Geodesic
Arbitrarily vertex decomposable caterpillars with four or five leaves
Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 2, pp. 291-305

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A graph G of order n is called arbitrarily vertex decomposable if for each sequence (a₁,...,aₖ) of positive integers such that a₁+...+aₖ = n there exists a partition (V₁,...,Vₖ) of the vertex set of G such that for each i ∈ 1,...,k, V_i induces a connected subgraph of G on a_i vertices. D. Barth and H. Fournier showed that if a tree T is arbitrarily vertex decomposable, then T has maximum degree at most 4. In this paper we give a complete characterization of arbitrarily vertex decomposable caterpillars with four leaves. We also describe two families of arbitrarily vertex decomposable trees with maximum degree three or four.
Keywords: arbitrarily vertex decomposable graphs, trees, caterpillars, star-like trees 
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     title = {Arbitrarily vertex decomposable caterpillars with four or five leaves},
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Cichacz, Sylwia; Görlich, Agnieszka; Marczyk, Antoni; Przybyło, Jakub; Woźniak, Mariusz. Arbitrarily vertex decomposable caterpillars with four or five leaves. Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 2, pp. 291-305. http://geodesic.mathdoc.fr/item/DMGT_2006_26_2_a10/