Inducibility of Topological Trees

TL;DR

This paper explores the inducibility of topological trees, extending the concept to trees without degree-2 vertices, and identifies extremal trees with maximum inducibility.

Contribution

It introduces the inducibility of topological trees, analyzes its properties, and characterizes trees with maximum inducibility of 1, such as stars and binary caterpillars.

Findings

Stars and binary caterpillars have inducibility 1.

Established a lower bound for inducibility of certain trees.

Connected inducibility with degree-restricted cases.

Abstract

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.