Related papers: On orientations maximizing total arc-connectivity

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…

Let $G$ be a connected graph and $\mathcal{P}(G)$ a graph parameter. We say that $\mathcal{P}(G)$ is feasible if $\mathcal{P}(G)$ satisfies the following properties: (I) $\mathcal{P}(G)\leq \mathcal{P}(G_{uv})$, if $G_{uv}=G[u\to v]$ for…

Let $D=(V,A)$ be a digraph. For an integer $k\geq 1$, a $k$-arc-connected flip is an arc subset of $D$ such that after reversing the arcs in it the digraph becomes (strongly) $k$-arc-connected. The first main result of this paper introduces…

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…

In 1960 Nash-Williams proved that an edge-connectivity of 2k is sufficient for a finite graph to have a k-arc-connected orientation. He then conjectured that the same is true for infinite graphs. In 2016, Thomassen, using his own results on…

We consider connectivity problems with orientation constraints. Given a directed graph $D$ and a collection of ordered node pairs $P$ let $P[D]=\{(u,v) \in P: D {contains a} uv{-path}}$. In the {\sf Steiner Forest Orientation} problem we…

Connectivity (or equivalently, unweighted maximum flow) is an important measure in graph theory and combinatorial optimization. Given a graph $G$ with vertices $s$ and $t$, the connectivity $\lambda(s,t)$ from $s$ to $t$ is defined to be…

For a digraph $D$ of order $n$ and an integer $1 \leq k \leq n-1$, the $k$-token digraph of $D$ is the graph whose vertices are all $k$-subsets of vertices of $D$ and, given two such $k$-subsets $A$ and $B$, $(A,B)$ is an arc in the…

For a connected graph $G$ with order $n$ and an integer $k\geq 1$, we denote by $$S_k(D(G))=\lambda_1(D(G))+\cdots+\lambda_k(D(G))$$ the sum of $k$ largest distance eigenvalues of $G$. In this paper, we consider the sharp upper bound and…

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 this paper, motivated by a problem of Scott and a conjecture of Lee, Loh and Sudakov we consider bisections of directed graphs. We prove that every directed graph with $m$ arcs and minimum semidegree at least $d$ admits a bisection in…

Let $G=(V,E)$ be a multigraph (it has multiple edges, but no loops). The edge connectivity, denoted by $\lambda(G)$, is the cardinality of a minimum edge-cut of $G$. We call $G$ maximally edge-connected if $\lambda(G)=\delta(G)$, and $G$…

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 nontrivial connected graph of order $n$, and $k$ 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 $G$ such…

Let $G$ be a connected graph with vertex set $V$. The distance, $d_G(u, v)$, between vertices $u$ and $v$ of $G$ is defined as the length of a shortest path between $u$ and $v$ in $G$. The distance matrix of $G$ is the matrix $\mathbf{D}(G)…

An instance of the Connected Maximum Cut problem consists of an undirected graph G = (V, E) and the goal is to find a subset of vertices S $\subseteq$ V that maximizes the number of edges in the cut \delta(S) such that the induced graph…

Connectivity of temporal graphs has been widely studied both as graph theory and as gossip theory. In particular, it is well known that in order to connect every vertex to every other, a temporal graph needs to have at least $2n-4$ edges…

A cut in a digraph $D=(V,A)$ is a set of arcs $\{uv \in A: u\in U, v\notin U\}$, for some $U\subseteq V$. It is known that the arc set $A$ is covered by $k$ cuts if and only if it admits a $k$-coloring such that no two consecutive arcs $uv,…

Given a graph G, of arbitrary size and unbounded vertex degree, denote by |G| the one-complex associated with $G$. The topological space |G| is n-arc connected (n-ac) if every set of no more than n points of |G| are contained in an arc (a…

We investigate problems addressing combined connectivity augmentation and orientations settings. We give a polynomial-time 6-approximation algorithm for finding a minimum cost subgraph of an undirected graph $G$ that admits an orientation…