Bounds for convex crossing numbers

Authors:
Farhad Shahrokhi;Ondrej Sykora;Laszlo A. Székely;Imrich Vrt'o
Affiliations:
Department of Computer Science, University of North Texas, Denton, TX;Department of Computer Science, Loughborough University, Loughborough, The United Kingdom;Department of Mathematics, University of South Carolina, Columbia, SC;Department of Informatics, Institute of Mathematics, Slovak Academy of Sciences, Bratislava, Slovak Republic
Venue:
COCOON'03 Proceedings of the 9th annual international conference on Computing and combinatorics
Year:
2003

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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 dv2) log n) crossings, where dv is the degree of v.