Analogies between the crossing number and the tangle crossing number

Tanglegrams are special graphs that consist of a pair of rooted binary trees with the same number of leaves, and a perfect matching between the two leaf-sets. These objects are of use in phylogenetics and are represented with straightline drawings where the leaves of the two plane binary trees are on two parallel lines and only the matching edges can cross. The tangle crossing number of a tanglegram is the minimum crossing number over all such drawings and is related to biologically relevant quantities, such as the number of times a parasite switched hosts. Our main results for tanglegrams which parallel known theorems for crossing numbers are as follows. The removal of a single matching edge in a tanglegram with $n$ leaves decreases the tangle crossing number by at most $n-3$, and this is sharp. Additionally, if $\gamma(n)$ is the maximum tangle crossing number of a tanglegram with $n$ leaves, we prove $\frac{1}{2}\binom{n}{2}(1-o(1))\le\gamma(n)<\frac{1}{2}\binom{n}{2}$. Further, we provide an algorithm for computing non-trivial lower bounds on the tangle crossing number in $O(n^4)$ time. This lower bound may be tight, even for tanglegrams with tangle crossing number $\Theta(n^2)$.Comment: 13 pages, 6 figure

An algorithm for estimating the crossing number of dense graphs, and continuous analogs of the crossing and rectilinear crossing numbers

We present a deterministic $n^{2+o(1)}$-time algorithm that approximates the crossing number of any graph $G$ of order $n$ up to an additive error of $o(n^4)$. We also provide a randomized polynomial-time algorithm that constructs a drawing of $G$ with $\text{cr}(G)+o(n^4)$ crossings. These results are made interesting by the well known fact that every dense $n$-vertex graph has crossing number $\Theta(n^4)$. Our work builds on a technique developed by Fox, Pach and S\'uk, who obtained very similar results for the rectilinear crossing number. The results by the aforementioned authors and in this paper imply that the (normalized) crossing and rectilinear crossing numbers are estimable parameters. Motivated by this, we introduce two graphon parameters, the crossing density and the rectilinear crossing density, and then we prove that, in a precise sense, these are the correct continuous analogs of the crossing and rectilinear crossing numbers of graphs.Comment: 23 pages, 4 figure

Central limit theorem for crossings in randomly embedded graphs

We consider the number of crossings in a random embedding of a graph, $G$, with vertices in convex position. We give explicit formulas for the mean and variance of the number of crossings as a function of various subgraph counts of $G$. Using Stein's method and size-bias coupling, we prove an upper bound on the Kolmogorov distance between the distribution of the number of crossings and a standard normal random variable. As an application, we establish central limit theorems, along with convergence rates, for the number of crossings in random matchings, path graphs, cycle graphs, and the disjoint union of triangles.Comment: 18 pages, 5 figures. This is a merger of arXiv:2104.01134 and arXiv:2205.0399

Approximating the rectilinear crossing number

A straight-line drawing of a graph G is a mapping which assigns to each vertex a point in the plane and to each edge a straight-line segment connecting the corresponding two points. The rectilinear crossing number of a graph G, (cr) over bar (G), is the minimum number of pairs of crossing edges in any straight-line drawing of G. Determining or estimating (cr) over bar (G) appears to be a difficult problem, and deciding if (cr) over bar (G) <= k is known to be NP-hard. In fact, the asymptotic behavior of (cr) over bar (K-n) is still unknown. In this paper, we present a deterministic n(2+o(1))-time algorithm that finds a straight-line drawing of any n-vertex graph G with ((cr) over barG)+ o(n(4)) pairs of crossing edges. Together with the well-known Crossing Lemma due to Ajtai et al. and Leighton, this result implies that for any dense n-vertex graph G, one can efficiently find a straight-line drawing of G with (1 + o(1))(cr) over bar (G) pairs of crossing edges. (C) 2019 Elsevier B.V. All rights reserved

Diszkrét és kombinatórikus geometriai kutatások = Topics in discrete and combinatorial geometry

