Crossing-Optimal Extension of Simple Drawings

TL;DR

This paper investigates the computational complexity of extending simple graph drawings with new edges while controlling crossings, providing fixed-parameter tractability results for various extension problems.

Contribution

It establishes fixed-parameter tractability for inserting edges into simple drawings based on the number of edges and crossing bounds, advancing understanding of crossing-sensitive extension problems.

Findings

Insertion problem is fixed-parameter tractable with parameters: number of edges and crossing limit.

Extended results for variants like $k$-plane drawings.

Single-edge insertion admits a single-exponential fixed-parameter algorithm.

Abstract

In extension problems of partial graph drawings one is given an incomplete drawing of an input graph $G$ and is asked to complete the drawing while maintaining certain properties. A prominent area where such problems arise is that of crossing minimization. For plane drawings and various relaxations of these, there is a number of tractability as well as lower-bound results exploring the computational complexity of crossing-sensitive drawing extension problems. In contrast, comparatively few results are known on extension problems for the fundamental and broad class of simple drawings, that is, drawings in which each pair of edges intersects in at most one point. In fact, only recently it has been shown that the extension problem of simple drawings is NP-hard even when the task is to insert a single edge. In this paper we present tractability results for the crossing-sensitive extension problem of simple drawings. In particular, we show that the problem of inserting edges into a simple drawing is fixed-parameter tractable when parameterized by the number of edges to insert and an upper bound on newly created crossings. Using the same proof techniques, we are also able to answer several closely related variants of this problem, among others the extension problem for $k$-plane drawings. Moreover, using a different approach, we provide a single-exponential fixed-parameter algorithm for the case in which we are only trying to insert a single edge into the drawing.