On k-planar crossing numbers
نویسندگان
چکیده
The k-planar crossing number of a graph is the minimum number of crossings of its edges over all possible drawings of the graph in k planes. We propose algorithms and methods for k-planar drawings of general graphs together with lower bound techniques. We give exact results for the k-planar crossing number ofK2k+1,q , for k 2.We prove tight bounds for complete graphs.We also study the rectilinear k-planar crossing number. © 2006 Elsevier B.V. All rights reserved.
منابع مشابه
One- and two-page crossing numbers for some types of graphs
The simplest graph drawing method is that of putting the vertices of a graph on a line (spine) and drawing the edges as half-circles on k half planes (pages). Such drawings are called k-page book drawings and the minimal number of edge crossings in such a drawing is called the k-page crossing number. In a one-page book drawing, all edges are placed on one side of the spine, and in a two-page bo...
متن کاملEdge Separators for Graphs of Bounded Genus with Applications
We prove that every n-vertex graph of genus 9 and maximal degree k has an edge separator of size O( ..;gTffl). The upper bound is best possible to within a constant factor. This extends known results on planar graphs and similar results about vertex separators. We apply the edge separatorto the isoperimetric number problem, graph embeddings and lower bounds for crossing numbers.
متن کاملCrossing and Weighted Crossing Number of Near-Planar Graphs
A nonplanar graph G is near-planar if it contains an edge e such that G− e is planar. The problem of determining the crossing number of a near-planar graph is exhibited from different combinatorial viewpoints. On the one hand, we develop min-max formulas involving efficiently computable lower and upper bounds. These min-max results are the first of their kind in the study of crossing numbers an...
متن کاملk-planar Crossing Number of Random Graphs and Random Regular Graphs
We give an explicit extension of Spencer’s result on the biplanar crossing number of the ErdősRényi random graph G(n, p). In particular, we show that the k-planar crossing number of G(n, p) is almost surely Ω((np)). Along the same lines, we prove that for any fixed k, the k-planar crossing number of various models of random d-regular graphs is Ω((dn)) for d > c0 for some constant c0 = c0(k).
متن کاملThe crossing numbers of join products of paths with graphs of order four
Kulli and Muddebihal [V.R. Kulli, M.H. Muddebihal, Characterization of join graphs with crossing number zero, Far East J. Appl. Math. 5 (2001) 87–97] gave the characterization of all pairs of graphs which join product is planar graph. The crossing number cr(G) of a graph G is the minimal number of crossings over all drawings of G in the plane. There are only few results concerning crossing numb...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Discrete Applied Mathematics
دوره 155 شماره
صفحات -
تاریخ انتشار 2007