856 1 archive 2 disk0/00/00/08/56 2008-06-24 2008-09-18 11:09:10 2008-09-18 11:09:10 confpaper show 0 Despite a long research effort, finding the minimum area for straight-line grid drawings of planar graphs is still an elusive goal. A long-standing lower bound on the area requirement for straight-line drawings of plane graphs was established in 1984 by Dolev, Leighton, and Trickey, who exhibited a family of graphs, known as nested triangles graphs, for which $(2n/3-1) times (2n/3-1)$ area is necessary. We show that nested triangles graphs can be drawn in $2n^2/9 + O(n)$ area when the outer face is not given, improving a previous $n^2/3$ area upper bound. Further, we show that $n^2/9 + Omega(n)$ area is necessary for any planar straight-line drawing of a nested triangles graph. Finally, we deepen our insight into the $4/9n^2-4/3n+1$ lower bound by Dolev, Leighton, and Trickey, which is conjectured to be tight, showing a family of plane graphs requiring more area. http://www.springerlink.com/openurl.asp?genre=article&issn=0302-9743&volume=4875&spage=339 Frati Fabrizio Patrignani Maurizio September 24-26, 2007 Graph Drawing Sidney, Australia Hong Seok-Hee Hong, Seok-Hee Nishizeki Takao Nishizeki, Takao Quan Wu Quan, Wu pub 339-344 FALSE Springer FALSE 1. N. Bonichon, S. Felsner, and M. Mosnah. Convex drawings of 3-connected plane graphs. In Proc. GD 2004, volume 3383 of LNCS, pages 60-70, 2004. 2. F. J. Brandenburg, D. Eppstein, M. T. Goodrich, S. G. Kobourov, G. Liotta, and P. Mutzel. Selected open problems in graph drawing. In Proc. GD 2003, volume 2912 of LNCS, pages 515-539, 2003. 3. M. Chrobak and S.-I. Nakano. Minimum-width grid drawing of plane graphs. Comp. Geom., (11):29-54, 1998. 4. H. de Fraysseix, J. Pach, and R. Pollack. How to draw a planar graph on a grid. Combinatorica, 10(1):41-51, 1990. 5. D. Dolev, F. T. Leighton, and H. Trickey. Planar embedding of planar graphs. Advances in Computing Research - Volume 2: VLSI Theory, 1984. 6. K. Miura, S. Nakano, and T. Nishizeki. Grid drawings of 4-connected plane graphs. Discr. Comp. Geom., (26):73-87, 2001. 7. P. Ossona de Mendez. Some problems: Grid drawings of planar graphs. http://www.ehess.fr/centres/cams/person/pom/langen/openpb. html. 8. W. Schnyder. Embedding planar graphs on the grid. In Proc. SODA 1990, pages 138-148, 1990. 9. H. Zhang and X. He. Compact visibility representation and straight-line grid embedding of plane graphs. In Proc. WADS 2003, volume 2748 of LNCS, pages 493-504, 2003. G.70 P.720 published 2008 none