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

A tanglegram Kuratowski theorem

A tanglegram consists of two rooted binary plane trees with the same number of leaves and a perfect matching between the two leaf sets. Tanglegrams are drawn with the leaves on two parallel lines, the trees on either side of the strip created by these lines, and the perfect matching inside the strip. If this can be done without any edges crossing, a tanglegram is called planar. We show that every non-planar tanglegram contains one of two non-planar 4-leaf tanglegrams as induced subtanglegram, which parallels Kuratowski's Theorem

On the inducibility of small trees

The quantity that captures the asymptotic value of the maximum number of appearances of a given topological tree (a rooted tree with no vertices of outdegree $1$) $S$ with $k$ leaves in an arbitrary tree with sufficiently large number of leaves is called the inducibility of $S$. Its precise value is known only for some specific families of trees, most of them exhibiting a symmetrical configuration. In an attempt to answer a recent question posed by Czabarka, Sz\'ekely, and the second author of this article, we provide bounds for the inducibility $J(A_5)$ of the $5$-leaf binary tree $A_5$ whose branches are a single leaf and the complete binary tree of height $2$. It was indicated before that $J(A_5)$ appears to be `close' to $1/4$. We can make this precise by showing that $0.24707\ldots \leq J(A_5) \leq 0.24745\ldots$. Furthermore, we also consider the problem of determining the inducibility of the tree $Q_4$, which is the only tree among $4$-leaf topological trees for which the inducibility is unknown

An infinite antichain of planar tanglegrams

Contrary to the expectation arising from the tanglegram Kuratowski theorem of \'E. Czabarka, L.A. Sz\'ekely and S. Wagner [SIAM J. Discrete Math. 31(3): 1732--1750, (2017)], we construct an infinite antichain of planar tanglegrams with respect to the induced subtanglegram partial order. R.E. Tarjan, R. Laver, D.A. Spielman and M. B\'ona, and possibly others, showed that the partially ordered set of finite permutations ordered by deletion of entries contains an infinite antichain, i.e. there exists an infinite collection of permutations, such that none of them contains another as a pattern. Our construction adds a twist to the construction of Spielman and B\'ona [Electr. J. Comb, Vol. 7. N2.

Inducibility of Topological Trees

Trees without vertices of degree $2$ are sometimes named topological trees. In this work, we bring forward the study of the inducibility of (rooted) topological trees with a given number of leaves. The inducibility of a topological tree $S$ is the limit superior of the proportion of all subsets of leaves of $T$ that induce a copy of $S$ as the size of $T$ grows to infinity. In particular, this relaxes the degree-restriction for the existing notion of the inducibility in $d$-ary trees. We discuss some of the properties of this generalised concept and investigate its connection with the degree-restricted inducibility. In addition, we prove that stars and binary caterpillars are the only topological trees that have an inducibility of $1$. We also find an explicit lower bound on the limit inferior of the proportion of all subsets of leaves of $T$ that induce either a star or a binary caterpillar as the size of $T$ tends to infinity.Comment: 15 page

Inducibility of d-ary trees

Imitating a recently introduced invariant of trees, we initiate the study of the inducibility of $d$-ary trees (rooted trees whose vertex outdegrees are bounded from above by $d\geq 2$) with a given number of leaves. We determine the exact inducibility for stars and binary caterpillars. For $T$ in the family of strictly $d$-ary trees (every vertex has $0$ or $d$ children), we prove that the difference between the maximum density of a $d$-ary tree $D$ in $T$ and the inducibility of $D$ is of order $\mathcal{O}(|T|^{-1/2})$ compared to the general case where it is shown that the difference is $\mathcal{O}(|T|^{-1})$ which, in particular, responds positively to an existing conjecture on the inducibility in binary trees. We also discover that the inducibility of a binary tree in $d$-ary trees is independent of $d$. Furthermore, we establish a general lower bound on the inducibility and also provide a bound for some special trees. Moreover, we find that the maximum inducibility is attained for binary caterpillars for every $d$

The minimum asymptotic density of binary caterpillars

Given $d\geq 2$ and two rooted $d$-ary trees $D$ and $T$ such that $D$ has $k$ leaves, the density $\gamma(D,T)$ of $D$ in $T$ is the proportion of all $k$-element subsets of leaves of $T$ that induce a tree isomorphic to $D$, after erasing all vertices of outdegree $1$. In a recent work, it was proved that the limit inferior of this density as the size of $T$ grows to infinity is always zero unless $D$ is the $k$-leaf binary caterpillar $F^2_k$ (the binary tree with the property that a path remains upon removal of all the $k$ leaves). Our main theorem in this paper is an exact formula (involving both $d$ and $k$) for the limit inferior of $\gamma(F^2_k,T)$ as the size of $T$ tends to infinity.Comment: 16 page

