Bounds for rectilinear crossing numbers

Abstract

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 2 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 w e 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. 0

Hanani-Tutte and Monotone Drawings

“…This variant of the weak Hanani-Tutte theorem was first proved by Pach and Tóth. 1 We give a new proof of this result as Theorem 1 in Section 2, which continues an elementary topological approach similar to earlier papers on the Hanani-Tutte theorem, e.g. [15].…”

“…Then there is a cycle C that contains e j and e k for some distinct k, j ∈ {1, 2, 3}, and so that 1 , e 2 , e 3 }, then we may assume that C contains vv L .…”

“…, where rcr 2 (G) allows straight-line edges with one bend [1]. Pach and Tóth in [14] recently proved that monotone crossing number for unleveled graphs is at most cr(G)…”

Monotone Crossing Number

“…We have cr(G) ≤ lin-cr(G). Bienstock and Dean [1] constructed a series of graphs with crossing number 4, whose rectilinear crossing numbers are arbitrarily large.…”

“…See Theorem 2.2. One of the key ideas of the construction proving Theorem 2, the use of "weighted" edges or repeated paths, goes back to the paper of Bienstock and Dean [1] mentioned above. This idea was further developed and applied to related problems by Pelsmajer, Schaefer, anď Stefankovič [13] and by Tóth [15].…”