Polyline Drawings with Topological Constraints

TL;DR

This paper investigates topological polyline drawings of graphs, establishing bounds on the number of bends per edge needed to preserve topology partially or fully, based on graph properties like connectivity and skewness.

Contribution

It introduces bounds on curve complexity for topological graph drawings that partially or fully preserve topology, depending on graph connectivity and skewness.

Findings

Connected crossing-free edges allow polyline drawings with at most three bends per edge.

For non-connected crossing-free edges, curve complexity can be as high as ()()()

Graphs with skewness k admit topologically preserving drawings with at most 2k bends per edge.

Abstract

Let $G$ be a simple topological graph and let $\Gamma$ be a polyline drawing of $G$. We say that $\Gamma$ \emph{partially preserves the topology} of $G$ if it has the same external boundary, the same rotation system, and the same set of crossings as $G$. Drawing $\Gamma$ fully preserves the topology of $G$ if the planarization of $G$ and the planarization of $\Gamma$ have the same planar embedding. We show that if the set of crossing-free edges of $G$ forms a connected spanning subgraph, then $G$ admits a polyline drawing that partially preserves its topology and that has curve complexity at most three (i.e., at most three bends per edge). If, however, the set of crossing-free edges of $G$ is not a connected spanning subgraph, the curve complexity may be $\Omega(\sqrt{n})$. Concerning drawings that fully preserve the topology, we show that if $G$ has skewness $k$, it admits one such drawing with curve complexity at most $2k$; for skewness-1 graphs, the curve complexity can be reduced to one, which is a tight bound. We also consider optimal $2$-plane graphs and discuss trade-offs between curve complexity and crossing angle resolution of drawings that fully preserve the topology.