We consider the problems of finding k-connected spanning subgraphs with minimum average weight. We show that the problems are NP-hard for k > 1. Approximation algorithms are given for four versions of the minimum average edge weight problem: 1. 3-approximation for k-edge-connectivity, 2. O(logk) approximation for k-node-connectivity 3. 2+ approximation for k-node-connectivity in Euclidian graph...

Let G be a (multi)graph of order n and let u,v vertices G. The maximum number internally disjoint u–v paths in is denoted by κG(u,v), the edge-disjoint λG(u,v). average connectivity defined κ¯(G)=∑κG(u,v)∕n2, edge-connectivity λ¯(G)=∑λG(u,v)∕n2, where both sums run over all unordered pairs {u,v}⊆V(G). A graph called ideally connected if κG(u,v)=min{deg(u),deg(v)} for {u,v} We prove that every m...

An edge cut that separates the connected graph into components with order at least two is restricted edge cut. The cardinality of minimum restricted edge cut is restricted edge connectivity. Denote byλ′ ( G) the restricted edge connectivity , then λ′( G) ≤ξ( G) where ξ( G) is the minimum edge degree. G is called maximal restricted edge connected if the equality in the previous inequality holds....

Preface Graphs are often used as a model for networks, for example transport networks, telecommunication systems, a network of servers and so on. Typically, the networks and thus the representing graphs are connected; that means, that there exists a path between all pairs of two vertices in the graph. But sometimes an element of the network fails, for the graphs that means the removal of a vert...

The restricted edge-connectivity of a graph is an important parameter to measure fault-tolerance of interconnection networks. This paper determines that the restricted edge-connectivity of the de Bruijn digraph B(d, n) is equal to 2d − 2 for d ≥ 2 and n ≥ 2 except B(2, 2). As consequences, the super edge-connectedness of B(d, n) is obtained immediately.

We survey results related to the data structures for dynamic graphs which can maintain global edge connectivity. We explain the major techniques used in these data structures and explore in detail the algorithm by Thorup for polylogarithmic edge-connectivity.

The problem of enumerating connected induced subgraphs a given graph is classical and studied well. It known that can be enumerated in constant time for each subgraph. In this paper, we focus on highly subgraphs. most major concept connectivity graphs vertex connectivity. For connectivity, some enumeration settings algorithms have been proposed, such as k-vertex spanning another edge-connectivi...

Two important branches of graph connectivity problems are connectivity augmentation, which consists of augmenting a graph by adding new edges so as to meet a specified target connectivity, and connectivity orientation, where the goal is to find an orientation of an undirected or mixed graph that satisfies some specified edge-connection property. In the present work an attempt is made to link th...

A fullerene graph is a planar cubic graph whose all faces are pentagonal and hexagonal. The structure of cyclic edge-cuts of fullerene graphs of sizes up to 6 is known. In the paper we study cyclic 7-edge connectivity of fullerene graphs, distinguishing between degenerated and non-degenerated cyclic edge-cuts, regarding the arrangement of the 12 pentagons. We prove that if there exists a non-de...