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Reconfiguration of Spanning Trees with Many or Few Leaves

Authors Nicolas Bousquet , Takehiro Ito , Yusuke Kobayashi , Haruka Mizuta, Paul Ouvrard, Akira Suzuki , Kunihiro Wasa

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ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • combinatorial reconfiguration
  • spanning trees
  • PSPACE
  • polynomial-time algorithms

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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.

Cite As Get BibTex

Nicolas Bousquet, Takehiro Ito, Yusuke Kobayashi, Haruka Mizuta, Paul Ouvrard, Akira Suzuki, and Kunihiro Wasa. Reconfiguration of Spanning Trees with Many or Few Leaves. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ESA.2020.24

Author Details

Nicolas Bousquet
  • CNRS, LIRIS, Université de Lyon, Université Claude Bernard Lyon 1, France
Takehiro Ito
  • Graduate School of Information Sciences, Tohoku University, Japan
Yusuke Kobayashi
  • Research Institute for Mathematical Sciences, Kyoto University, Japan
Haruka Mizuta
  • Graduate School of Information Sciences, Tohoku University, Japan
Paul Ouvrard
  • Univ. Bordeaux, Bordeaux INP, CNRS, LaBRI, UMR5800, Talence, France
Akira Suzuki
  • Graduate School of Information Sciences, Tohoku University, Japan
Kunihiro Wasa
  • Department of Computer Science and Engineering, Toyohashi University of Technology, Japan

Funding

Partially supported by JSPS and MEAE-MESRI under the Japan-France Integrated Action Program (SAKURA).
  • Bousquet, Nicolas: This work was supported by ANR project GrR (ANR-18-CE40-0032).
  • Ito, Takehiro: Partially supported by JSPS KAKENHI Grant Numbers JP18H04091 and JP19K11814, Japan.
  • Kobayashi, Yusuke: Supported by JSPS KAKENHI Grant Numbers JP17K19960, JP18H05291, and JP20K11692, Japan.
  • Mizuta, Haruka: Partially supported by JSPS KAKENHI Grant Number JP19J10042, Japan.
  • Ouvrard, Paul: This work was supported by ANR project GrR (ANR-18-CE40-0032).
  • Suzuki, Akira: Partially supported by JSPS KAKENHI Grant Numbers JP18H04091 and JP20K11666, Japan.
  • Wasa, Kunihiro: Partially supported by JST CREST Grant Numbers JPMJCR18K3 and JPMJCR1401, and JSPS KAKENHI Grant Number JP19K20350, Japan.

References

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