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.

FAULT-TOLERANT MAXIMAL LOCAL-CONNECTIVITY ON CAYLEY GRAPHS GENERATED BY TRANSPOSITION TREES

2009 ◽  
Vol 10 (03) ◽  
pp. 253-260 ◽  
Author(s):  
LUN-MIN SHIH ◽  
CHIEH-FENG CHIANG ◽  
LIH-HSING HSU ◽  
JIMMY J. M. TAN
Keyword(s):  
Cayley Graph ◽  
Fault Tolerant ◽  
Minimum Degree ◽  
Cayley Graphs ◽  
Disjoint Paths ◽  
Local Connectivity ◽  
Vertex Disjoint

The local connectivity of two vertices is defined as the maximum number of internally vertex-disjoint paths between them. In this paper, we define two vertices as maximally local-connected, if the maximum number of internally vertex-disjoint paths between them equals the minimum degree of these two vertices. Moreover, we show that an (n-1)-regular Cayley graph generated by transposition tree is maximally local-connected, even if there are at most (n-3) faulty vertices in it, and prove that it is also (n-1)-fault-tolerant one-to-many maximally local-connected.

Fault-Tolerant Maximal Local-Connectivity on Cayley Graphs Generated by Transpositions

2019 ◽  
Vol 30 (08) ◽  
pp. 1301-1315 ◽  
Author(s):  
Liqiong Xu ◽  
Shuming Zhou ◽  
Weihua Yang
Keyword(s):  
Fault Tolerance ◽  
Interconnection Networks ◽  
Fault Tolerant ◽  
Interconnection Network ◽  
Cayley Graphs ◽  
Disjoint Paths ◽  
Local Connectivity ◽  
Communication Links ◽  
Restricted Condition ◽  
Vertex Disjoint

An interconnection network is usually modeled as a graph, in which vertices and edges correspond to processors and communication links, respectively. Connectivity is an important metric for fault tolerance of interconnection networks. A graph [Formula: see text] is said to be maximally local-connected if each pair of vertices [Formula: see text] and [Formula: see text] are connected by [Formula: see text] vertex-disjoint paths. In this paper, we show that Cayley graphs generated by [Formula: see text]([Formula: see text]) transpositions are [Formula: see text]-fault-tolerant maximally local-connected and are also [Formula: see text]-fault-tolerant one-to-many maximally local-connected if their corresponding transposition generating graphs have a triangle, [Formula: see text]-fault-tolerant one-to-many maximally local-connected if their corresponding transposition generating graphs have no triangles. Furthermore, under the restricted condition that each vertex has at least two fault-free adjacent vertices, Cayley graphs generated by [Formula: see text]([Formula: see text]) transpositions are [Formula: see text]-fault-tolerant maximally local-connected if their corresponding transposition generating graphs have no triangles.

The (E)FTSM-(edge) Connectivity of Cayley Graphs Generated by Transposition Trees

2021 ◽  
pp. 1-11
Author(s):  
Pingshan Li ◽  
Rong Liu ◽  
Xianglin Liu
Keyword(s):  
Cayley Graph ◽  
Fault Tolerant ◽  
Cayley Graphs ◽  
Disjoint Paths ◽  
Star Graph ◽  
Edge Connectivity ◽  
Short Edge ◽  
Edge Disjoint

The Cayley graph generated by a transposition tree [Formula: see text] is a class of Cayley graphs that contains the star graph and the bubble sort graph. A graph [Formula: see text] is called strongly Menger (SM for short) (edge) connected if each pair of vertices [Formula: see text] are connected by [Formula: see text] (edge)-disjoint paths, where [Formula: see text] are the degree of [Formula: see text] and [Formula: see text] respectively. In this paper, the maximally edge-fault-tolerant and the maximally vertex-fault-tolerant of [Formula: see text] with respect to the SM-property are found and thus generalize or improve the results in [19, 20, 22, 26] on this topic.

scholarly journals Strong Menger Connectedness of Augmented k-ary n-cubes

2020 ◽  
Author(s):  
Mei-Mei Gu ◽  
Jou-Ming Chang ◽  
Rong-Xia Hao
Keyword(s):  
Connected Graph ◽  
Fault Tolerant ◽  
Disjoint Paths ◽  
Edge Connectivity ◽  
Arbitrary Vertex ◽  
Edge Disjoint ◽  
Vertex Set ◽  
Delta G ◽  
Disjoint Edge

