Reconfiguration of the Union of Arborescences

Authors Yusuke Kobayashi , Ryoga Mahara , Tamás Schwarcz

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Author Details

Yusuke Kobayashi
  • Research Institute for Mathematical Sciences, Kyoto University, Japan
Ryoga Mahara
  • Department of Mathematical Informatics, University of Tokyo, Japan
Tamás Schwarcz
  • MTA-ELTE Momentum Matroid Optimization Research Group, Department of Operations Research, ELTE Eötvös Loránd University, Budapest, Hungary


The authors thank members of the project "Fusion of Computer Science, Engineering and Mathematics Approaches for Expanding Combinatorial Reconfiguration" for discussion on this topic. The authors are grateful to András Frank for bringing the paper [Francisco Barahona and William R Pulleyblank, 1987] to their attention.

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Yusuke Kobayashi, Ryoga Mahara, and Tamás Schwarcz. Reconfiguration of the Union of Arborescences. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 48:1-48:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


An arborescence in a digraph is an acyclic arc subset in which every vertex except a root has exactly one incoming arc. In this paper, we show the reconfigurability of the union of k arborescences for fixed k in the following sense: for any pair of arc subsets that can be partitioned into k arborescences, one can be transformed into the other by exchanging arcs one by one so that every intermediate arc subset can also be partitioned into k arborescences. This generalizes the result by Ito et al. (2023), who showed the case with k = 1. Since the union of k arborescences can be represented as a common matroid basis of two matroids, our result gives a new non-trivial example of matroid pairs for which two common bases are always reconfigurable to each other.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Combinatorial optimization
  • Arborescence packing
  • common matroid basis
  • combinatorial reconfiguration


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