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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)