Number Of Edge-disjoint Paths Research Articles

A Note of Reliability Analysis of SM-λ in Folded-Crossed Hypercube with Conditional Faults

The fault tolerance of an interconnection network of parallel and distributed systems can be evaluated by various topological parameters of its underlying graph [Formula: see text], with strong Menger edge connectivity being a vital parameter in this regard. A connected graph [Formula: see text] is called strongly Menger edge connected (SM-[Formula: see text]) if it connects any pair of vertices [Formula: see text] and [Formula: see text] with [Formula: see text] number of edge-disjoint paths. Under the uniform distribution of faults in a large interconnection network, it is improbable that each faulty edge incident to a vertex will occur simultaneously. Thus, [Formula: see text]-strongly Menger edge connected of order [Formula: see text] was introduced in 2018 by He et al. Here, [Formula: see text] is called as [Formula: see text]-strongly Menger edge connected of order [Formula: see text], if [Formula: see text] remains SM-[Formula: see text], where [Formula: see text] is an arbitrary edge set in a graph [Formula: see text] with [Formula: see text] and the minimum degree of the remaining graph [Formula: see text]. The largest [Formula: see text] keeping the property of [Formula: see text] being SM-[Formula: see text] is denoted as [Formula: see text]. Among variants of hypercube, the [Formula: see text]-dimensional folded-crossed hypercube [Formula: see text] attracts attention in recent years. In this paper, we focus on calculating the exact value of the maximum conditional edge-fault-tolerant number of order [Formula: see text] of [Formula: see text], [Formula: see text] for two integers [Formula: see text] and [Formula: see text].

Exponential type of many-to-many edge disjoint paths on ternary n-cubes

One of most central issues in various interconnection networks of parallel and distributed systems is to find edge-disjoint paths concerned with an information transmission through links. It is universally acknowledged that an interconnection network can be modeled as an undirected simple graph G=(V,E) with processors and physical links between the processors represented as vertices and edges of the G, respectively. The studies on the edge-disjoint paths are closely related to the edge-connectivity. By the well-known Menger theorem, the maximum number of edge disjoint paths connecting any two disjoint connected subgraphs with g vertices in G can also define by the minimum modified edge-cut, called the g-extra edge-connectivity of Gλg(G). It is the cardinality of the minimum set of edges in G, if such a set exists, whose deletion disconnects G and leaves every remaining component with at least g vertices. The k-ary n-cube network is known as one of the most attractive interconnection networks for parallel and distributed systems. This article intends to show that for 3⌈n2⌉+r−⌊32r+e+12⌋≤g≤3⌈n2⌉+r, the g-extra edge-connectivity of ternary n-cube (also called 3-ary n-cube) (n≥3) exists a concentration behavior, and prove that these corresponding g-extra edge-connectivities of ternary n-cubes are the constants 2(⌊n2⌋−r)3⌈n2⌉+r, where r=0,1,2,⋯,⌊n2⌋−1, and e=1 if n is odd and e=0 if n is even. The above upper and lower bounds of g are sharp. As the integer g varies within the range between 3⌈n2⌉+r−⌊32r+e+12⌋ and 3⌈n2⌉+r, a necessary and sufficient condition of λg-optimality is found. Our results improve the several previous results.

Average Edge-Connectivity of Cubic Graphs

In this paper, we consider the concept of the average edge-connectivity [Formula: see text] of a graph [Formula: see text], defined to be the average, over all pairs of vertices, of the maximum number of edge-disjoint paths connecting these vertices. Kim and O previously proved that [Formula: see text] for any connected cubic graph on [Formula: see text] vertices. We refine their result by showing that [Formula: see text] We also characterize the graphs where equality holds.

Number of vertices of degree three in spanning 3-trees in square graphs

In this paper, we show that the square graph of a tree T has a spanning tree of maximum degree at most three and with at most max{0,∑x∈W≥3(T)(tT(x)−2)−2} vertices of degree three, where W≥3(T)={x∈V(T): there are at least three edge-disjoint paths of length at least two that start x} and tT(x) is the number of edge-disjoint paths with length at least two that start at a vertex x.

Multipath Discovery Algorithms for VoD Streaming In Wireless Mesh Network

Transmission and routing of video data over wireless network is a challenging task because of wireless interferences. To improve the performance of video on demand transmission over wireless networks multipath algorithms are used. IPD/S (Iterative path discovery/selection) PPD/S (Parallel Path discovery/selection) are two algorithms which is used for discovering maximum number of edge disjoint paths from source to destination, for each VoD request by considering the effects of wireless interferences. In this paper performance evaluation of these multipath discovery algorithms for VoD (Video on demand) streaming in wireless mesh network is presented. These algorithms are evaluated on the bases of Number of Path discovers, Packet drop ratio and delay. Simulation result shows that PPD/S works batter as compared to IPD/S because it’s able to discover more paths than IPD/S under same circumstances.

Robust Network Coding in the Presence of Untrusted Nodes

While network coding can be an efficient means of information dissemination in networks, it is highly susceptible to "pollution attacks," as the injection of even a single erroneous packet has the potential to corrupt each and every packet received by a given destination. Even when suitable error-control coding is applied, an adversary can, in many interesting practical situations, overwhelm the error-correcting capability of the code. To limit the power of potential adversaries, a broadcast transformation is introduced, in which nodes are limited to just a single (broadcast) transmission per generation. Under this broadcast transformation, the multicast capacity of a network is changed (in general reduced) from the number of edge-disjoint paths between source and sink to the number of internally-disjoint paths. Exploiting this fact, we propose a family of networks whose capacity is largely unaffected by a broadcast transformation. This results in a significant achievable transmission rate for such networks, even in the presence of adversaries.

Optimal Construction of Edge-Disjoint Paths in Random Graphs

Given a graph G=(V,E) with n vertices, m edges, and a family of $\kappa$ pairs of vertices in $V$, we are interested in finding for each pair (ai, bi) a path connecting ai to bi such that the set of $\kappa$ paths so found is edge disjoint. (For arbitrary graphs the problem is ${\cal NP}$-complete, although it is in ${\cal P}$ if $\kappa$ is fixed.) We present a polynomial time randomized algorithm for finding the optimal number of edge disjoint paths (up to constant factors) in the random graph Gn,m for all edge densities above the connectivity threshold. (The graph is chosen first; then an adversary chooses the pairs of endpoints.) Our results give the first tight bounds for the edge-disjoint paths problem for any nontrivial class of graphs.

Paths and edge-connectivity in graphs III. Six-terminalk paths

Suppose thatk ? 1 is an odd integer, (s 1,t 1),..., (s k> ,t k ) are pairs of vertices of a graphG and?(s i ,t i ) is the maximal number of edge-disjoint paths betweens i andt i . We prove that if?(s i ,t i )? k (1≤ i ≤ k) and |{s 1,...s k ,t 1,...,t k }| ≤ 6, then there exist edge-disjoint pathsP 1,...,P k such thatP i has endss i andt i (1≤ i ≤ k).