scholarly journals The l-good-neighbor edge connectivity of graphs

2021 ◽  
Vol 2132 (1) ◽  
pp. 012027
Author(s):  
Shumin Zhang ◽  
Yalan Li ◽  
Chengfu Ye
Keyword(s):  
Interconnection Networks ◽  
Minimum Degree ◽  
Type Problem ◽  
Edge Connectivity ◽  
Good Neighbor

Abstract The l -good-neighbor edge connectivity is an useful parameter to measure the reliability and tolerance of interconnection networks. For a graph H with order p and an integer l (l ≥ 0), an edge subset X ⸦ E(H) is called a l-good-neighbor edge-cut if H − X is disconnected and the minimum degree of every component of H − X is at least £. The order of the minimum l-good-neighbor edge-cut of H is called the l-good-neighbor edge connectivity of H, denoted by λ l (H). In this paper, we show λ(H) ≤ λ l+1(H), obtain the bounds of λl (H) when 0 ≤ l ≤ [p-2/2], character some graphs with the small λl (H) and get some results about the Erdös-Gallai-type problem about λl (H).

ON EDGE-CONNECTIVITY AND SUPER EDGE-CONNECTIVITY OF INTERCONNECTION NETWORKS MODELED BY PRODUCT GRAPHS

2010 ◽  
Vol 02 (02) ◽  
pp. 143-150
Author(s):  
CHUNXIANG WANG
Keyword(s):  
Connected Graph ◽  
Interconnection Networks ◽  
Minimum Degree ◽  
Edge Connectivity ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Product Graph ◽  
Product Graphs

The super edge-connectivity λ′ of a connected graph G is the minimum cardinality of an edge-cut F in G such that every component of G–F contains at least two vertices. Let two connected graphs Gm and Gp have m and p vertices, minimum degree δ(Gm) and δ(Gp), edge-connectivity λ(Gm) and λ(Gp), respectively. This paper shows that min {pλ(Gm), λ(Gp) + δ(Gm), δ(Gm)(λ(Gp) + 1), (δ(Gm) + 1)λ(Gp)} ≤ λ(Gm * Gp) ≤ δ(Gm) + δ(Gp), where the product graph Gm * Gp of two given graphs Gm and Gp, defined by J. C. Bermond et al. [J. Combin. Theory B36 (1984) 32–48] in the context of the so-called (△, D)-problem, is one interesting model in the design of large reliable networks. Moreover, this paper determines λ′(Gm * Gp) ≤ min {pδ(Gm), ξ(Gp) + 2δ(Gm)} and λ′(G1 ⊕ G2) ≥ min {n, λ1 + λ2} if δ1 = δ2.

Reliability Evaluation of Augmented Cubes on Degree

2021 ◽  
Author(s):  
Mingzu Zhang ◽  
Xiaoli Yang ◽  
Xiaomin He ◽  
Zhuangyan Qin ◽  
Yongling Ma
Keyword(s):  
Fault Tolerance ◽  
Interconnection Networks ◽  
Minimum Degree ◽  
Tolerance Analysis ◽  
Reliability Evaluation ◽  
Affirmative Answer ◽  
Edge Connectivity ◽  
Minimum Number ◽  
Edge Fault Tolerance ◽  
Augmented Cube

The [Formula: see text]-dimensional augmented cube [Formula: see text], proposed by Choudum and Sunitha in 2002, is one of the most famous interconnection networks of the distributed parallel system. Reliability evaluation of underlying topological structures is vital for fault tolerance analysis of this system. As one of the most extensively studied parameters, the [Formula: see text]-conditional edge-connectivity of a connected graph [Formula: see text], [Formula: see text], is defined as the minimum number of the cardinality of the edge-cut of [Formula: see text], if exists, whose removal disconnects this graph and keeps each component of [Formula: see text] having minimum degree at least [Formula: see text]. Let [Formula: see text], [Formula: see text] and [Formula: see text] be three integers, where [Formula: see text], if [Formula: see text] and [Formula: see text], if [Formula: see text]. In this paper, we determine the exact value of the [Formula: see text]-conditional edge-connectivity of [Formula: see text], [Formula: see text] for each positive integer [Formula: see text] and [Formula: see text], and give an affirmative answer to Shinde and Borse’s corresponding conjecture on this topic in [On edge-fault tolerance in augmented cubes, J. Interconnection Netw. 20(4) (2020), DOI:10.1142/S0219265920500139].

