TY - JOUR

T1 - Reconfiguration of Spanning Trees with Many or Few Leaves

AU - Bousquet, Nicolas

AU - Ito, Takehiro

AU - Kobayashi, Yusuke

AU - Mizuta, Haruka

AU - Ouvrard, Paul

AU - Suzuki, Akira

AU - Wasa, Kunihiro

N1 - Publisher Copyright:
Copyright © 2020, The Authors. All rights reserved.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/6/25

Y1 - 2020/6/25

N2 - 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.

AB - 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.

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