Related papers: Pendant 3-tree Connectivity of Augmented Cubes

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…

Let $D=(V(D),A(D))$ be a digraph with a terminal vertex subset $S\subseteq V(D)$ such that $|S|=k\geq 2$. An out-tree $T$ of $D$ rooted at $r$ is called a directed pendant $(S,r)$-Steiner tree (or, pendant $(S,r)$-tree for short) if $r\in…

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…

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…

We consider the problem of packing edge-disjoint Steiner forests in a graph. The input consists of a multi-graph $G=(V,E)$ and a collection of $h$ vertex subsets $S = \{S_1,S_2,\ldots,S_h\}$. A Steiner forest for $S$, also called an…

Connectivity is a cornerstone concept in graph theory, essential for evaluating the robustness of networks against failures. To better capture fault tolerance in complex systems, researchers have extended classical connectivity notions, one…

For a digraph $D=(V(D),A(D))$ and a set $S\subseteq V(D)$ with $|S|\geq 2$ and $r\in S$, a directed pendant $(S,r)$-Steiner tree (or, simply, a pendant $(S,r)$-tree) is an out-tree $T$ rooted at $r$ such that $S\subseteq V(T)$ and each…

Let T1, T2,.... Tk be spanning trees in a graph G. If for any pair of vertices u and v of G, the paths between u and v in every Ti( 0 < i < k+1) do not contain common edges then T1, T2,.... Tk are called edge-disjoint spanning trees in G.…

In Connectivity Augmentation problems we are given a graph $H=(V,E_H)$ and an edge set $E$ on $V$, and seek a min-size edge set $J \subseteq E$ such that $H \cup J$ has larger edge/node connectivity than $H$. In the Edge-Connectivity…

The $S$-Steiner tree packing problem provides mathematical foundations for optimizing multi-path information transmission, particularly in designing fault-tolerant parallelized routing architectures for massive-scale network…

For a connected graph, a {\em minimum vertex separator} is a minimum set of vertices whose removal creates at least two connected components. The vertex connectivity of the graph refers to the size of the minimum vertex separator and a…

Let $S\subseteq V(G)$ and $\pi_{G}(S)$ denote the maximum number $t$ of edge-disjoint paths $P_{1},P_{2},\ldots,P_{t}$ in a graph $G$ such that $V(P_{i})\cap V(P_{j})=S$ for any $i,j\in\{1,2,\ldots,t\}$ and $i\neq j$. If $S=V(G)$, then…

Let $G$ be a graph and $S\subseteq V(G)$ with $|S|\geq 2$. Then the trees $T_1, T_2, \cdots, T_\ell$ in $G$ are \emph{internally disjoint Steiner trees} connecting $S$ (or $S$-Steiner trees) if $E(T_i) \cap E(T_j )=\emptyset$ and…

Strengthening the classical concept of Steiner trees, West and Wu [J. Combin. Theory Ser. B 102 (2012), 186--205] introduced the notion of a $T$-connector in a graph $G$ with a set $T$ of terminals. They conjectured that if the set $T$ is…

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…

The Steiner Tree problem is a classical problem in combinatorial optimization: the goal is to connect a set $T$ of terminals in a graph $G$ by a tree of minimum size. Karpinski and Zelikovsky (1996) studied the $\delta$-dense version of…

Given a simple connected undirected graph G = (V, E), a set X \subseteq V(G), and integers k and p, STEINER SUBGRAPH EXTENSION problem asks if there exists a set S \supseteq X with at most k vertices such that G[S] is p-edge-connected. This…

For a connected graph, a vertex separator is a set of vertices whose removal creates at least two components and a minimum vertex separator is a vertex separator of least cardinality. The vertex connectivity refers to the size of a minimum…

A connected graph $G$ is called strongly Menger (edge) connected if for any two distinct vertices $x,y$ of $G$, there are $\min \{{\rm deg}_G(x), {\rm deg}_G(y)\}$ vertex(edge)-disjoint paths between $x$ and $y$. In this paper, we consider…

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…