Related papers: Vulnerability of super edge-connected graphs

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 $G=(V,E)$ 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 $\Delta$-edge stability number ${\rm es}_{\Delta}(G)$ of a graph $G$ is the minimum number of edges of $G$ whose removal results in a subgraph $H$ with $\Delta(H) = \Delta(G)-1$. Sets whose removal results in a subgraph with smaller…

In this paper, we consider the question of computing sparse subgraphs for any input directed graph $G=(V,E)$ on $n$ vertices and $m$ edges, that preserves reachability and/or strong connectivity structures. We show $O(n+\min\{|{\cal…

Let $\kappa'(G)$ be the edge connectivity of $G$ and $G\times H$ the direct product of $G$ and $H$. Let $H$ be an arbitrary dense graph with minimal degree $\delta(H)>|H|/2$. We prove that for any graph $G$, $\kappa'(G\times…

Let $G$ be a connected graph. The edge-connectivity of $G$, denoted by $\lambda(G)$, is the minimum number of edges whose removal renders $G$ disconnected. Let $\delta(G)$ be the minimum degree of $G$. It is well-known that $\lambda(G) \leq…

An edge-cut $R$ of an edge-colored connected graph is called a rainbow-cut if no two edges in the edge-cut are colored the same. An edge-colored graph is rainbow disconnected if for any two distinct vertices $u$ and $v$ of the graph, there…

Let $G$ and $H$ be simple 3-connected graphs such that $G$ has an $H$-minor. An edge $e$ in $G$ is called {\it $H$-deletable} if $G\backslash e$ is 3-connected and has an $H$-minor. The main result in this paper establishes that, if $G$ has…

A vertex cut $S$ of a connected graph $G$ is a subset of vertices of $G$ whose deletion makes $G$ disconnected. A super vertex cut $S$ of a connected graph $G$ is a subset of vertices of $G$ whose deletion makes $G$ disconnected and there…

For a given multigraph H, a graph G is H-linked, if |G| \geq |H| and for every injective map {\tau}: V (H) \rightarrow V (G), we can find internally disjoint paths in G, such that every edge from uv in H corresponds to a {\tau} (u) - {\tau}…

Let $\delta$ and $\Delta$ be the minimum and the maximum degree of the vertices of a simple connected graph $G$, respectively. The distinguishing index of a graph $G$, denoted by $D'(G)$, is the least number of labels in an edge labeling of…

Let $G = (V(G), E(G))$ be a graph. The maximum cardinality of a set $M_k \subseteq E(G)$ such that $M_k$ contains exactly $k$-pairs of adjacent edges of $G$ is called the $k$-nearly edge independence number of $G$, and is denoted by…

The Tur\'{a}n number of a graph $H$, $\text{ex}(n,H)$, is the maximum number of edges in an $n$-vertex graph that does not contain $H$ as a subgraph. For a vertex $v$ and a multi-set $\mathcal{F}$ of graphs, the suspension $\mathcal{F}+v$…

Given a non-trivial graph $G$, the minimum cardinality of a set of edges $F$ in $G$ such that $\chi'(G \setminus F)<\chi'(G)$ is called the chromatic edge stability index of $G$, denoted by $es_{\chi'}(G)$, and such a (smallest) set $F$ is…

A dominating set in a graph $G$ is a set $S$ of vertices of $G$ such that every vertex outside $S$ is adjacent to a vertex in $S$. A connected dominating set in $G$ is a dominating set $S$ such that the subgraph $G[S]$ induced by $S$ is…

The edge-fault-tolerance of networks is of great significance to the design and maintenance of networks. For any pair of vertices $u$ and $v$ of the connected graph $G$, if they are connected by $\min \{ \deg_G(u),\deg_G(v)\}$ edge-disjoint…

Let $G$ be a multigraph. A subset $F$ of $E(G)$ is an edge cover of $G$ if every vertex of $G$ is incident to an edge of $F$. The cover index, $\xi(G)$, is the largest number of edge covers into which the edges of $G$ can be partitioned.…

Let $G$ be a connected graph with minimum degree $\delta(G)$ and vertex-connectivity $\kappa(G)$. The graph $G$ is $k$-connected if $\kappa(G)\geq k$, maximally connected if $\kappa(G) = \delta(G)$, and super-connected (or super-$\kappa$)…

Given a graph G, a non-negative integer h and a set of vertices S, the h-extra connectivity of G is the cardinality of a minimum set S such that G-S is disconnected and each component of G-S has at least h+1 vertices. The 2-extra…

An edge-coloured graph $G$ is called $properly$ $connected$ if every two vertices are connected by a proper path. The $proper$ $connection$ $number$ of a connected graph $G$, denoted by $pc(G)$, is the smallest number of colours that are…