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Reconfiguration of Spanning Trees with Degree Constraints or Diameter Constraints

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Abstract

We investigate the complexity of finding a transformation from a given spanning tree in a graph to another given spanning tree in the same graph via a sequence of edge flips. The exchange property of the matroid bases immediately yields that such a transformation always exists if we have no constraints on spanning trees. In this paper, we wish to find a transformation which passes through only spanning trees satisfying some constraint. Our focus is bounding either the maximum degree or the diameter of spanning trees, and we give the following results. The problem with a lower bound on maximum degree is solvable in polynomial time, while the problem with an upper bound on maximum degree is PSPACE-complete. The problem with a lower bound on diameter is NP-hard, while the problem with an upper bound on diameter is solvable in polynomial time.

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Notes

  1. Formally, we perturb the length by defining \(\ell (e)\) as the vector \((1, \chi _{i(e)}) \in {\mathbb {R}} \times {\mathbb {R}}^{|E|}\), where \(\chi _{j} \in {\mathbb {R}}^{|E|}\) is the characteristic vector of \(\chi _{j}\) (see e.g., [3]). Then, we use the lexicographical order to compare the lengths of paths. With this notation, for \(\alpha \in {\mathbb {R}} \times {\mathbb {R}}^{|E|}\), we can formally define \({\bar{\alpha }} \in {\mathbb {R}}\) as the first coordinate of \(\alpha \). However, to simplify the notation, we define \(\ell (e)\) as a real number in this paper.

  2. More precisely, if \(e_1\) and \(e_2\) share an end vertex, then \(b_\top \) (resp. \(b_\bot \)) may be contained in \(e_1\) or \(e_2\). In such a case, \({V_\top } \subseteq V\) (resp. \({V_\bot } \subseteq V\)) is defined as the vertex set of the connected component of \(Q-\left\{ e_1, e_2\right\} \) that does not contain \(b_\bot \) (resp. \(b_\top \)).

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Funding

This work is partially supported by JSPS and MEAE-MESRI under the Japan-France Integrated Action Program (SAKURA). Nicolas Bousquet was supported by ANR project GrR (ANR-18-CE40-0032). Takehiro Ito is partially supported by JSPS KAKENHI Grant Numbers JP18H04091, JP19K11814 and JP20H05793, Japan. Yusuke Kobayashi is partially supported by JSPS KAKENHI Grant Numbers JP18H05291, JP20K11692, and JP20H05795, Japan. Paul Ouvrard was supported by ANR project GrR (ANR-18-CE40-0032). Akira Suzuki is partially supported by JSPS KAKENHI Grant Numbers JP18H04091, JP20K11666 and JP20H05794, Japan. Kunihiro Wasa is partially supported by JST CREST Grant Numbers JPMJCR18K3 and JPMJCR1401, and JSPS KAKENHI Grant Numbers JP19K20350 and JP20H05793, Japan.

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Authors and Affiliations

  1. CNRS, LIRIS, Université de Lyon, Lyon, France

    Nicolas Bousquet

  2. Graduate School of Information Sciences, Tohoku University, Sendai, Japan

    Takehiro Ito, Haruka Mizuta & Akira Suzuki

  3. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan

    Yusuke Kobayashi

  4. Université de Bordeaux, Bordeaux, France

    Paul Ouvrard

  5. Faculty of Science and Engineering, Hosei University, Tokyo, Japan

    Kunihiro Wasa

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  1. Nicolas Bousquet
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  2. Takehiro Ito
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  3. Yusuke Kobayashi
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  4. Haruka Mizuta
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  6. Akira Suzuki
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  7. Kunihiro Wasa
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Correspondence to Akira Suzuki.

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Bousquet, N., Ito, T., Kobayashi, Y. et al. Reconfiguration of Spanning Trees with Degree Constraints or Diameter Constraints. Algorithmica 85, 2779–2816 (2023). https://doi.org/10.1007/s00453-023-01117-z

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  • DOI: https://doi.org/10.1007/s00453-023-01117-z

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