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arXiv · 1712.01942

Leaf realization problem, caterpillar graphs and prefix normal words

Alexandre Blondin Massé·Julien de Carufel·Alain Goupil·Mélodie Lapointe·Émile Nadeau·Élise Vandomme
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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.

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Alexandre Blondin Massé, Julien de Carufel, Alain Goupil, Mélodie Lapointe, Émile Nadeau, Élise Vandomme. 2017-12-05. Leaf realization problem, caterpillar graphs and prefix normal words. https://doi.org/10.1016/j.tcs.2018.04.019

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