Polyline drawings with topological constraints

Emilio Di Giacomo,Peter Eades,Giuseppe Liotta,Henk Meijer,Fabrizio Montecchiani

doi:10.1016/j.tcs.2019.12.016

Emilio Di Giacomo, Peter Eades + Show 3 more

Open Access

https://doi.org/10.1016/j.tcs.2019.12.016

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Abstract

We study the problem of representing topological graphs as polyline drawings with few bends per edge and such that the topology of the graph is either fully or partially preserved. More formally, let G be a simple topological graph and let Γ be a polyline drawing of G. Drawing Γ partially preserves the topology of G if it has the same external boundary, the same circular order of the edges around each vertex, and the same set of crossings as G, while it fully preserves the topology of G if the planarization of G and the planarization of Γ have the same planar embedding. We prove that if the set of crossing-free edges of G forms a biconnected (connected) spanning subgraph, then G admits a polyline drawing that partially preserves its topology and that has curve complexity at most one (three), i.e., with at most one (three) bend(s) 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), while it is O(1) if the number of connected components is O(1). Concerning drawings that fully preserve the topology, we show that if G is k-skew (i.e., it becomes planar after removing k suitably chosen edges), it admits one such drawing with curve complexity at most 2k; for 1-skew graphs, the curve complexity can be reduced to one, which is a tight bound. We also consider optimal 2-plane graphs (i.e., with at most two crossings per edge and maximum edge density), for which we discuss trade-offs between curve complexity and crossing angle resolution of drawings that fully preserve the topology.

Highlights

A fundamental result in graph drawing is the so-called “stretchability theorem” [12, 17, 18]: Every planar simple topological graph admits a straight-line drawing that preserves its topology
Let G be a simple topological graph and let Γ be a polyline drawing of G. (Note that, by definition of simple topological graph, G has neither multiple edges nor self-loops; see Section 2 for formal definitions.) Drawing Γ fully preserves the topology of G if the planarization of G and the planarization of Γ have the same planar embedding
A polyline drawing of a simple topological graph G partially preserves the topology of G if it has the same rotation system, the same external boundary, and the same set of crossings as G, while it may not preserve the order of the crossings along an edge

Summary

Introduction

A fundamental result in graph drawing is the so-called “stretchability theorem” [12, 17, 18]: Every planar simple topological graph admits a straight-line drawing that preserves its topology. A polyline drawing of a simple topological graph G partially preserves the topology of G if it has the same rotation system, the same external boundary, and the same set of crossings as G, while it may not preserve the order of the crossings along an edge. 1-plane graphs (every edge can be crossed at most once) admit a polyline drawing with curve complexity one that fully preserves the topology and such that any crossing angle is π 2. Missing details can be found in [13]

Preliminaries

Polyline Drawings that Partially Preserve the Topology

Polyline Drawings that Fully Preserve the Topology

Open Problems