Notes on equitable partitions into matching forests in mixed graphs and into b-branchings in digraphs

• Main Author: Takazawa, K. (Author)
• Format: Article
• Language: English
• Published: Discrete Mathematics and Theoretical Computer Science 2022
• Subjects: branching; Branching; Decomposition property; delta-matroid; Delta-matroid; Directed graphs; Equitable partitions; Forestry; In-Degree; integer decomposition property; Integer decomposition property; matching; Matchings; Mixed graph; Polytopes; Simple++
• Online Access: View Fulltext in Publisher

 An equitable partition into branchings in a digraph is a partition of the arc set into branchings such that the sizes of any two branchings differ at most by one. For a digraph whose arc set can be partitioned into k branchings, there always exists an equitable partition into k branchings. In this paper, we present two extensions of equitable partitions into branchings in digraphs: those into matching forests in mixed graphs; and into b-branchings in digraphs. For matching forests, Király and Yokoi (2022) considered a tricriteria equitability based on the sizes of the matching forest, and the matching and branching therein. In contrast to this, we introduce a single-criterion equitability based on the number of covered vertices, which is plausible in the light of the delta-matroid structure of matching forests. While the existence of this equitable partition can be derived from a lemma in Király and Yokoi, we present its direct and simpler proof. For b-branchings, we define an equitability notion based on the size of the b-branching and the indegrees of all vertices, and prove that an equitable partition always exists. We then derive the integer decomposition property of the associated polytopes. © 2022 by the author(s).