Reconfiguration of Spanning Trees with Many or Few Leaves
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Let G be a graph and T_1,T_2 be two spanning trees of G. We say that T_1 can be transformed into T_2 via an edge flip if there exist two edges e ∈ T_1 and f in T_2 such that T_2= (T_1 ∖ 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 by Ito et al. We investigate the problem of determining, given two spanning trees T_1,T_2 with an additional property Π, if there exists an edge flip transformation from T_1 to T_2 keeping property Π all along. First we show that determining if there exists a transformation from T_1 to T_2 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_1 to T_2 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.
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