The 3-Good Neighbor Edge-Fault-Tolerant Strong Menger Edge Connectivity on the Class of BC Networks

2021 ◽  
Author(s):  
Rong Liu ◽  
Pingshan Li
Keyword(s):  
Fault Tolerant ◽  
Minimum Degree ◽  
Disjoint Paths ◽  
Edge Connectivity ◽  
Restricted Condition ◽  
Good Neighbor ◽  
Twisted Cubes ◽  
Free Edges

A graph [Formula: see text] is called strongly Menger edge connected (SM-[Formula: see text] for short) if the number of disjoint paths between any two of its vertices equals the minimum degree of these two vertices. In this paper, we focus on the maximally edge-fault-tolerant of the class of BC-networks (contain hypercubes, twisted cubes, Möbius cubes, crossed cubes, etc.) concerning the SM-[Formula: see text] property. Under the restricted condition that each vertex is incident with at least three fault-free edges, we show that even if there are [Formula: see text] faulty edges, all BC-networks still have SM-[Formula: see text] property and the bound [Formula: see text] is sharp.

Semi-Deterministic Construction of Scale-Free Networks with Designated Parameters

2018 ◽  
Vol 18 (01) ◽  
pp. 1850001
Author(s):  
NAOKI TAKEUCHI ◽  
SATOSHI FUJITA
Keyword(s):  
Degree Distribution ◽  
Interconnection Networks ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Scale Free ◽  
Scale Free Networks ◽  
Ideal Number ◽  
Message Propagation ◽  
Deterministic Construction ◽  
The Ideal

Scale-free networks have several favorable properties as the topology of interconnection networks such as the short diameter and the quick message propagation. In this paper, we propose a method to construct scale-free networks in a semi-deterministic manner. The proposed algorithm extends the Bulut's algorithm for constructing scale-free networks with designated minimum degree k and maximum degree m, in such a way that: (1) it determines the ideal number of edges derived from the ideal degree distribution; and (2) after connecting each new node to k existing nodes as in the Bulut’s algorithm, it adjusts the number of edges to the ideal value by conducting add/removal of edges. We prove that such an adjustment is always possible if the number of nodes in the network exceeds [Formula: see text]. The performance of the algorithm is experimentally evaluated.

Component Edge Connectivity of Hypercubes

2018 ◽  
Vol 29 (06) ◽  
pp. 995-1001 ◽  
Author(s):  
Shuli Zhao ◽  
Weihua Yang ◽  
Shurong Zhang ◽  
Liqiong Xu
Keyword(s):  
Fault Tolerance ◽  
Complete Graph ◽  
Interconnection Networks ◽  
Interconnection Network ◽  
Optimal Solutions ◽  
Edge Connectivity ◽  
Important Measure ◽  
Minimum Number ◽  
Text Component

Fault tolerance is an important issue in interconnection networks, and the traditional edge connectivity is an important measure to evaluate the robustness of an interconnection network. The component edge connectivity is a generalization of the traditional edge connectivity. The [Formula: see text]-component edge connectivity [Formula: see text] of a non-complete graph [Formula: see text] is the minimum number of edges whose deletion results in a graph with at least [Formula: see text] components. Let [Formula: see text] be an integer and [Formula: see text] be the decomposition of [Formula: see text] such that [Formula: see text] and [Formula: see text] for [Formula: see text]. In this note, we determine the [Formula: see text]-component edge connectivity of the hypercube [Formula: see text], [Formula: see text] for [Formula: see text]. Moreover, we classify the corresponding optimal solutions.

