Maximizing Ink in Partial Edge Drawings of k-plane Graphs

TL;DR

This paper investigates the computational complexity of maximizing ink in partial edge drawings of k-plane graphs, proving NP-hardness for certain cases and providing efficient algorithms for graphs with bounded treewidth.

Contribution

It establishes NP-hardness for ink maximization in 3-plane and 4-plane graphs and offers efficient algorithms for graphs with intersection graphs of bounded treewidth.

Findings

NP-hardness for 3-plane and 4-plane graphs.

Efficient algorithms for graphs with bounded treewidth.

Experimental evaluation of the algorithms.

Abstract

Partial edge drawing (PED) is a drawing style for non-planar graphs, in which edges are drawn only partially as pairs of opposing stubs on the respective end-vertices. In a PED, by erasing the central parts of edges, all edge crossings and the resulting visual clutter are hidden in the undrawn parts of the edges. In symmetric partial edge drawings (SPEDs), the two stubs of each edge are required to have the same length. It is known that maximizing the ink (or the total stub length) when transforming a straight-line graph drawing with crossings into a SPED is tractable for 2-plane input drawings, but NP-hard for unrestricted inputs. We show that the problem remains NP-hard even for 3-plane input drawings and establish NP-hardness of ink maximization for PEDs of 4-plane graphs. Yet, for k-plane input drawings whose edge intersection graph forms a collection of trees or, more generally, whose intersection graph has bounded treewidth, we present efficient algorithms for computing maximum-ink PEDs and SPEDs. We implemented the treewidth-based algorithms and show a brief experimental evaluation.