Crossing Minimization in Perturbed Drawings

Abstract

Due to data compression or low resolution, nearby vertices and edges of a graph drawing 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 ψε : G → R 2 with ϕ − ψε < 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.

Tightening Curves on Surfaces Monotonically with Applications

“…The proof of Theorem 1.1 can be viewed as an amalgamation of the curve shortening algorithm of de Graaf and Schrijver [38], the cluster and pipe expansion technique from graph drawings [15,19,30], and the crossing minimization algorithm for flat braids originated from Geck and Pfeiffer in the context of word problem over symmetric groups [32,38]. The first step relies on hyperbolic geometry, which is very relevant to our tightening problem for the following reasons: (1) any (multi)curve on a surface endowed with a hyperbolic metric is homotopic to a unique (multi)geodesic, and (2) a primitive (multi)geodesic is in minimal position.…”

“…We pick an open neighborhood-called a pipe system-of some underlying skeleton graph, such that γ can be drawn in proper ways respecting the pipe system. We then describe a way to morph the pipe systems using cluster and pipe expansions, a technique introduced by Cortese et al [19] in graph drawings (and later on applied to weak embeddings [2,3,15] and crossing numbers [30]), so that the multicurve inside the pipe system can be canonicalized using polynomially many monotonic homotopy moves. Conceptually the expansion operations can be viewed as ways to morph the metric on surface Σ, so that curves on Σ get transformed closer and closer to the geodesic with respect to the morphing metric.…”

“…We define two operations called the cluster expansion and pipe expansion performed on a multicurve γ lying in a pipe system Π in this subsection. Such operations have been used to study clustered planarity in graph drawings [19,31], weakly simple polygons [2,15], weak embeddings of graphs [3], and crossing numbers [30]. Our definition most closely resembles the one in Fulek and Tóth [30]; both allow the edges of skeleton graph G to cross in the drawing.…”

“…Such operations have been used to study clustered planarity in graph drawings [19,31], weakly simple polygons [2,15], weak embeddings of graphs [3], and crossing numbers [30]. Our definition most closely resembles the one in Fulek and Tóth [30]; both allow the edges of skeleton graph G to cross in the drawing. The main differences are, instead of preserving crossing numbers, we need to argue that the tightening problem remains the same before and after the expansion; and unlike the previous papers where expansions can be done instantly, we have to implement each expansion operation using monotonic homotopy moves.…”

“…For sake of analysis, we want to make sure that the pipe expansions we performed actually improve the quality of the multicurve in a pipe system. This motivates the following definitions [14,19,30]. Cluster u is a base of pipe uv if every vertex in C(u) is incident to some edge in C(uv).…”