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 phi:G -> R-2. We wish to perturb phi by an arbitrarily small epsilon>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 epsilon-perturbation, for every epsilon>0, is given by a piecewise linear map psi epsilon:G -> R-2 with ||phi-psi epsilon||<epsilon is the uniform norm (i.e.,supnorm). We present a polynomial-time solution for this optimization problem whenGis a cycle and the map phi has nospurs(i.e., no two adjacent edges are mapped to overlapping arcs). We also show that the problem becomes NP-complete (i) whenGis an arbitrary graph and phi has no spurs, and (ii) when phi may have spurs andGis a cycle or a union of disjoint paths.