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Bounds for rectilinear crossing numbers
Author(s) -
Bienstock Daniel,
Dean Nathaniel
Publication year - 1993
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.3190170308
Subject(s) - crossing number (knot theory) , combinatorics , mathematics , graph , line segment , point (geometry) , quadratic equation , discrete mathematics , geometry , intersection (aeronautics) , engineering , aerospace engineering
A rectilinear drawing of a graph is one where each edge is drawn as a straight‐line segment, and the rectilinear crossing number of a graph is the minimum number of crossings over all rectilinear drawings. We describe, for every integer k ≥ 4, a class of graphs of crossing number k , but unbounded rectilinear crossing number. This is best possible since the rectilinear crossing number is equal to the crossing number whenever the latter is at most 3. Further, if we consider drawings where each edge is drawn as a polygonal line segment with at most one break point, then the resulting crossing number is at most quadratic in the regular crossing number. © 1993 John Wiley & Sons, Inc.