Notes on Equitable Partitions into Matching Forests in Mixed Graphs and into $b$-branchings in Digraphs

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