Reconfiguration of Spanning Trees with Many or Few Leaves

Let G be a graph and T1, T2 be two spanning trees of G. We say that T1 can be transformed into T2 via an edge flip if there exist two edges e ∈ T1 and f in T2 such that T2 = (T1 \ 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 [12]. We investigate the problem of determining, given two spanning trees T1, T2 with an additional property Π, if there exists an edge flip transformation from T1 to T2 keeping property Π all along. First we show that determining if there exists a transformation from T1 to T2 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 T1 to T2 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.