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A tree T is arbitrarily vertex decomposable if for any sequence of positive integers adding up to the order of T there is a sequence of vertex-disjoint subtrees of T whose orders are given by. An on-line version of the problem of characterizing arbitrarily vertex decomposable trees is completely solved here.

An edge colouring of a graph is said to be neighbour-distinguishing if any two adjacent vertices have distinct sets of colours of their incident edges. In this paper the list version of the problem of determining the minimum number of colours in a neighbour-distinguishing colouring of a given graph is considered.

A k-ranking of a graph G is a colouring ϕ : V (G) → {1,. .. , k} such that any path in G with endvertices x, y fulfilling ϕ(x) = ϕ(y) contains an internal vertex z with ϕ(z) > ϕ(x). On-line ranking number χ * r (G) of a graph G is a minimum k such that G has a k-ranking constructed step by step if vertices of G are coming and coloured one by one in an… (More)

A tree T is arbitrarily vertex decomposable if for any sequence τ of positive integers adding up to the order of T there is a sequence of vertex-disjoint subtrees of T whose orders are given by τ ; from a result by Barth and Fournier it follows that ∆(T) ≤ 4. A necessary and a sufficient condition for being an arbitrarily vertex decomposable star-like tree… (More)