Progress on Partial Edge Drawings

Abstract

Recently, a new way of avoiding crossings in straight-line drawings of non-planar graphs has been introduced. 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 symmetric partial edge drawings (SPEDs) that require 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 $K_{165}$ 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 ${\rm M{\small AX} SPED}$ 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 of minimizing the total amount of erased ink.

Maximizing Ink in Partial Edge Drawings of k-plane Graphs

“…We extended the work by Bruckdorfer et al [6] and showed NP-hardness for the MaxPED and MaxSPED problems, as well as polynomial-time algorithms for the case of the intersection graph of the input drawing being a tree or having bounded treewidth. For the latter, our proof-of-concept implementation worked reasonably well for small to medium-size instances.…”

“…As a first result, Bruckdorfer and Kaufmann [7] presented an integer linear program for solving MaxSPED on general input drawings. Later, Bruckdorfer et al [6] gave an O(n log n)-time algorithm for MaxSPED on the class of 2-plane input drawings (no edge has more than two crossings), where n is the number of vertices. They also described an efficient 2-approximation algorithm for the dual problem of minimizing the amount of erased ink for arbitrary input drawings.…”

“…There are a number of additional results for PEDs without a given input drawing, i.e., having the additional freedom of placing the vertices in the plane. For example, the existence or non-existence of SHPEDs with a specified stub ratio δ for certain graph classes such as complete graphs, complete bipartite graphs, or graphs of bounded bandwidth has been investigated [6,7]. From a practical perspective, Bruckdorfer et al [8] presented a force-directed layout algorithm to compute SHPEDs for stubs of 1/4 of the total edge length, but without a guarantee that all crossings are eliminated.…”

Dynamic graph exploration by interactively linked node-link diagrams and matrix visualizations

“…The layout of a graph G is the positioning of the given vertices V to n (normally) distinct locations in the display space, whereas the edges in between are drawn as either straight, curved, orthogonal [ 29 ], or partial links [ 30 , 31 ], depending on several esthetic graph drawing criteria to be followed [ 32 ].…”

Visual analytics of large dynamic digraphs

“…Although we decide to use the node-link visual metaphor in this work, we argue that additional enhancements to the link representation can become a powerful concept to represent long graph sequences. In this article, we argue that we do not need the complete link in a node-link diagram to show connectedness of objects as evaluated by Burch et al, 12 invented by Becker et al, 24 and mathematically modeled as partial edge drawing by Bruckdorfer et al 25 Although there are many other link representation styles like curved, orthogonal, or tapered ones which are evaluated by Holten and Van Wijk, 26 we use traditional straight link drawings with equally thick lines. But, as an enhancement to the parallel edge splatting concept introduced by Burch et al, 11 we focus on space-efficiency aspects.…”