Connectivity and Nature Diagnosability of Leaf-Sort Graphs

2020 ◽  
Vol 20 (03) ◽  
pp. 2050011
Author(s):  
JUTAO ZHAO ◽  
SHIYING WANG
Keyword(s):  
Interconnection Networks ◽  
Interconnection Network ◽  
Research Topic ◽  
Multiprocessor System ◽  
Important Research ◽  
Edge Connectivity ◽  
Topology Structure ◽  
Pmc Model ◽  
Important Research Topic

The connectivity and diagnosability of a multiprocessor system or an interconnection network is an important research topic. The system and interconnection network has a underlying topology, which usually presented by a graph. As a famous topology structure of interconnection networks, the n-dimensional leaf-sort graph CFn has many good properties. In this paper, we prove that (a) the restricted edge connectivity of CFn (n ≥ 3) is 3n − 5 for odd n and 3n − 6 for even n; (b) CFn (n ≥ 5) is super restricted edge-connected; (c) the nature diagnosability of CFn (n ≥ 4) under the PMC model is 3n − 4 for odd n and 3n − 5 for even n; (d) the nature diagnosability of CFn (n ≥ 5) under the MM* model is 3n − 4 for odd n and 3n − 5 for even n.

scholarly journals The g-Good-Neighbor Diagnosability of Bubble-Sort Graphs under Preparata, Metze, and Chien’s (PMC) Model and Maeng and Malek’s (MM)* Model

Information ◽  
2019 ◽  
Vol 10 (1) ◽  
pp. 21
Author(s):  
Shiying Wang ◽  
Zhenhua Wang
Keyword(s):  
Fault Diagnosis ◽  
Interconnection Networks ◽  
Multiprocessor System ◽  
Topology Structure ◽  
Free Node ◽  
Pmc Model ◽  
Good Neighbor

Diagnosability of a multiprocessor system is an important topic of study. A measure for fault diagnosis of the system restrains that every fault-free node has at least g fault-free neighbor vertices, which is called the g-good-neighbor diagnosability of the system. As a famous topology structure of interconnection networks, the n-dimensional bubble-sort graph B n has many good properties. In this paper, we prove that (1) the 1-good-neighbor diagnosability of B n is 2 n − 3 under Preparata, Metze, and Chien’s (PMC) model for n ≥ 4 and Maeng and Malek’s (MM) ∗ model for n ≥ 5 ; (2) the 2-good-neighbor diagnosability of B n is 4 n − 9 under the PMC model and the MM ∗ model for n ≥ 4 ; (3) the 3-good-neighbor diagnosability of B n is 8 n − 25 under the PMC model and the MM ∗ model for n ≥ 7 .

Super-connectivity and super-edge-connectivity for some interconnection networks

2003 ◽  
Vol 140 (2-3) ◽  
pp. 245-254 ◽  
Author(s):  
Y-Chuang Chen ◽  
Jimmy J.M. Tan ◽  
Lih-Hsing Hsu ◽  
Shin-Shin Kao
Keyword(s):  
Interconnection Networks ◽  
Edge Connectivity ◽  
Super Connectivity

Minimum degree, edge-connectivity and radius

2012 ◽  
Vol 26 (3) ◽  
pp. 585-591 ◽  
Author(s):  
Baoyindureng Wu ◽  
Xinhui An ◽  
Guojie Liu ◽  
Guiying Yan ◽  
Xiaoping Liu
Keyword(s):  
Minimum Degree ◽  
Edge Connectivity

scholarly journals Edge-fault-tolerant strong Menger edge connectivity of bubble-sort graphs

2021 ◽  
Vol 6 (12) ◽  
pp. 13210-13221
Author(s):  
Yanling Wang ◽  
◽  
Shiying Wang
Keyword(s):  
Interconnection Networks ◽  
Fault Tolerant ◽  
Edge Connectivity ◽  
Special Cases

<abstract><p>This paper studies the edge-fault-tolerant strong Menger edge connectivity of $ n $-dimensional bubble-sort graph $ B_{n} $. We give the values of faulty edges that $ B_{n} $ can tolerant when $ B_{n} $ is strongly Menger edge connected under two conditions. When there are $ (n-3) $ faulty edges removed from $ B_{n} $, the $ B_{n} $ network is still working and it is strongly Menger edge connected. When the condition of any vertex in $ B_{n} $ has at least two neighbors is imposed, the number of faulty edges that can removed from $ B_{n} $ is $ (2n-6) $ when $ B_{n} $ is also strongly Menger edge connected. And two special cases are used to illustrate the correctness of the conclusions. The conclusions can help improve the reliability of the interconnection networks.</p></abstract>

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