Related papers: Directed Steiner path packing and directed path co…

Consider a setting where possibly sensitive information sent over a path in a network is visible to every {neighbor} of the path, i.e., every neighbor of some node on the path, thus including the nodes on the path itself. The exposure of a…

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…

In the Directed Steiner Network problem we are given an arc-weighted digraph $G$, a set of terminals $T \subseteq V(G)$, and an (unweighted) directed request graph $R$ with $V(R)=T$. Our task is to output a subgraph $G' \subseteq G$ of the…

Let $G$ be a directed planar graph of complexity $n$, each arc having a nonnegative length. Let $s$ and $t$ be two distinct faces of $G$; let $s_1,...,s_k$ be vertices incident with $s$; let $t_1,...,t_k$ be vertices incident with $t$. We…

Let $D$ be a digraph. A $k$-container of $D$ between $u$ and $v$, $C(u,v)$, is a set of $k$ internally disjoint paths between $u$ and $v$. A $k$-container $C(u,v)$ of $D$ is a strong (resp. weak) $k^{*}$-container if there is a set of $k$…

The Induced Disjoint Paths problem is to test whether a graph G with k distinct pairs of vertices (s_i,t_i) contains paths P_1,...,P_k such that P_i connects s_i and t_i for i=1,...,k, and P_i and P_j have neither common vertices nor…

Given a directed graph $G$ and a list $(s_1,t_1),\dots,(s_d,t_d)$ of terminal pairs, the Directed Steiner Network problem asks for a minimum-cost subgraph of $G$ that contains a directed $s_i\to t_i$ path for every $1\le i \le k$. The…

The $k$ disjoint shortest paths problem ($k$-DSPP) on a graph with $k$ source-sink pairs $(s_i, t_i)$ asks for the existence of $k$ pairwise edge- or vertex-disjoint shortest $s_i$-$t_i$-paths. It is known to be NP-complete if $k$ is part…

An out-(in-)branching B_s^+ (B_s^-) rooted at s in a digraph D is a connected spanning subdigraph of D in which every vertex x != s has precisely one arc entering (leaving) it and s has no arcs entering (leaving) it. We settle the…

In the Directed Steiner Network problem, the input is a directed graph G, a subset T of k vertices of G called the terminals, and a demand graph D on T. The task is to find a subgraph H of G with the minimum number of edges such that for…

In a digraph $D=(V,A)$, an oriented path is a sequence $P=x_1x_2\dots x_p$ of distinct vertices such that either $x_ix_{i+1}\in A$ or $x_{i+1}x_{i}\in A$ or both for every $i\in [p-1]$. If $x_ix_{i+1}\in A$ in $P$, then $x_ix_{i+1}$ is a…

Let $D=(V,A)$ be a digraph of order $n$, $S$ a subset of $V$ of size $k$ and $2\le k\leq n$. A strong subgraph $H$ of $D$ is called an $S$-strong subgraph if $S\subseteq V(H)$. A pair of $S$-strong subgraphs $D_1$ and $D_2$ are said to be…

In the Vertex Disjoint Paths with Congestion problem, the input consists of a digraph $D$, an integer $c$ and $k$ pairs of vertices $(s_i, t_i)$, and the task is to find a set of paths connecting each $s_i$ to its corresponding $t_i$,…

Cartesian product networks are always regarded as a tool for ``combining'' two given networks with established properties to obtain a new one that inherits properties from both. For a graph $F=(V,E)$ and a set $S\subseteq V(F)$ of at least…

The {\sc weak 2-linkage} problem for digraphs asks for a given digraph and vertices $s_1,s_2,t_1,t_2$ whether $D$ contains a pair of arc-disjoint paths $P_1,P_2$ such that $P_i$ is an $(s_i,t_i)$-path. This problem is NP-complete for…

An antidirected trail in a digraph is a trail (a walk with no arc repeated) in which the arcs alternate between forward and backward arcs. An antidirected path is an antidirected trail where no vertex is repeated. We show that it is…

Many of the tools developed for the theory of tree-decompositions of graphs do not work for directed graphs. In this paper we show that some of the most basic tools do work in the case where the model digraph is a directed path. Using these…

The well-known Disjoint Paths problem is to decide if a graph contains k pairwise disjoint paths, each connecting a different terminal pair from a set of k distinct pairs. We determine, with an exception of two cases, the complexity of the…

Given any digraph $D$ on $n$ vertices, let $\mathcal{P}(D)$ be the family of all directed paths in $D$, and let $H$ be a digraph with the arc set $A(H)=\{a_1, \ldots, a_k\}$. The digraph $D$ is called arbitrary Hamiltonian $H$-linked if for…

For distinct vertices $u$ and $v$ in a graph $G$, the {\em connectivity} between $u$ and $v$, denoted $\kappa_G(u,v)$, is the maximum number of internally disjoint $u$--$v$ paths in $G$. The {\em average connectivity} of $G$, denoted…