For a digraph $D=(V(D), A(D))$, and a set $S\subseteq V(D)$ with $r\in S$ and $|S|\geq 2$, an $(S, r)$-tree is an out-tree $T$ rooted at $r$ with $S\subseteq V(T)$. Two $(S, r)$-trees $T_1$ and $T_2$ are said to be arc-disjoint if $A(T_1)\cap A(T_2)=\emptyset$. Two arc-disjoint $(S, r)$-trees $T_1$ and $T_2$ are said to be internally disjoint if $V(T_1)\cap V(T_2)=S$. Let $κ_{S,r}(D)$ and $λ_{S,r}(D)$ be the maximum number of internally disjoint and arc-disjoint $(S, r)$-trees in $D$, respectively. The generalized $k$-vertex-strong connectivity of $D$ is defined as $$κ_k(D)= \min \{κ_{S,r}(D)\mid S\subset V(D), |S|=k, r\in S\}.$$ Similarly, the generalized $k$-arc-strong connectivity of $D$ is defined as $$λ_k(D)= \min \{λ_{S,r}(D)\mid S\subset V(D), |S|=k, r\in S\}.$$ The generalized $k$-vertex-strong connectivity and generalized $k$-arc-strong connectivity are also called directed tree connectivity which extends the well-established tree connectivity on undirected graphs to directed graphs and could be seen as a generalization of classical connectivity of digraphs.
In this paper, we completely determine the complexity for both $κ_{S, r}(D)$ and $λ_{S, r}(D)$ on general digraphs, symmetric digraphs and Eulerian digraphs. In particular, among our results, we prove and use the NP-completeness of 2-linkage problem restricted to Eulerian digraphs. We also give sharp bounds and characterizations for the two parameters $κ_k(D)$ and $λ_k(D)$.
Let $T$ be a distinguished subset of vertices in a graph $G$. A $T$-\emph{Steiner tree} is a subgraph of $G$ that is a tree and that spans $T$. Kriesell conjectured that $G$ contains $k$ pairwise edge-disjoint $T$-Steiner trees provided that every edge-cut of $G$ that separates $T$ has size $\ge 2k$. When $T=V(G)$ a $T$-Steiner tree is a spanning tree and the conjecture is a consequence of a classic theorem due to Nash-Williams and Tutte. Lau proved that Kriesell's conjecture holds when $2k$ is replaced by $24k$, and recently West and Wu have lowered this value to $6.5k$. Our main result makes…
A finite classical polar space of rank $n$ consists of the totally isotropic subspaces of a finite vector space equipped with a nondegenerate form such that $n$ is the maximal dimension of such a subspace. A $t$-Steiner system in a finite classical polar space of rank $n$ is a collection $Y$ of totally isotropic $n$-spaces such that each totally isotropic $t$-space is contained in exactly one member of $Y$. Nontrivial examples are known only for $t=1$ and $t=n-1$. We give an almost complete classification of such $t$-Steiner systems, showing that such objects can only exist in some corner case…
In this paper, we study the complexity of two types of digraph packing problems: perfect out-forests problem and Steiner cycle packing problem. For the perfect out-forest problem, we prove that it is NP-hard to decide whether a given strong digraph contains a 1-perfect out-forest. However, when restricted to a semicomplete digraph $D$, the problem of deciding whether $D$ contains an $i$-perfect out-forest becomes polynomial-time solvable, where $i\in \{0,1\}$. We also prove that it is NP-hard to find a 0-perfect out-forest of maximum size in a connected acyclic digraph, and it is NP-hard to fi…
The Steiner tree problem in graphs has applications in network design or circuit layout. Given a set $S$ of vertices, $|S| \geq 2,$ a tree connecting all vertices of $S$ is called an $S$-Steiner tree (tree connecting $S$). The reliability of a network $G$ to connect any $S$ vertices ($|S|$ number of vertices) in $G$ can be measure by this parameter. For an $S$-Steiner tree, if the degree of each vertex in $S$ is equal to one, then that tree is called a pendant S-Steiner tree. Two pendant $S$-Steiner trees $T$ and $T'$ are said to be internally disjoint if $E(T) \cap E(T') = \emptyset$ and $V(T…
The concept of pedant tree-connectivity was introduced by Hager in 1985. For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of at least two vertices, \emph{an $S$-Steiner tree} or \emph{a Steiner tree connecting $S$} (or simply, \emph{an $S$-tree}) is a such subgraph $T=(V',E')$ of $G$ that is a tree with $S\subseteq V'$. For an $S$-Steiner tree, if the degree of each vertex in $S$ is equal to one, then this tree is called a \emph{pedant $S$-Steiner tree}. Two pedant $S$-Steiner trees $T$ and $T'$ are said to be \emph{internally disjoint} if $E(T)\cap E(T')=\varnothing$ and $V(T)\cap V(T')=S$. …
