Crossing minimization in perturbed drawings

Abstract

Due to data compression or low resolution, nearby vertices and edges of a graph drawn in the plane may be bundled to a common node or arc. We model such a “compromised” drawing by a piecewise linear map $\varphi :G\rightarrow {\mathbb {R}}^2$. We wish to perturb $\varphi$ by an arbitrarily small $\varepsilon >0$ into a proper drawing (in which the vertices are distinct points, any two edges intersect in finitely many points, and no three edges have a common interior point) that minimizes the number of crossings. An $\varepsilon$-perturbation, for every $\varepsilon >0$, is given by a piecewise linear map $\psi _\varepsilon :G\rightarrow {\mathbb {R}}^2$ with $\Vert \varphi -\psi _\varepsilon \Vert <\varepsilon$, where $\Vert .\Vert$ is the uniform norm (i.e., $\sup$ norm). We present a polynomial-time solution for this optimization problem when G is a cycle and the map $\varphi$ has no spurs (i.e., no two adjacent edges are mapped to overlapping arcs). We also show that the problem becomes NP-complete (i) when G is an arbitrary graph and $\varphi$ has no spurs, and (ii) when $\varphi$ may have spurs and G is a cycle or a union of disjoint paths.