arXiv subjects

Laszlo Szekely

Publications and source records attributed to Laszlo Szekely.

7 recordsLinked to original sources

arXiv2025

Reconstruction of caterpillar tanglegrams

A tanglegram consists of two rooted binary trees with the same number of leaves and a perfect matching between the leaves of the trees. Given a size-$n$ tanglegram, i.e., a tanglegram for two trees with $n$ leaves, a multiset of induced size-$(n-1)$ tanglegrams is obtained by deleting a pair of matched leaves in every possible way. Here, we analyze whether a size-$n$ tanglegram is uniquely encoded by this multiset of size-$(n-1)$ tanglegrams. We answer this question affirmatively in the case that at least one of the two trees of the tanglegram is a caterpillar tree.

math.CO↗

arXiv2023

Decks of rooted binary trees

We consider extremal problems related to decks and multidecks of rooted binary trees (a.k.a. rooted phylogenetic tree shapes). Here, the deck (resp. multideck) of a tree $T$ refers to the set (resp. multiset) of leaf induced binary subtrees of $T$. On the one hand, we consider the reconstruction of trees from their (multi)decks. We give lower and upper bounds on the minimum (multi)deck size required to uniquely encode a rooted binary tree on $n$ leaves. On the other hand, we consider problems related to deck cardinalities. In particular, we characterize trees with minimum-size as well as maximum-size decks. Finally, we present some exhaustive computations for $k$-universal trees, i.e., rooted binary trees that contain all $k$-leaf rooted binary trees as induced subtrees.

math.CO↗

arXiv2023

Universal rooted phylogenetic tree shapes and universal tanglegrams

We provide an $Ω(n\log n) $ lower bound and an $O(n^2)$ upper bound for the smallest size of rooted binary trees (a.k.a. phylogenetic tree shapes), which are universal for rooted binary trees with $n$ leaves, i.e., contain all of them as induced binary subtrees. We explicitly compute the smallest universal trees for $n\leq 11$. We also provide an $Ω(n^2) $ lower bound and an $O(n^4)$ upper bound for the smallest size of tanglegrams, which are universal for size $n$ tanglegrams, i.e., which contain all of them as induced subtanglegrams. Some of our results generalize to rooted $d$-ary trees and to $d$-ary tanglegrams.

math.CO↗

arXiv2018

Using Block Designs in Crossing Number Bounds

The crossing number ${\mbox {cr}}(G)$ of a graph $G=(V,E)$ is the smallest number of edge crossings over all drawings of $G$ in the plane. For any $k\ge 1$, the $k$-planar crossing number of $G$, ${\mbox {cr}}_k(G)$, is defined as the minimum of ${\mbox {cr}}(G_1)+{\mbox {cr}}(G_2)+\ldots+{\mbox {cr}}(G_{k})$ over all graphs $G_1, G_2,\ldots, G_{k}$ with $\cup_{i=1}^{k}G_i=G$. Pach et al. [\emph{Computational Geometry: Theory and Applications} {\bf 68} 2--6, (2018)] showed that for every $k\ge 1$, we have ${\mbox {cr}}_k(G)\le \left(\frac{2}{k^2}-\frac1{k^3}\right){\mbox {cr}}(G)$ and that this bound does not remain true if we replace the constant $\frac{2}{k^2}-\frac1{k^3}$ by any number smaller than $\frac1{k^2}$. We improve the upper bound to $\frac{1}{k^2}(1+o(1))$ as $k\rightarrow \infty$. For the class of bipartite graphs, we show that the best constant is exactly $\frac{1}{k^2}$ for every $k$. The results extend to the rectilinear variant of the $k$-planar crossing number.

math.CO↗

arXiv2017

The k-planar crossing number of random graphs and random regular graphs

We give an explicit extension of Spencer's result on the biplanar crossing number of the Erdos-Renyi random graph $G(n,p)$. In particular, we show that the k-planar crossing number of $G(n,p)$ is almost surely $Ω((n^2p)^2)$. Along the same lines, we prove that for any fixed $k$, the $k$-planar crossing number of various models of random $d$-regular graphs is $Ω((dn)^2)$ for $d > c_0$ for some constant $c_0=c_0(k)$.

math.CO↗

arXiv2009

An improved bound on the Maximum Agreement Subtree problem

We improve the lower bound on the extremal version of the Maximum Agreement Subtree problem. Namely we prove that two binary trees on the same $n$ leaves have subtrees with the same $\geq c\log\log n$ leaves which are homeomorphic, such that homeomorphism is identity on the leaves.

q-bio.PE↗

arXiv2008

Phylogenetic information complexity: Is testing a tree easier than finding it?

Phylogenetic trees describe the evolutionary history of a group of present-day species from a common ancestor. These trees are typically reconstructed from aligned DNA sequence data. In this paper we analytically address the following question: is the amount of sequence data required to accurately reconstruct a tree significantly more than the amount required to test whether or not a candidate tree was the `true' tree? By `significantly', we mean that the two quantities behave the same way as a function of the number of species being considered. We prove that, for a certain type of model, the amount of information required is not significantly different; while for another type of model, the information required to test a tree is independent of the number of leaves, while that required to reconstruct it grows with this number. Our results combine probabilistic and combinatorial arguments.

q-bio.PE↗