Crossing minimization in perturbed drawings

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 φ :G→ℝ^2 . We wish to perturb φ by an arbitrarily small ε >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 ε -perturbation, for every ε >0 , is given by a piecewise linear map ψ _ε :G→ℝ^2 with ‖φ -ψ _ε‖ <ε , where ‖ .‖ 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 φ 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 φ has no spurs, and (ii) when φ may have spurs and G is a cycle or a union of disjoint paths.