2023/04/26 by Kobayashi, Yusuke, Mahara, Ryoga, Schwarcz, Tamás · 1 citation
#Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics
paper · doi:10.48550/arxiv.2304.13217
An arborescence in a digraph is an acyclic arc subset in which every vertex execpt a root has exactly one incoming arc. In this paper, we reveal 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.