A most lezárult OTKA grant, 8 résztvevő diszkrét geometriai kutatását támogatta. Itt a témák ilusztrálására kiemelünk néhányat az elért 72 publikációból. 1. Jelentős eredmények születtek (8 cikk) gráfok síkba rajzolhatóságáról, például az úgynevezett metszési számról. 2. Többek között sikerült igazolni Katchalski és Lewis 20 éves sejtését, mely szerint diszjunkt egységkörökből álló rendszereknél ha bármely három körnek van közös metsző egyenese akkor van olyan egyenes, amely legfeljebb 2 kör kivételével valamennyit metsz. 3. Littlewood (1964) problémájaként ismert volt az a kérdés, hogy hány henger érintheti kölcsönösen egymást? Viszonylag alacsony felső korlátot találtunk és egy régóta ismert elhelyzés valótlanságát igazoltuk. 4. Többszörös fedések egyszerű fedésekre való szétbontását vizsgáltuk és értünk el lényeges előrelépést. 5. A Borsuk-féle darabolási problémanak azt a variánsát vizsgáltuk, amelyben a darabolást u. n. hengeres darabolásra korlátozták. 6. Bebizonyítottuk, hogy ''nem nagyon elnyúlt'' ellipszisek esetében a sík legritkább fedésének meghatározásánál el lehet tekinteni az u.n. nem-keresztezési feltételtől. 7. A sejtetthez nagyon közeli korlátot találtunk arra a problémára, hogy az n-dimenziós térben legfeljebb hány homotetikus konvex test helyezhető el úgy, hogy bármely kettő érintse egymást. | Discrete geometry in Hungary flourished since the sixties as a result of the work of László Fejes Tóth. The supported research of 8 participant also belongs to this area. Here we illustrate the achieved 72 publications by mentioning a few results. 1. Important theorems (8 papers) were proved concerning graph drawing. 2. Among others, a 20 year old problem of Katchalsky was proved, stating that in a packing of congruent circles, if any three has a common transversal, then there is a line, which avoids at most two of the circles. 3. Concerning a conjecture of Littlewood we found a small upper bound for the number of infinite cylinders which mutually touch each other. 4. We studied decomposability of multiple coverings into single coverings. 5. We studied that variant of the famous Borsuk problem where the partitions are restricted to cylindrical partitions. 6. We proved that in case of ellipses which are not ''too long'' at determining the thinnest covering one can omit the usually needed noncrossing condition. 7. A bound close to the conjectured bound was found concerning the number of n-dimensional homothetic convex solids which mutually touch each other

Smoothed Analysis on Connected Graphs

The main paradigm of smoothed analysis on graphs suggests that for any large graph G in a certain class of graphs, perturbing slightly the edges of G at random (usually adding few random edges to G) typically results in a graph having much "nicer" properties. In this work we study smoothed analysis on trees or, equivalently, on connected graphs. Given an n-vertex connected graph G, form a random supergraph of G* of G by turning every pair of vertices of G into an edge with probability epsilon/n, where epsilon is a small positive constant. This perturbation model has been studied previously in several contexts, including smoothed analysis, small world networks, and combinatorics. Connected graphs can be bad expanders, can have very large diameter, and possibly contain no long paths. In contrast, we show that if G is an n-vertex connected graph then typically G* has edge expansion Omega(1/(log n)), diameter O(log n), vertex expansion Omega(1/(log n)), and contains a path of length Omega(n), where for the last two properties we additionally assume that G has bounded maximum degree. Moreover, we show that if G has bounded degeneracy, then typically the mixing time of the lazy random walk on G* is O(log^2(n)). All these results are asymptotically tight

On Graph Crossing Number and Edge Planarization

Given an n-vertex graph G, a drawing of G in the plane is a mapping of its vertices into points of the plane, and its edges into continuous curves, connecting the images of their endpoints. A crossing in such a drawing is a point where two such curves intersect. In the Minimum Crossing Number problem, the goal is to find a drawing of G with minimum number of crossings. The value of the optimal solution, denoted by OPT, is called the graph's crossing number. This is a very basic problem in topological graph theory, that has received a significant amount of attention, but is still poorly understood algorithmically. The best currently known efficient algorithm produces drawings with $O(\log^2 n)(n + OPT)$ crossings on bounded-degree graphs, while only a constant factor hardness of approximation is known. A closely related problem is Minimum Edge Planarization, in which the goal is to remove a minimum-cardinality subset of edges from G, such that the remaining graph is planar. Our main technical result establishes the following connection between the two problems: if we are given a solution of cost k to the Minimum Edge Planarization problem on graph G, then we can efficiently find a drawing of G with at most \poly(d)\cdot k\cdot (k+OPT) crossings, where $d$ is the maximum degree in G. This result implies an O(n\cdot \poly(d)\cdot \log^{3/2}n)-approximation for Minimum Crossing Number, as well as improved algorithms for special cases of the problem, such as, for example, k-apex and bounded-genus graphs

A note on the width of sparse random graphs

In this note, we consider the width of a supercritical random graph according to some commonly studied width measures. We give short, direct proofs of results of Lee, Lee and Oum, and of Perarnau and Serra, on the rank- and tree-width of the random graph $G(n,p)$ when $p= \frac{1+\epsilon}{n}$ for $\epsilon > 0$ constant. Our proofs avoid the use, as a black box, of a result of Benjamini, Kozma and Wormald on the expansion properties of the giant component in this regime, and so as a further benefit we obtain explicit bounds on the dependence of these results on $\epsilon$. Finally, we also consider the width of the random graph in the weakly supercritical regime, where $\epsilon = o(1)$ and $\epsilon^3n \to \infty$. In this regime, we determine, up to a constant multiplicative factor, the rank- and tree-width of $G(n,p)$ as a function of $n$ and $\epsilon$.Comment: 18 page