Progress on Partial Edge Drawings

TL;DR

This paper investigates partial edge drawings (PED) as a crossing-avoidance technique in graph visualization, analyzing the existence of symmetric and homogeneous models, and providing algorithms for maximizing stub lengths in specific graph classes.

Contribution

It introduces the symmetric and homogeneous partial edge drawing models, establishes non-existence results for certain graphs, and develops algorithms for optimizing stub lengths in specific graph drawings.

Findings

K_{241} lacks a 1/4-SHPED

Bandwidth-k graphs have a a(1/\u221a{k})-SHPED

Efficient algorithms for 2-planar drawings

Abstract

Recently, a new way of avoiding crossings in straight-line drawings of non-planar graphs has been investigated. The idea of partial edge drawings (PED) is to drop the middle part of edges and rely on the remaining edge parts called stubs. We focus on a symmetric model (SPED) that requires the two stubs of an edge to be of equal length. In this way, the stub at the other endpoint of an edge assures the viewer of the edge's existence. We also consider an additional homogeneity constraint that forces the stub lengths to be a given fraction $\delta$ of the edge lengths ($\delta$-SHPED). Given length and direction of a stub, this model helps to infer the position of the opposite stub. We show that, for a fixed stub--edge length ratio $\delta$, not all graphs have a $\delta$-SHPED. Specifically, we show that $K_{241}$ does not have a 1/4-SHPED, while bandwidth-$k$ graphs always have a $\Theta(1/\sqrt{k})$-SHPED. We also give bounds for complete bipartite graphs. Further, we consider the problem \textsc{MaxSPED} where the task is to compute the SPED of maximum total stub length that a given straight-line drawing contains. We present an efficient solution for 2-planar drawings and a 2-approximation algorithm for the dual problem.