Related papers: Constructing edge-disjoint Steiner trees in Cartes…

Let $G$ be a graph, $S$ be a set of vertices of $G$, and $\lambda(S)$ be the maximum number $\ell$ of pairwise edge-disjoint trees $T_1, T_2,..., T_{\ell}$ in $G$ such that $S\subseteq V(T_i)$ for every $1\leq i\leq \ell$. The generalized…

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. An edge subset $F\subseteq E(G)$ is called a restricted edge-cut if $G-F$ is disconnected and has no isolated vertices. The restricted edge-connectivity $\lambda'(G)$ of $G$ is…

The restricted edge-connectivity of a connected graph $G$, denoted by $\lambda^{\prime}(G)$, if it exists, is the minimum cardinality of a set of edges whose deletion makes $G$ disconnected and each component with at least 2 vertices. It…

A tree $T$ in an edge-colored graph is called a {\it proper tree} if no two adjacent edges of $T$ receive the same color. Let $G$ be a connected graph of order $n$ and $k$ be an integer with $2\leq k \leq n$. For $S\subseteq V(G)$ and $|S|…

The generalized $k$-edge-connectivity $\lambda_k(G)$ of a graph $G$ is a generalization of the concept of edge-connectivity. The lexicographic product of two graphs $G$ and $H$, denoted by $G\circ H$, is an important graph product. In this…

The direct product of graphs $G=(V(G),E(G))$ and $H=(V(H),E(H))$ is the graph, denoted as $G\times H$, with vertex set $V(G\times H)=V(G)\times V(H)$, where vertices $(x_1,y_1)$ and $(x_2,y_2)$ are adjacent in $G\times H$ if $x_1x_2\in…

The concept of maximum local connectivity $\bar {\kappa}$ of a graph was introduced by Bollob\'{a}s. One of the problems about it is to determine the largest number of edges $f(n;\bar{\kappa}\leq \ell)$ for graphs of order $n$ that have…

An edge subset \( S \subseteq E(G) \) is called a 3-restricted edge-cut if \( G - S \) is disconnected and each component of \( G - S \) contains at least three vertices. The 3-restricted edge-connectivity of a graph \( G \), denoted by \(…

The conditional $h$-vertex($h$-edge) connectivity of a connected graph $H$ of minimum degree $ k > h$ is the size of a smallest vertex(edge) set $F$ of $H$ such that $H - F$ is a disconnected graph of minimum degree at least $h.$ Let $G$ be…

Let $G$ be a nontrivial connected graph of order $n$ and let $k$ be an integer with $2\leq k\leq n$. For a set $S$ of $k$ vertices of $G$, let $\kappa (S)$ denote the maximum number $\ell$ of edge-disjoint trees $T_1,T_2,...,T_\ell$ in $G$…

The generalized $k$-connectivity $\kappa_{k}(G)$ of a graph $G$, which was introduced by Chartrand et al.(1984) is a generalization of the concept of vertex connectivity. Let $G$ and $H$ be nontrivial connected graphs. Recently, Li et al.…

Let $G$ be a nontrivial connected graph of order $n$ and let $k$ be an integer with $2\leq k\leq n$. For a set $S$ of $k$ vertices of $G$, let $\kappa (S)$ denote the maximum number $\ell$ of edge-disjoint trees $T_1,T_2,...,T_\ell$ in $G$…

An edge subset \( S \subseteq E(G) \) is called a 3-restricted edge-cut if $G-S$ is disconnected and each component of \( G - S \) contains at least three vertices. The 3-restricted edge-connectivity of a graph \( G \), denoted by \(…

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…

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…

The generalized $k$-connectivity $\kappa_{k}(G)$ of a graph $G$ is a parameter that can measure the reliability of a network $G$ to connect any $k$ vertices in $G$, which is proved to be NP-complete for a general graph $G$. Let $S\subseteq…

Let $G$ be a connected graph and $\ell : E(G) \to \mathbb{R}^+$ a length-function on the edges of $G$. The Steiner distance $\mathrm{sd}_G(A)$ of $A \subseteq V(G)$ within $G$ is the minimum length of a connected subgraph of $G$ containing…

The generalized connectivity of a graph $G$ was introduced by Chartrand et al. Let $S$ be a nonempty set of vertices of $G$, and $\kappa(S)$ be defined as the largest number of internally disjoint trees $T_1, T_2, \cdots, T_k$ connecting…

We consider Directed Steiner Forest (DSF), a fundamental problem in network design. The input to DSF is a directed edge-weighted graph $G = (V, E)$ and a collection of vertex pairs $\{(s_i, t_i)\}_{i \in [k]}$. The goal is to find a minimum…

We initiate the study of degree-bounded network design problems in the online setting. The degree-bounded Steiner tree problem { which asks for a subgraph with minimum degree that connects a given set of vertices { is perhaps one of the…