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The problem of matroid-reachability-based packing of arborescences was solved by Kir\'aly. Here we solve the corresponding decomposition problem that turns out to be more complicated. The result is obtained from the solution of the more general problem of matroid-reachability-based $(\ell,\ell')$-limited packing of arborescences where we are given a lower bound $\ell$ and an upper bound $\ell'$ on the total number of arborescences in the packing. The problem is considered for branchings and in directed hypergraphs as well.

The seminal papers of Edmonds \cite{Egy}, Nash-Williams \cite{NW} and Tutte \cite{Tu} have laid the foundations of the theories of packing arborescences and packing trees. The directed version has been extensively investigated, resulting in a great number of generalizations. In contrast, the undirected version has been marginally considered. The aim of this paper is to further develop the theories of packing trees and forests. Our main result on graphs characterizes the existence of a packing of $k$ forests, $F_1, \ldots, F_k$, in a graph $G$ such that each vertex of $G$ belongs to exactly $h$ of the forests, and in addition, each $F_i$ has between $\ell(i)$ and $\ell'(i)$ connected components and the total number of connected components in the packing is between $\alpha$ and $\beta$. Finally, we extend this result to hypergraphs and dypergraphs, the latter giving a generalization of a theorem of B\'erczi and Frank \cite{BF3}., Comment: 18 pages

The aim of this paper is twofold. We first provide a new orientation theorem which gives a natural and simple proof of a result of Gao, Yang \cite{GY} on matroid-reachability-based packing of mixed arborescences in mixed graphs by reducing it to the corresponding theorem of Cs. Kir\'aly \cite{cskir} on directed graphs. Moreover, we extend another result of Gao, Yang \cite{GY2} by providing a new theorem on mixed hypergraphs having a packing of mixed hyperarborescences such that their number is at least $\ell$ and at most $\ell'$, each vertex belongs to exactly $k$ of them, and each vertex $v$ is the root of least $f(v)$ and at most $g(v)$ of them., Comment: 14 pages

We show that if the arc-connectivity of a directed graph $D$ is at most $\lfloor\frac{k+1}{2}\rfloor$ and the reorientation of an arc set $F$ in $D$ results in a $k$-arc-connected directed graph then we can reorient one arc of $F$ without decreasing the arc-connectivity of $D.$ This improves a result of Fukuda, Prodon, Sakuma and one of Ito et al. for $k\in\{2,3\}$., Comment: 8 pages

The orientation theorem of Nash-Williams states that an undirected graph admits a $k$-arc-connected orientation if and only if it is $2k$-edge-connected. Recently, Ito et al. showed that any orientation of an undirected $2k$-edge-connected graph can be transformed into a $k$-arc-connected orientation by reorienting one arc at a time without decreasing the arc-connectivity at any step, thus providing an algorithmic proof of Nash-Williams' theorem. We generalize their result to hypergraphs and therefore provide an algorithmic proof of the characterization of hypergraphs with a $k$-hyperarc-connected orientation originally given by Frank et al. We prove that any orientation of an undirected $(k,k)$-partition-connected hypergraph can be transformed into a $k$-hyperarc-connected orientation by reorienting one hyperarc at a time without decreasing the hyperarc-connectivity in any step. Furthermore, we provide a simple combinatorial algorithm for computing such a transformation in polynomial time., Comment: 18 pages, 3 figures, 3 algorithms

We say that a tree $T$ is an $S$-Steiner tree if $S \subseteq V(T)$ and a hypergraph is an $S$-Steiner hypertree if it can be trimmed to an $S$-Steiner tree. We prove that it is NP-complete to decide, given a hypergraph $\mathcal{H}$ and some $S \subseteq V(\mathcal{H})$, whether there is a subhypergraph of $\mathcal{H}$ which is an $S$-Steiner hypertree. As corollaries, we give two negative results for two Steiner orientation problems in hypergraphs. Firstly, we show that it is NP-complete to decide, given a hypergraph $\mathcal{H}$, some $r \in V(\mathcal{H})$ and some $S \subseteq V(\mathcal{H})$, whether this hypergraph has an orientation in which every vertex of $S$ is reachable from $r$. Secondly, we show that it is NP-complete to decide, given a hypergraph $\mathcal{H}$ and some $S \subseteq V(\mathcal{H})$, whether this hypergraph has an orientation in which any two vertices in $S$ are mutually reachable from each other. This answers a longstanding open question of the Egerv\'ary Research group. We further show that it is NP-complete to decide if a given hypergraph has a well-balanced orientation. On the positive side, we show that the problem of finding a Steiner hypertree and the first orientation problem can be solved in polynomial time if the number of terminals $|S|$ is fixed.

We present a slight generalization of the result of Kamiyama and Kawase \cite{kamkaw} on packing time-respecting arborescences in acyclic pre-flow temporal networks. Our main contribution is to provide the first results on packing time-respecting arborescences in non-acyclic temporal networks. As negative results, we prove the NP-completeness of the decision problem of the existence of 2 arc-disjoint spanning time-respecting arborescences and of a related problem proposed in this paper.

We show that the problem of deciding whether a given graph $G$ has a well-balanced orientation $\vec{G}$ such that $d_{\vec{G}}^+(v)\leq \ell(v)$ for all $v \in V(G)$ for a given function $\ell:V(G)\rightarrow \mathbb{Z}_{\geq 0}$ is NP-complete. We also prove a similar result for best-balanced orientations. This improves a result of Bern\' ath, Iwata, Kir\' aly, Kir\'aly and Szigeti and answers a question of Frank.

