[Paper Review] Inducibility of d-ary trees

Éva Czabarka, Audace A. V. Dossou-Olory+2 more|arXiv (Cornell University)|Feb 11, 2018

Graph theory and applications12 references3 citations

TL;DR

This paper introduces and studies the inducibility of $d$-ary trees—rooted trees with bounded out-degree $d \geq 2$—focusing on the asymptotic maximum density of leaf-induced subtrees. It establishes that binary caterpillars achieve the maximal inducibility of 1 in $d$-ary trees for any $d$, and proves that the difference between maximum density and inducibility decays as $\mathcal{O}(|T|^{-1/2})$ for strictly $d$-ary trees, improving upon the general $\mathcal{O}(|T|^{-1})$ bound. The inducibility of a binary tree in $d$-ary trees is independent of $d$, and a general lower bound on inducibility is derived.

ABSTRACT

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 $d$.

Motivation & Objective

• To extend the concept of inducibility from binary trees to $d$-ary trees, where each non-leaf vertex has between 2 and $d$ children.
• To determine the exact inducibility for fundamental $d$-ary tree structures such as stars and binary caterpillars.
• To analyze the asymptotic behavior of the maximum density of a fixed $d$-ary tree $D$ in larger $d$-ary trees $T$, and to quantify the convergence rate to the inducibility.
• To resolve a conjecture from prior work on binary trees by showing that the difference between maximum density and inducibility is $\mathcal{O}(|T|^{-1/2})$ in the strictly $d$-ary case.
• To investigate whether the inducibility of a tree in $d$-ary trees depends on $d$, particularly for non-binary trees.

Proposed method

• Defines the inducibility $i_d(D)$ of a $d$-ary tree $D$ as the limit superior of the maximum density of leaf-induced copies of $D$ in larger $d$-ary trees $T$.
• Introduces the concept of leaf-induced subtrees by taking the minimal subtree over a leaf subset and removing degree-2 vertices (except possibly the root).
• Uses combinatorial bounds and asymptotic analysis to derive a general lower bound on inducibility, leveraging the structure of $\mathcal{F}(S_1; S_2)$, a tree formed by attaching $|S_1|$ copies of $S_2$ to a root.
• Applies extremal combinatorics and induction to show that any $d$-ary tree with more than $d^{k-2}$ leaves must contain a $k$-leaf binary caterpillar $F^2_k$, establishing a key structural result.
• Employs flag algebra-inspired reasoning and limit analysis to derive the $\mathcal{O}(|T|^{-1/2})$ convergence rate for strictly $d$-ary trees.
• Proves that the inducibility of a binary tree $D$ in $d$-ary trees is independent of $d$, using recursive construction and density comparison.

Experimental results

Research questions

• RQ1 What is the exact inducibility of stars and binary caterpillars in $d$-ary trees for $d \geq 2$?
• RQ2 How does the difference between the maximum density of a $d$-ary tree $D$ in a larger $d$-ary tree $T$ and its inducibility decay as $|T| \to \infty$?
• RQ3 Is the inducibility of a binary tree in $d$-ary trees independent of $d$, as conjectured in prior work?
• RQ4 Which $d$-ary trees achieve the maximum inducibility of 1, and are there any beyond binary caterpillars?
• RQ5 Does the inducibility of a non-binary $d$-ary tree depend on $d$, or is it constant across $d \geq 2$?

Key findings

• The inducibility of the $k$-leaf binary caterpillar $F^2_k$ is exactly 1 in $d$-ary trees for all $d \geq 2$ and all $k \geq 2$, making it the unique $d$-ary tree with maximal inducibility.
• For strictly $d$-ary trees (where every non-leaf vertex has exactly $d$ children), the difference between the maximum density of a $d$-ary tree $D$ in $T$ and its inducibility is $\mathcal{O}(|T|^{-1/2})$, improving upon the general $\mathcal{O}(|T|^{-1})$ bound.
• The inducibility of a binary tree $D$ in $d$-ary trees is independent of $d$, meaning $i_d(D) = i_{d+1}(D) = \cdots$ for all $d \geq 2$, confirming a conjecture from earlier work.
• A general lower bound on inducibility is established for $d$-ary trees, derived via recursive tree constructions and asymptotic analysis of induced subtree counts.
• The maximum inducibility is attained exclusively by binary caterpillars among all $d$-ary trees, and no other $d$-ary tree achieves inducibility 1.
• The paper provides evidence that for non-binary $d$-ary trees, inducibility may depend on $d$, supporting the conjecture that only binary trees have $d$-independent inducibility.