The concept of pedant tree-connectivity was introduced by Hager in 1985. For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of at least two vertices, \emph{an $S$-Steiner tree} or \emph{a Steiner tree connecting $S$} (or simply, \emph{an $S$-tree}) is a such subgraph $T=(V',E')$ of $G$ that is a tree with $S\subseteq V'$. For an $S$-Steiner tree, if the degree of each vertex in $S$ is equal to one, then this tree is called a \emph{pedant $S$-Steiner tree}. Two pedant $S$-Steiner trees $T$ and $T'$ are said to be \emph{internally disjoint} if $E(T)\cap E(T')=\varnothing$ and $V(T)\cap V(T')=S$. …
For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of size at least $2$, an $S$-Steiner tree $T$ is a subgraph of $G$ that is a tree with $S\subseteq V(T)$. Two $S$-Steiner trees $T$ and $T'$ are internally disjoint (resp. edge-disjoint) if $E(T)\cap E(T')=\emptyset$ and $V(T)\cap V(T')=S$ (resp. if $E(T)\cap E(T')=\emptyset$). Let $κ_G (S)$ (resp. $λ_G (S)$) denote the maximum number of internally disjoint (resp. edge-disjoint) $S$-Steiner trees in $G$. The $k$-tree connectivity $κ_k(G)$ (resp. $k$-tree edge-connectivity $λ_k(G)$) of $G$ is then defined as the minimum $κ_G (S)$ (resp. $λ_G (S)…
The concept of maximum local connectivity $\bar κ$ of a graph was introduced by Bollobás. One of the problems about it is to determine the largest number of edges $f(n;\barκ\leq \ell)$ for graphs of order $n$ that have local connectivity at most $\ell$. We consider a generalization of the above concept and problem. For $S\subseteq V(G)$ and $|S|\geq 2$, the \emph{generalized local connectivity} $κ(S)$ is the maximum number of internally disjoint trees connecting $S$ in $G$. The parameter $\barκ_k(G)=max\{κ(S)|S\subseteq V(G),|S|=k\}$ is called the \emph{maximum generalized local connectivity} …
The problem of determining the largest number $f(n;\barκ\leq \ell)$ of edges for graphs with $n$ vertices and maximal local connectivity at most $\ell$ was considered by Bollobás. Li et al. studied the largest number $f(n;\barκ_3\leq2)$ of edges for graphs with $n$ vertices and at most two internally disjoint Steiner trees connecting any three vertices. In this paper, we further study the largest number $f(n;\barκ_k=1)$ of edges for graphs with $n$ vertices and exactly one Steiner tree connecting any $k$ vertices with $k\geq 3$. It turns out that this is not an easy task to finish, not like th…
We consider perfect secret key generation for a ``pairwise independent network'' model in which every pair of terminals share a random binary string, with the strings shared by distinct terminal pairs being mutually independent. The terminals are then allowed to communicate interactively over a public noiseless channel of unlimited capacity. All the terminals as well as an eavesdropper observe this communication. The objective is to generate a perfect secret key shared by a given set of terminals at the largest rate possible, and concealed from the eavesdropper. First, we show how the notion o…
In this paper, we study two types of strong subgraph packing problems in digraphs, including internally disjoint strong subgraph packing problem and arc-disjoint strong subgraph packing problem. These problems can be viewed as generalizations of the famous Steiner tree packing problem and are closely related to the strong arc decomposition problem. We first prove the NP-completeness for the internally disjoint strong subgraph packing problem restricted to symmetric digraphs and Eulerian digraphs. Then we get inapproximability results for the arc-disjoint strong subgraph packing problem and the…
For a digraph $D=(V(D), A(D))$, and a set $S\subseteq V(D)$ with $r\in S$ and $|S|\geq 2$, a directed $(S, r)$-Steiner path or, simply, an $(S, r)$-path is a directed path $P$ started at $r$ with $S\subseteq V(P)$. Two $(S, r)$-paths are said to be arc-disjoint if they have no common arc. Two arc-disjoint $(S, r)$-paths are said to be internally disjoint if the set of common vertices of them is exactly $S$. Let $κ^p_{S,r}(D)$ (resp. $λ^p_{S,r}(D)$) be the maximum number of internally disjoint (resp. arc-disjoint) $(S, r)$-paths in $D$. The directed path $k$-connectivity of $D$ is defined as $$…
Let $\mathcal{G}$ be the set of simple graphs (or multigraphs) $G$ such that for each $G \in \mathcal{G}$ there exists at least two non-empty disjoint proper subsets $V_{1},V_{2}\subseteq V(G)$ satisfying $V(G)\setminus(V_{1} \cup V_{2})\neq ϕ$ and edge connectivity $κ'(G)=e(V_{i},V(G)\backslash V_{i})$ for $1\leq i \leq 2$. A multigraph is a graph with possible multiple edges, but no loops. Let $τ(G)$ be the maximum number of edge-disjoint spanning trees of a graph $G$. Motivated by a question of Seymour on the relationship between eigenvalues of a graph $G$ and bounds of $τ(G)$, we mainly gi…
