Leaf realization problem, caterpillar graphs and prefix normal words

Abstract Given a simple graph G with n vertices and a natural number i ≤ 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 leaf realization problem, which consists in deciding whether, for a given sequence of n + 1 natural numbers ( l 0 , l 1 , … , l n ) , there exists a simple graph G with n vertices such that l i = L G ( i ) for i = 0 , 1 , … , 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 ( Δ L G ( i ) ) 1 ≤ i ≤ n − 3 and the set of prefix normal words.

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