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.