Simple realizability of complete abstract topological graphs simplified

An abstract topological graph (briefly an AT-graph) is a pair $A=(G,\mathcal{X})$ where $G=(V,E)$ is a graph and $\mathcal{X}\subseteq {E \choose 2}$ is a set of pairs of its edges. The AT-graph $A$ is simply realizable if $G$ can be drawn in the plane so that each pair of edges from $\mathcal{X}$ crosses exactly once and no other pair crosses. We show that simply realizable complete AT-graphs are characterized by a finite set of forbidden AT-subgraphs, each with at most six vertices. This implies a straightforward polynomial algorithm for testing simple realizability of complete AT-graphs, which simplifies a previous algorithm by the author. We also show an analogous result for independent $\mathbb{Z}_2$-realizability, where only the parity of the number of crossings for each pair of independent edges is specified.Comment: 26 pages, 17 figures; major revision; original Section 5 removed and will be included in another pape

Crossing-Optimal Extension of Simple Drawings

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, the extension problem of simple drawings has only recently been shown to be NP-hard even for inserting 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

Simple Drawings of Kn from Rotation Systems

A complete rotation system on n vertices is a collection of n cyclic permutations of the elements [n]\{i}, for i∈[n]. If D is a drawing of a labelled graph, then a rotation at vertex v is the cyclic ordering of the edges at v. In particular, the collection of all vertex rotations of a simple drawing of Kn is a complete rotation system. Can we characterize when a complete rotation system can be represented as a simple drawing of Kn (a.k.a. realizable)? This thesis is motivated by two specific results on complete rotation systems. The first motivating theorem was published by Kyncl in 2011, who, using homotopy, proved as a corollary that if all complete 6-vertex rotation systems of a complete n-vertex rotation system H are realizable, then H is realizable. Combined with communications with Aichholzer, Kyncl determined that complete realizable n-vertex rotation systems are characterized by their complete 5-vertex rotation systems. The second motivating theorem was published by Gioan in 2005, he proved that if two simple drawings of the complete graph D and D′ have the same rotation system, then there is a sequence of Reidemeister III moves that transforms D into D′. Motivated by these results, we prove both facts combinatorially by sequentially drawing the edge crossings of an edge to form a simple drawing. Such a method can be used to prove both theorems, generate every simple drawing of a complete rotation system, or find a non-realizable complete 5-vertex rotation system in any complete rotation system (when one exists)