Leaf realization problem, caterpillar graphs and prefix normal words

TL;DR

This paper introduces the leaf realization problem, exploring sequences of maximum leaves in induced subtrees, and establishes a connection between caterpillar graphs' leaf sequences and prefix normal words.

Contribution

It defines the leaf realization problem, analyzes sequence structures for graphs and trees, and links caterpillar graph leaf sequences to prefix normal words.

Findings

Characterization of leaf sequences for general graphs and trees

A bijection between derivatives of leaf sequences in caterpillar graphs and prefix normal words

Foundational observations on the structure of leaf realization sequences

Abstract

Given a simple graph $G$ with $n$ vertices and a natural number $i \leq n$, let $L_G(i)$ be the maximum number of leaves that can be realized by an induced subtree $T$ of $G$ with $i$ vertices. We introduce a problem that we call the \emph{leaf realization problem}, which consists in deciding whether, for a given sequence of $n+1$ natural numbers $(\ell_0, \ell_1, \ldots, \ell_n)$, there exists a simple graph $G$ with $n$ vertices such that $\ell_i = L_G(i)$ for $i = 0, 1, \ldots, n$. We present basic observations on the structure of these sequences for general graphs and trees. In the particular case where $G$ is a caterpillar graph, we exhibit a bijection between the set of the discrete derivatives of the form $(\Delta L_G(i))_{1 \leq i \leq n - 3}$ and the set of prefix normal words.