We consider the following problem: given a k-(node) connected graph G find a smallest set F of new edges so that the graph G + F is (k + 1)-connected. The complexity status of this problem is an open question. The problem admits a 2approximation algorithm. Another algorithm due to Jordán computes an augmenting edge set with at most d(k− 1)/2e edges over the optimum. C ⊂ V (G) is a k-separator (...

A set of vertices $S$ in a connected graph $G$ is a different-distance set if, for any vertex $w$ outside $S$, no two vertices in $S$ have the same distance to $w$.The lower and upper different-distance number of a graph are the order of a smallest, respectively largest, maximal different-distance set.We prove that a different-distance set induces either a special type of path or an independent...

Let $G=(V,E)$ be a simple connected graph. A matching $M$ in a graph $G$ is a collection of edges of $G$ such that no two edges from $M$ share a vertex. A matching $M$ is maximal if it cannot be extended to a larger matching in $G$. The cardinality of any smallest maximal matching in $G$ is the saturation number of $G$ and is denoted by $s(G)$. In this paper we study the saturation numbe...

A fundamental result by Karger [10] states that for any λ-edgeconnected graph with n nodes, independently sampling each edge with probability p = Ω(logn/λ) results in a graph that has edge connectivity Ω(λp), with high probability. This paper proves the analogous result for vertex connectivity, when sampling vertices. We show that for any k-vertex-connected graph G with n nodes, if each node is...

Let m and n be positive integers with n − 1 ≤ m ≤ ( n 2 ) and CG(m,n) be the set of all non-isomorphic connected graphs of order n and size m. The vertex-connectivity and the edge-connectivity of a graph G are denoted by κ(G) and λ(G), respectively. We prove that if π ∈ {κ, λ}, then there exist positive integers a and b such that {π(G) : G ∈ CG(m,n)} = {x ∈ Z : a ≤ x ≤ b}. Thus {π(G) : G ∈ CG(m...

The vertex-connectivity and edge-connectivity of the zero-divisor graph associated to a finite commutative ring are studied. It is shown that the edgeconnectivity of ΓR always coincides with the minimum degree. When R is not local, it is shown that the vertex-connectivity also equals the minimum degree, and when R is local, various upper and lower bounds are given for the vertex-connectivity.

We present faster algorithms for computing the 2-edge and 2-vertex strongly connected components of a directed graph, which are straightforward generalizations of strongly connected components. While in undirected graphs the 2-edge and 2-vertex connected components can be found in linear time, in directed graphs only rather simple O(mn)-time algorithms were known. We use a hierarchical sparsifi...

The relationship between graph coloring and the immersion order is considered. Vertex connectivity, edge connectivity and related issues are explored. These lead to the conjecture that, if G requires at least t colors, then G must have immersed within it Kt, the complete graph on t vertices. Evidence in support of such a proposition is presented. For each fixed value of t, there can be only a f...

Throughout this paper, R will denote a commutative ring with identity and M is a unitary R- module and Z will denote the ring of integers. We introduce the graph Ω(M) of module M with the set of vertices contain all nontrivial non-essential submodules of M. We investigate the interplay between graph-theoretic properties of Ω(M) and algebraic properties of M. Also, we assign the values of natura...

‎The textit{metric dimension} of a connected graph $G$ is the minimum number of vertices in a subset $B$ of $G$ such that all other vertices are uniquely determined by their distances to the vertices in $B$‎. ‎In this case‎, ‎$B$ is called a textit{metric basis} for $G$‎. ‎The textit{basic distance} of a metric two dimensional graph $G$ is the distance between the elements of $B$‎. ‎Givi...