Further results on the inducibility of dd-ary trees

A subset of leaves of a rooted tree induces a new tree in a natural way. The density of a tree $D$ inside a larger tree $T$ is the proportion of such leaf-induced subtrees in $T$ that are isomorphic to $D$ among all those with the same number of leaves as $D$. The inducibility of $D$ measures how large this density can be as the size of $T$ tends to infinity. In this paper, we explicitly determine the inducibility in some previously unknown cases and find general upper and lower bounds, in particular in the case where $D$ is balanced, i.e., when its branches have at least almost the same size. Moreover, we prove a result on the speed of convergence of the maximum density of $D$ in strictly $d$-ary trees $T$ (trees where every internal vertex has precisely $d$ children) of a given size $n$ to the inducibility as $n \to \infty$, which supports an open conjecture

Results on Select Combinatorial Problems With an Extremal Nature

This dissertation is split into three sections, each containing new results on a particular combinatorial problem. In the first section, we consider the set of 3-connected quadrangulations on n vertices and the set of 5-connected triangulations on n vertices. In each case, we find the minimum Wiener index of any graph in the given class, and identify graphs that obtain this minimum value. Moreover, we prove that these graphs are unique up to isomorphism. In the second section, we work with structures emerging from the biological sciences called tanglegrams. In particular, our work pertains to an invariant of tanglegrams called the tangle crossing number, an invariant which is NP-hard to compute. Czabarka, Székely, and Wagner found a finite characterization of tanglegrams with tangle crossing number equal to 0, which motivated the work here. In particular, our aim was to find a similar finite (and minimal) characterization of tanglegrams with tangle crossing number at least k, for any fixed k ≥ 2. We set out to prove this using an elegant order-theoretic argument, but came to another surprising result instead; we proved that the set of tanglegrams with the induced subtanglegram relation is not a well partial order. In the final section, we work on the problem of finding an upper bound on the diameter of graphs with particular properties. It was proven independently by several groups that for fixed minimum degree $\delta\ge 2$, every connected graph $G$ of order $n$ satisfies diam$(G)\le \dfrac{3n}{\delta + 1} + O(1)$ as $n\rightarrow \infty$. Erd\H{o}s, Pach, Pollack, and Tuza noticed that the graphs which achieve the aforementioned bound all contain complete subgraphs whose order increases with $n$, and conjectured that if we disallowed complete subgraphs of a given fixed size, then we could improve the bound. Czabarka, Singgih, and Sz\\u27ekely recently found a counterexample to part of the conjecture of Erd\H{o}s \emph{et al.} and formulated a new conjecture. Under a stronger assumption, we verify two cases of this new conjecture using a novel unified duality approach

Reassessment of the evolutionary history of the late Triassic and early Jurassic sauropodomorph dinosaurs through comparative cladistics and the supermatrix approach

Non-sauropod sauropodomorphs, also known as 'basal sauropodomorphs' or 'prosauropods', have been thoroughly studied in recent years. Several hypotheses on the interrelationships within this group have been proposed, ranging from a complete paraphyly, where the group represents a grade from basal saurischians to Sauropoda, to a group on its own. The grade-like hypothesis is the most accepted; however, the relationships between the different taxa are not consistent amongst the proposed scenarios. These inconsistencies have been attributed to missing data and unstable (i.e., poorly preserved) taxa, nevertheless, an extensive comparative cladistic analysis has found that these inconsistencies instead come from the character coding and character selection, plus the strategies on merging data sets. Furthermore, a detailed character analysis using information theory and mathematical topology as an approach for character delineation is explored here to operationalise characters and reduce the potential impact of missing data. This analysis also produced the largest and most comprehensive matrix after the reassessment and operationalisation of every character applied to this group far. Additionally, partition analyses performed on this data set have found consistencies in the interrelationships within non-sauropod Sauropodomorpha and has found strong support for smaller clades such as Plateosauridae, Riojasauridae, Anchisauridae, Massospondylinae and Lufengosarinae. The results of these analyses also highlight a different scenario on how quadrupedality evolved, independently originating twice within the group, and provide a better framework to understand the palaeo-biogeography and diversification rate of the first herbivore radiation of dinosaurs