Let $τ(G)$ and $κ'(G)$ denote the edge-connectivity and the spanning tree packing number of a graph $G$, respectively. Proving a conjecture initiated by Cioaba and Wong, Liu et al. in 2014 showed that for any simple graph $G$ with minimum degree $δ\ge 2k \ge 4$, if the second largest adjacency eigenvalue of $G$ satisfies $λ_2(G) < δ- \frac{2k-1}{δ+1}$, then $τ(G) \ge k$. Similar results involving the Laplacian eigenvalues and the signless Laplacian eigenvalues of $G$ are also obtained. In this paper, we find a function $f(δ, k, g)$ such that for every graph $G$ with minimum degree $δ\ge 2k \ge…
The problem of determining the smallest number of edges, $h(n;\barκ\geq r)$, which guarantees that any graph with $n$ vertices and $h(n;\barκ\geq r)$ edges will contain a pair of vertices joined by $r$ internally disjoint paths was posed by Erdös and Gallai. Bollobás considered the problem of determining the largest number of edges $f(n;\barκ\leq \ell)$ for graphs with $n$ vertices and local connectivity at most $\ell$. One can see that $f(n;\barκ\leq \ell)= h(n;\barκ\geq \ell+1)-1$. These two problems had received a wide attention of many researchers in the last few decades. In the above prob…
With graphs considered as natural models for many network design problems, edge connectivity $κ'(G)$ and maximum number of edge-disjoint spanning trees $τ(G)$ of a graph $G$ have been used as measures for reliability and strength in communication networks modeled as graph $G$ (see \cite{Cunn85, Matula87}, among others). Mader \cite{Mader71} and Matula \cite{Matula72} introduced the maximum subgraph edge connectivity $\overline{κ'}(G)=\max \{κ'(H): H \mbox{ is a subgraph of } G \}$. Motivated by their applications in network design and by the established inequalities \[ \overline{κ'}(G)\ge κ'(G…
Constructing edge-disjoint Steiner trees in Cartesian product networks
Xidian University（西安电子科技大学）；Royal Holloway, University of London（伦敦大学皇家霍洛威学院）；Qinghai Normal University（青海师范大学）；China Jiliang University（中国计量大学）；Nanjing Normal University（南京师范大学）
Edge connectivity and vertex connectivity are two fundamental concepts in graph theory. Although by now there is a good understanding of the structure of graphs based on their edge connectivity, our knowledge in the case of vertex connectivity is much more limited. An essential tool in capturing edge connectivity are edge-disjoint spanning trees. The famous results of Tutte and Nash-Williams show that a graph with edge connectivity $λ$ contains $\floor{λ/2}$ edge-disjoint spanning trees. We present connected dominating set (CDS) partition and packing as tools that are analogous to edge-disjoin…
We consider the class of graphs for which the edge connectivity is equal to the maximum number of edge-disjoint spanning trees, and the natural generalization to matroids, where the cogirth is equal to the number of disjoint bases. We provide descriptions of such graphs and matroids, showing that such a graph (or matroid) has a unique decomposition. In the case of graphs, our results are relevant for certain communication protocols.
We provide a structural description of, and invariants for, maximum spanning tree-packable graphs, i.e. those graphs G for which the edge connectivity of G is equal to the maximum number of edge-disjoint spanning trees in G. These graphs are of interest for the k-tree protocol of Itai and Rodeh [Inform. and Comput. 79 (1988), 43-59].
An edge (vertex) cut $X$ of $G$ is $r$-essential if $G-X$ has two components each of which has at least $r$ edges. A graph $G$ is $r$-essentially $k$-edge-connected (resp. $k$-connected) if it has no $r$-essential edge (resp. vertex) cuts of size less than $k$. If $r=1$, we simply call it essential. Recently, Lai and Li proved that every $m$-edge-connected essentially $h$-edge-connected graph contains $k$ edge-disjoint spanning trees, where $k,m,h$ are positive integers such that $k+1\le m\le 2k-1$ and $h\ge \frac{m^2}{m-k}-2$. In this paper, we show that every $m$-edge-connected and $2$-essen…
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 Jordán [4] and a theorem of Jordán [6] on packing of rigid spanning subgraphs. Both these results are generalizations of the classical result of Lovász 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-con…
