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Suppose we are given a graph G together with two proper vertex k-colourings of G, α and β. How easily can we decide whether it is possible to transform α into β by recolouring vertices of G one at a time, making sure we always have a proper k-colouring of G? This decision problem is trivial for k = 2, and decidable in polynomial time for k = 3. Here we(More)
We study the unsplittable flow problem on a path P. We are given a set of n tasks. Each task is specified by a sub path of P , a demand, and a profit. Moreover, each edge of P has a given capacity. The aim is to find a subset of the tasks with maximum profit, for which the given demands can be simultaneously routed along P , subject to the capacities. The(More)
We study the following independent set reconfiguration problem, called TAR-Reachabi-lity: given two independent sets I and J of a graph G, both of size at least k, is it possible to transform I into J by adding and removing vertices one-by-one, while maintaining an independent set of size at least k throughout? This problem is known to be PSPACE-hard in(More)
It is known that graphs on n vertices with minimum degree at least 3 have spanning trees with at least n/4 + 2 leaves and that this can be improved to (n + 4)/3 for cubic graphs without the diamond K 4 − e as a subgraph. We generalize the second result by proving that every graph with minimum degree at least 3, without diamonds and certain subgraphs called(More)
  • Paul S Bonsma, W H M Zijm, Ir J P M Van Den Heuvel, H L Kratsch, E C Bodlaender, Van Berkum +1 other
  • 2006
The graph shown on the cover of this thesis is one of the graphs used in the N P-completeness proofs of Chapter 3. The bold edges indicate a matching-cut in this graph, consisting of two of the six minimal matching-cuts possible in this graph. The cut consisting of the eleven dashed edges separates the graph into two parts with 56 vertices. This is a(More)
We present a polynomial-time algorithm that given two independent sets in a claw-free graph G decides whether one can be transformed into the other by a sequence of elementary steps. Each elementary step is to remove a vertex v from the current independent set S in the sequence and to add a new vertex w (not in S) such that the set S −v +w is independent in(More)