We consider two possible extensions of a theorem of Thomassen characterizing the graphs admitting a 2-vertex-connected orientation. First, we show that the problem of deciding whether a mixed graph has a 2-vertex-connected orientation is NP-hard. This answers a question of Bang-Jensen, Huang and Zhu. For the second part, we call a directed graph $D=(V,A)$ $2T$-connected for some $T \subseteq V$ if $D$ is 2-arc-connected and $D-v$ is strongly connected for all $v \in T$. We deduce a characterization of the graphs admitting a $2T$-connected orientation from the theorem of Thomassen.

Given a mixed hypergraph $\mathcal{F}=(V,\mathcal{A}\cup \mathcal{E})$, functions $f,g:V\rightarrow \mathbb{Z}_+$ and an integer $k$, a packing of $k$ spanning mixed hyperarborescences is called $(k,f,g)$-flexible if every $v \in V$ is the root of at least $f(v)$ and at most $g(v)$ of the mixed hyperarborescences. We give a characterization of the mixed hypergraphs admitting such packings. This generalizes results of Frank and, more recently, Gao and Yang. Our approach is based on matroid intersection, generalizing a construction of Edmonds. We also obtain an algorithm for finding a minimum weight solution to the above mentioned problem.

We attempt to generalize a theorem of Nash-Williams stating that a graph has a $k$-arc-connected orientation if and only if it is $2k$-edge-connected. In a strongly connected digraph we call an arc {\it deletable} if its deletion leaves a strongly connected digraph. Given a $3$-edge-connected graph $G$, we define its Frank number $f(G)$ to be the minimum number $k$ such that there exist $k$ orientations of $G$ with the property that every edge becomes a deletable arc in at least one of these orientations. We are interested in finding a good upper bound for the Frank number. We prove that $f(G)\leq 7$ for every $3$-edge-connected graph. On the other hand, we show that a Frank number of $3$ is attained by the Petersen graph. Further, we prove better upper bounds for more restricted classes of graphs and establish a connection to the Berge-Fulkerson conjecture. We also show that deciding whether all edges of a given subset can become deletable in one orientation is NP-complete., Comment: to appear in European Journal of Combinatorics

Fortier et al. proposed several research problems on packing arborescences. Some of them were settled in that article and others were solved later by Matsuoka and Tanigawa and by Gao and Yang. The last open problem is settled in this article. We show how to turn an inductive idea used in the latter two articles into a simple proof technique that allows to relate previous results on arborescence packings. We show how a strong version of Edmonds' theorem on packing spanning arborescences implies Kamiyama, Katoh and Takizawa's result on packing reachability arborescences and how Durand de Gevigney, Nguyen and Szigeti's theorem on matroid-based packing of arborescences implies Kir\'aly's result on matroid-reachability-based packing of arborescences. Finally, we deduce a new result on matroid-reachability-based packing of mixed hyperarborescences from a theorem on matroid-based packing of mixed hyperarborescences due to Fortier et al.. In the last part of the article, we deal with the algorithmic aspects of the problems considered. We first obtain algorithms to find the desired packings of arborescences in all settings and then apply Edmonds' weighted matroid intersection algorithm to also find solutions minimizing a given weight function.

Durand de Gevigney and Szigeti \cite{DgGSz} have recently given a min-max theorem for the $(2,k)$-connectivity augmentation problem. This article provides an $O(n^3(m+ n \textrm{ }log\textrm{ }n))$ algorithm to find an optimal solution for this problem.

A graph G is called (2k, k)-connected if G is 2k-edge-connected and G-v is k-edge-connected for every vertex v. The study of (2k, k)-connected graphs is motivated by a conjecture of Frank which states that a graph has a 2-vertex-connected orientation if and only if it is (4, 2)-connected. In this paper, we provide a construction of the family of (2k, k)-connected graphs for k even which generalizes the construction given by Jord\'an for k = 2. We also solve the corresponding connectivity augmentation problem: given a graph G and an integer k \geq 2, what is the minimum number of edges to be added to make G (2k, k)-connected. Both these results are based on a new splitting-off theorem for (2k, k)-connected graphs., Comment: 13 pages

We provide the directed counterpart of a slight extension of Katoh and Tanigawa's result on rooted-tree decompositions with matroid constraints. Our result characterises digraphs having a packing of arborescences with matroid constraints. It is a proper extension of Edmonds' result on packing of spanning arborescences and implies - using a general orientation result of Frank - the above result of Katoh and Tanigawa. We also give a complete description of the convex hull of the incidence vectors of the basic packings of arborescences and prove that the mimimum cost version of the problem can be solved in polynomial time.

We prove that every (6k + 2l, 2k)-connected simple graph contains k rigid and l connected edge-disjoint spanning subgraphs. This implies a theorem of Jackson and Jord\'an [4] and a theorem of Jord\'an [6] on packing of rigid spanning subgraphs. Both these results are generalizations of the classical result of Lov\'asz and Yemini [9] saying that every 6-connected graph is rigid for which our approach provides a transparent proof. Our result also gives two improved upper bounds on the connectivity of graphs that have interesting properties: (1) every 8-connected graph packs a spanning tree and a 2-connected spanning subgraph; (2) every 14-connected graph has a 2-connected orientation.