Inducibility in Binary Trees and Crossings in Random Tanglegrams

Abstract

Abstract. In analogy to other concepts of a similar nature, we define the inducibility of a rooted binary tree. Given a fixed rooted binary tree B with k leaves, we let γ(B, T ) be the proportion of all subsets of k leaves in T that induce a tree isomorphic to B. The inducibility of B is lim sup |T |→∞ γ(B, T ). We determine the inducibility in some special cases, show that every binary tree has positive inducibility and prove that caterpillars are the only binary trees with inducibility 1. We also formulate some open problems and conjectures on the inducibility. Finally, we present an application to crossing numbers of random tanglegrams.

Inducibility of topological trees

“…To this day, there is substantial activity regarding this concept. In analogy to (Pippenger and Golumbic, 1975), the inducibility of a rooted tree S with k leaves is defined as the maximum frequency at which S can appear as a subtree induced by k leaves of an arbitrary rooted tree whose number of leaves tends to infinity (Czabarka et al, 2017(Czabarka et al, , 2020Dossou-Olory and Wagner, 2019). Bubeck and Linial (2016) defined the inducibility of a tree S with k vertices as the maximum proportion of S as a subtree among all k-vertex subtrees of a tree whose number of vertices tends to infinity.…”

“…The concept of inducibility of a tree with k leaves is still new and the precise value of the inducibility is known only for a few classes of trees, most of them exhibiting a symmetrical configuration. The recent paper (Czabarka et al, 2017) raised some questions on the inducibility of binary trees, one of which is discussed and approximately solved within this note. The present paper also covers a related problem concerning the inducibility of a ternary tree with four leaves.…”

A tanglegram Kuratowski theorem

“…Note that the (graph) crossing number cr( ) of a tanglegram is less or equal to the (tanglegram) crossing number crt( ) of , since the tanglegram layout is more restrictive than the graph drawing. The following proposition that provides an almost obvious characterization of planarity of tanglegram, was found by [25] and later rediscovered [8]:…”

“…Otherwise it is called nonplanar . In an earlier article , we showed that the tanglegram crossing number of a randomly and uniformly selected tanglegram of size n is with high probability, i.e. as large as it can be within a constant multiplicative factor.…”

“…Note that the (graph) crossing number of a tanglegram T is less or equal to the (tanglegram) crossing number of T , since the tanglegram layout is more restrictive than the graph drawing. The following proposition that provides an almost obvious characterization of planarity of tanglegram, was found by and later rediscovered : Proposition Assume that a tanglegram T is represented by the layout , and let the roots of L and R be r and ρ. Let denote the graph in which the underlying graph of T (consisting of the two binary trees and the matching edges) is augmented by the edge .…”