Abstract
A convex drawing of an n-vertex graph G = (V,E) is a drawing in which the vertices are placed on the corners of a convex n-gon in the plane and each edge is drawn using one straight line segment. We derive a general lower bound on the number of crossings in any convex drawings of G, using isoperimetric properties of G. The result implies that convex drawings for many graphs, including the planar 2-dimensional grid on n vertices have at least Ω(n log n) crossings. Moreover, for any given arbitrary drawing of G with c crossings in the plane, we construct a convex drawing with at most O((c +Σv∈V d 2v ) log n) crossings, where d v is the degree of v.
This research was supported by the NSF grant CCR9988525.
This research was supported by the EPSRC grant GR/R37395/01.
This author was visiting the National Center for Biotechnology Information, NLM, NIH, with the support of the Oak Ridge Institute for Science and Education. This research was supported by the NSF contract 007 2187.
This research was supported by the VEGA grant No. 02/3164/23.
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Shahrokhi, F., Sýkora, O., Székely, L.A., Vrt’o, I. (2003). Bounds for Convex Crossing Numbers. In: Warnow, T., Zhu, B. (eds) Computing and Combinatorics. COCOON 2003. Lecture Notes in Computer Science, vol 2697. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45071-8_49
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DOI: https://doi.org/10.1007/3-540-45071-8_49
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