Augmenting a hypergraph to have a matroid-based $(f,g)$-bounded $(\alpha,\beta)$-limited packing of rooted hypertrees

TL;DR

This paper extends the theory of packing rooted trees in graphs and hypergraphs, providing new characterizations and augmentation methods for matroid-based packings with bounded root counts.

Contribution

It generalizes classical tree packing results to include bounds on root counts and develops augmentation techniques for such packings in hypergraphs.

Findings

Extended Nash-Williams and Tutte theorems for rooted tree packings.

Characterized existence conditions for bounded root packings in graphs and hypergraphs.

Provided algorithms for minimal augmentation to achieve desired packings.

Abstract

The aim of this paper is to further develop the theory of packing trees in a graph. We first prove the classic result of Nash-Williams \cite{NW} and Tutte \cite{Tu} on packing spanning trees by adapting Lov\'asz' proof \cite{Lov} of the seminal result of Edmonds \cite{Egy} on packing spanning arborescences in a digraph. Our main result on graphs extends the theorem of Katoh and Tanigawa \cite{KT} on matroid-based packing of rooted trees by characterizing the existence of such a packing satisfying the following further conditions: for every vertex $v$, there are a lower bound $f(v)$ and an upper bound $g(v)$ on the number of trees rooted at $v$ and there are a lower bound $\alpha$ and an upper bound $\beta$ on the total number of roots. We also answer the hypergraphic version of the problem. Furthermore, we are able to solve the augmentation version of the latter problem, where the goal is to add a minimum number of edges to have such a packing. The methods developed in this paper to solve these problems may have other applications in the future.