Abstract A connected graph $G$ is called strongly Menger (edge) connected if for any two distinct vertices $x,y$ of $G$, there are $\min \{\textrm{deg}_G(x), \textrm{deg}_G(y)\}$ internally disjoint (edge disjoint) paths between $x$ and $y$. Motivated by parallel routing in networks with faults, Oh and Chen (resp., Qiao and Yang) proposed the (fault-tolerant) strong Menger (edge) connectivity as follows. A graph $G$ is called $m$-strongly Menger (edge) connected if $G-F$ remains strongly Menger (edge) connected for an arbitrary vertex set $F\subseteq V(G)$ (resp. edge set $F\subseteq E(G)$) with $|F|\leq m$. A graph $G$ is called $m$-conditional strongly Menger (edge) connected if $G-F$ remains strongly Menger (edge) connected for an arbitrary vertex set $F\subseteq V(G)$ (resp. edge set $F\subseteq E(G)$) with $|F|\leq m$ and $\delta (G-F)\geq 2$. In this paper, we consider strong Menger (edge) connectedness of the augmented $k$-ary $n$-cube $AQ_{n,k}$, which is a variant of $k$-ary $n$-cube $Q_n^k$. By exploring the topological proprieties of $AQ_{n,k}$, we show that $AQ_{n,3}$ (resp. $AQ_{n,k}$, $k\geq 4$) is $(4n-9)$-strongly (resp. $(4n-8)$-strongly) Menger connected for $n\geq 4$ (resp. $n\geq 2$) and $AQ_{n,k}$ is $(4n-4)$-strongly Menger edge connected for $n\geq 2$ and $k\geq 3$. Moreover, we obtain that $AQ_{n,k}$ is $(8n-10)$-conditional strongly Menger edge connected for $n\geq 2$ and $k\geq 3$. These results are all optimal in the sense of the maximum number of tolerated vertex (resp. edge) faults.

Fault-Tolerant Maximal Local-Edge-Connectivity of Augmented Cubes

2020 ◽  
Vol 30 (03) ◽  
pp. 2040001
Author(s):  
Liyang Zhai ◽  
Liqiong Xu ◽  
Weihua Yang
Keyword(s):  
Regular Graph ◽  
Interconnection Networks ◽  
Fault Tolerant ◽  
Interconnection Network ◽  
Disjoint Paths ◽  
Communication Links ◽  
Restricted Condition ◽  
Local Edge ◽  
Augmented Cube ◽  
Cube Formula

An interconnection network is usually modeled as a graph, in which vertices and edges correspond to processors and communication links, respectively. Connectivity is an important metric for fault tolerance of interconnection networks. A connected graph [Formula: see text] is said to be maximally local-edge-connected if each pair of vertices [Formula: see text] and [Formula: see text] of [Formula: see text] are connected by [Formula: see text] pairwise edge-disjoint paths. In this paper, we show that the [Formula: see text]-dimensional augmented cube [Formula: see text] is [Formula: see text]-edge-fault-tolerant maximally local-edge-connected and the bound [Formula: see text] is sharp; under the restricted condition that each vertex has at least three fault-free adjacent vertices, [Formula: see text] is [Formula: see text]-edge-fault-tolerant maximally local-edge-connected and the bound [Formula: see text] is sharp; and under the restricted condition that each vertex has at least [Formula: see text] fault-free adjacent vertices, [Formula: see text] is [Formula: see text]-edge-fault-tolerant maximally local-edge-connected. Furthermore, we show that a [Formula: see text]-regular graph [Formula: see text] is [Formula: see text]-fault-tolerant one-to-many maximally local-connected if [Formula: see text] does not contain [Formula: see text] and is super [Formula: see text]-vertex-connected of order 1, a [Formula: see text]-regular graph [Formula: see text] is [Formula: see text]-fault-tolerant one-to-many maximally local-connected if [Formula: see text] does not contain [Formula: see text] and is super [Formula: see text]-vertex-connected of order 1.

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).

Conditional Edge Connectivity of the Locally Twisted Cubes

2019 ◽  
Vol 7 (3) ◽  
pp. 501-509
Author(s):  
Hui Shang ◽  
Eminjan Sabir ◽  
Ji-Xiang Meng
Keyword(s):  
Edge Connectivity ◽  
Locally Twisted Cubes ◽  
Twisted Cubes
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