Zhou X, Hikino T, Nishizeki N (2010) Grid drawings of planar graphs with balanced bipartition. In: Proc of WALCOM 2010. LNCS, vol 5942, pp 47–57
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This work is supported in part by a Grant-in-Aid for Scientific Research (C) 19500001 from Japan Society for the Promotion of Science (JSPS).
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Zhou, X., Hikino, T. & Nishizeki, T. Small grid drawings of planar graphs with balanced partition. J Comb Optim 24, 99–115 (2012). https://doi.org/10.1007/s10878-011-9381-7
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DOI: https://doi.org/10.1007/s10878-011-9381-7