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 [Takehiro Ito et al., 2011]. 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.