Shooting Stars in Simple Drawings of Km,nK_{m,n}

Simple drawings are drawings of graphs in which two edges have at most one common point (either a common endpoint, or a proper crossing). It has been an open question whether every simple drawing of a complete bipartite graph $K_{m,n}$ contains a plane spanning tree as a subdrawing. We answer this question to the positive by showing that for every simple drawing of $K_{m,n}$ and for every vertex $v$ in that drawing, the drawing contains a shooting star rooted at $v$, that is, a plane spanning tree containing all edges incident to $v$.Comment: Appears in the Proceedings of the 30th International Symposium on Graph Drawing and Network Visualization (GD 2022

On short edges in complete topological graphs

Let $h(n)$ be the minimum integer such that every complete $n$-vertex simple topological graph contains an edge that crosses at most $h(n)$ other edges. In 2009, Kyn\v{c}l and Valtr showed that $h(n) = O(n^2/\log^{1/4} n)$, and in the other direction, gave constructions showing that $h(n) = \Omega(n^{3/2})$. In this paper, we prove that $h(n) = O(n^{7/4})$. Along the way, we establish a new variant of Chazelle and Welzl's matching theorem for set systems with bounded VC-dimension, which we believe to be of independent interest

Unavoidable patterns in complete simple topological graphs

In this paper, we show that every complete $n$-vertex simple topological graph contains a topological subgraph on at least $(\log n)^{1/4 - o(1)}$ vertices that is weakly isomorphic to the complete convex geometric graph or the complete twisted graph. This improves the previously known bound of $\Omega(\log^{1/8}n)$ due to Pach, Solymosi, and T\'oth. We also show that every complete $n$-vertex simple topological graph contains a planar path of length at least $(\log n)^{1 - o(1)}$

On plane subgraphs of complete topological drawings

Topological drawings are representations of graphs in the plane, where vertices are represented by points, and edges by simple curves connecting the points. A drawing is simple if two edges intersect at most in a single point, either at a common endpoint or at a proper crossing. In this paper we study properties of maximal plane subgraphs of simple drawings Dnof the complete graph Knon n vertices. Our main structural result is that maximal plane subgraphs are 2-connected and what we call essentially 3-edge-connected. Besides, any maximal plane subgraph contains at least [3n/2] edges. We also address the problem of obtaining a plane subgraph of Dnwith the maximum number of edges, proving that this problem is NP-complete. However, given a plane spanning connected subgraph of Dn, a maximum plane augmentation of this subgraph can be found in O(n3) time. As a side result, we also show that the problem of finding a largest compatible plane straight-line graph of two labeled point sets is NP-complete. © 2021 Society of Mathematicians, Physicists and Astronomers of Slovenia. All rights reserved

Saturated simple and k-simple topological graphs

A simple topological graph $G$ is a graph drawn in the plane so that any pair of edges have at most one point in common, which is either an endpoint or a proper crossing. $G$ is called saturated if no further edge can be added without violating this condition. We construct saturated simple topological graphs with $n$ vertices and $O(n)$ edges. For every $k>1$, we give similar constructions for $k$-simple topological graphs, that is, for graphs drawn in the plane so that any two edges have at most $k$ points in common. We show that in any $k$-simple topological graph, any two independent vertices can be connected by a curve that crosses each of the original edges at most $2k$ times. Another construction shows that the bound $2k$ cannot be improved. Several other related problems are also considered.Comment: 25 pages, 17 figures, added some new results and improvement

Towards Crossing-Free Hamiltonian Cycles in Simple Drawings of Complete Graphs

It is a longstanding conjecture that every simple drawing of a complete graph on $n \geq 3$ vertices contains a crossing-free Hamiltonian cycle. We strengthen this conjecture to "there exists a crossing-free Hamiltonian path between each pair of vertices" and show that this stronger conjecture holds for several classes of simple drawings, including strongly c-monotone drawings and cylindrical drawings. As a second main contribution, we give an overview on different classes of simple drawings and investigate inclusion relations between them up to weak isomorphism.Comment: Final version as published in the journal Computing in Geometry and Topology. (30 pages, 22 figures

Many disjoint edges in topological graphs

A monotone cylindrical graph is a topological graph drawn on an open cylinder with an infinite vertical axis satisfying the condition that every vertical line intersects every edge at most once. It is called simple if any pair of its edges have at most one pdint in common: an endpoint or a point at which they properly cross. We say that two edges are disjoint if they do not intersect. We show that every simple complete monotone cylindrical graph on n vertices contains Omega (n(1-epsilon)) pairwise disjoint edges for any epsilon > 0. As a consequence, we show that every simple complete topological graph (drawn in the plane) With n vertices contains Omega(n(1/2-epsilon)) pairwise disjoint edges for any epsilon > 0. This improves the previous lower bound of Omega(n(1/3)) by Suk which was reproved by Fulek and Ruiz-Vargas. We remark that our proof implies a polynomial time algorithm for finding this set of pairwise disjoint edges. (C) 2016 Elsevier B.V. All rights reserved