Shortest Reconfiguration of Colorings Under Kempe Changes

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Reconfiguration of Spanning Trees with Degree Constraint or Diameter Constraint

We investigate the complexity of finding a transformation from a given spanning tree in a graph to another given spanning tree in the same graph via a sequence of edge flips. The exchange property of the matroid bases immediately yields that such a transformation always exists if we have no constraints on spanning trees. In this paper, we wish to find a transformation which passes through only spanning trees satisfying some constraint. Our focus is bounding either the maximum degree or the diameter of spanning trees, and we give the following results. The problem with a lower bound on maximum degree is solvable in polynomial time, while the problem with an upper bound on maximum degree is PSPACE-complete. The problem with a lower bound on diameter is NP-hard, while the problem with an upper bound on diameter is solvable in polynomial time

Reconfiguration of Spanning Trees with Many or Few Leaves

Let G be a graph and T?,T? be two spanning trees of G. We say that T? can be transformed into T? via an edge flip if there exist two edges e ? T? and f in T? such that T? = (T??e) ? f. Since spanning trees form a matroid, one can indeed transform a spanning tree into any other via a sequence of edge flips, as observed in [Takehiro Ito et al., 2011]. We investigate the problem of determining, given two spanning trees T?,T? with an additional property ?, if there exists an edge flip transformation from T? to T? keeping property ? all along. First we show that determining if there exists a transformation from T? to T? such that all the trees of the sequence have at most k (for any fixed k ? 3) leaves is PSPACE-complete. We then prove that determining if there exists a transformation from T? to T? such that all the trees of the sequence have at least k leaves (where k is part of the input) is PSPACE-complete even restricted to split, bipartite or planar graphs. We complete this result by showing that the problem becomes polynomial for cographs, interval graphs and when k = n-2