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URN: urn:nbn:de:0030-drops-128909
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### Reconfiguration of Spanning Trees with Many or Few Leaves

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### Abstract

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.

### BibTeX - Entry

```@InProceedings{bousquet_et_al:LIPIcs:2020:12890,
author =	{Nicolas Bousquet and Takehiro Ito and Yusuke Kobayashi and Haruka Mizuta and Paul Ouvrard and Akira Suzuki and Kunihiro Wasa},
title =	{{Reconfiguration of Spanning Trees with Many or Few Leaves}},
booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
pages =	{24:1--24:15},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-162-7},
ISSN =	{1868-8969},
year =	{2020},
volume =	{173},
editor =	{Fabrizio Grandoni and Grzegorz Herman and Peter Sanders},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},