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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}, Vi induces a connected subgraph of G on ai 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 292 S. Cichacz, A. Görlich, A. Marczyk, J. PrzybyÃlo and ... arbitrarily vertex decomposable trees with maximum degree three or four.

Arbitrarily vertex decomposable caterpillars with four or five leaves