Polyline Drawings with Topological Constraints

09/21/2018
by   Emilio Di Giacomo, et al.
Università Perugia
University College Roosevelt
0

Let G be a simple topological graph and let Γ be a polyline drawing of G. We say that Γ partially preserves the topology of G if it has the same external boundary, the same rotation system, and the same set of crossings as G. Drawing Γ fully preserves the topology of G if the planarization of G and the planarization of Γ have the same planar embedding. We show that if the set of crossing-free edges of G forms a connected spanning subgraph, then G admits a polyline drawing that partially preserves its topology and that has curve complexity at most three (i.e., at most three bends per edge). If, however, the set of crossing-free edges of G is not a connected spanning subgraph, the curve complexity may be Ω(√(n)). Concerning drawings that fully preserve the topology, we show that if G has skewness k, it admits one such drawing with curve complexity at most 2k; for skewness-1 graphs, the curve complexity can be reduced to one, which is a tight bound. We also consider optimal 2-plane graphs and discuss trade-offs between curve complexity and crossing angle resolution of drawings that fully preserve the topology.

READ FULL TEXT

page 1

page 2

page 3

page 4

12/19/2017

On Fan-Crossing Graphs

A fan is a set of edges with a single common endpoint. A graph is fan-cr...
08/25/2020

Simple Topological Drawings of k-Planar Graphs

Every finite graph admits a simple (topological) drawing, that is, a dra...
01/09/2020

RAC Drawings in Subcubic Area

In this paper, we study tradeoffs between curve complexity and area of R...
08/24/2021

Quasi-upward Planar Drawings with Minimum Curve Complexity

This paper studies the problem of computing quasi-upward planar drawings...
04/25/2020

Extending Partial 1-Planar Drawings

Algorithmic extension problems of partial graph representations such as ...
08/23/2019

Variants of the Segment Number of a Graph

The segment number of a planar graph is the smallest number of line segm...
06/25/2021

Shortcut Hulls: Vertex-restricted Outer Simplifications of Polygons

Let P be a crossing-free polygon and 𝒞 a set of shortcuts, where each sh...

Please sign up or login with your details

Forgot password? Click here to reset