Bubble-sort Graphs Research Articles

Fault-tolerant strong Menger connectivity of modified bubble-sort graphs

An interconnection network is usually modeled as a graph, in which vertices and edges correspond to processors and communication links, respectively. For the study of graphs, connectivity and edge connectivity are the basic issues, and the discussion of fault tolerance is based on connectivity. A connected graph G is strongly Menger connected if each pair of vertices u and v of G, there are min{dG(u),dG(v)} vertex-disjoint paths connecting u and v. G is m-fault-tolerant strongly Menger connected if G−F remains strongly Menger connected for any F⊆V(G) with |F|≤m. G is m-conditional fault-tolerant strongly Menger connected if G−F remains strongly Menger connected for any F⊆V(G) with |F|≤m and δ(G−F)≥2. In this paper, we demonstrate that n-dimensional modified bubble-sort graphs MBn is (n−2)-fault-tolerant strongly Menger connected and (n−2)-fault-tolerant one-to-many strongly Menger connected for n≥4. Moreover, under the restricted condition that each vertex has at least two fault-free adjacent vertices, MBn is (2n−5)-conditional fault-tolerant strongly Menger connected for n≥4.

Hyper star fault tolerance of bubble sort networks

Let H and M be two connected subgraphs of an interconnection network G. If the removal of any minimum H-structure-cut (resp. minimum H-substructure-cut) splits G into exactly two components, one of which is isomorphic to M, then G is said to be hyperH|M-connected (resp. hyper sub-H|M-connected). The bubble sort graph Bn is one of alternative interconnection networks for multiprocessor systems. Let n≥7 and 2≤r≤4. Denote by K1 the complete graph on one vertex and denote by K1,r a star on 1+r vertices. In this paper, we prove that (i) both the K1,r-structure connectivity and the sub-K1,r-structure connectivity of Bn are ⌈n−12⌉; and (ii)Bn is hyper K1,r|K1-connected and hyper sub-K1,r|K1-connected.

On the reliability of modified bubble-sort graphs

On the reliability of modified bubble-sort graphs

Reliability evaluation of Modified bubble-sort graph networks based on structure fault pattern

Let G be a graph and H be a fixed connected subgraph. Let H={H1,H2,⋯,Hk} be a set of connected subgraphs of G. The H-structure connectivity (resp. H-substructure connectivity) κ(G;H) (resp. κs(G;H)) is defined as the least cardinality of H such that Hi is isomorphic to H (resp. a connected subgraph of H) for any 1≤i≤k, and H’s deletion makes G disconnected or trivial. As an extension of the classic connectivity, the H-structure (resp. H-substructure) connectivity can better evaluate the fault-tolerance of an interconnection network. In this paper, we focus on the n-dimensional modified bubble-sort graph MBn. We determine κ(MBn;Pl) (resp. κs(MBn;Pl), where n≥5, 2≤l≤2n, and Pl is a path on l vertices; κ(MBn;C2l) (resp. κs(MBn;C2l)), where 6≤2l≤n, and C2l is a cycle on 2l vertices; κ(MBn;T2l) (resp. κs(MBn;T2l)), where 1≤l≤n−2 and T2l is an l-leaves 2-step star. In addition, we give the upper bound of κ(MBn;K1,l) (resp. κs(MBn;K1,l)), where n≥4 and 2≤l≤n and K1,l is a star on l+1 vertices, and prove that the upper bounds are sharp for l=2,3,4.

The Diagnosability of Interconnection Networks with Missing Edges and Broken-Down Nodes Under the PMC and MM* Models

Abstract Diagnosability is often considered as an important factor for measuring the self-diagnostic ability of network systems. However, classic system-level diagnosis focuses only on processor faults and ignores the objective reality of communication faults. Under real circumstances, missing edges and node failures usually occur simultaneously in multiprocessor systems (called hybrid fault circumstances). Therefore, it is important to study the diagnosability of multiprocessor systems under hybrid fault circumstances. In this paper, we propose several diagnosabilities of interconnection networks with missing edges and faulty nodes. By exploring some important relationships between diagnosability and the minimum degree of a network under hybrid fault circumstances, we present and prove the diagnosability of several classic interconnection networks, including BC (bijective connection) networks, star graphs, folded hypercubes, exchanged hypercubes, exchanged crossed cubes, k-ary n-cubes, bubble-sort star graphs and balanced hypercubes, with missing edges and broken-down nodes under the PMC (Preparata, Metze and Chien) and MM* (Maeng and Malek) models.

The non-inclusive g-good-neighbor diagnosability of interconnection networks

Motivated by the definitions of g-good-neighbor diagnosability and non-inclusive diagnosability, we propose a new diagnosability—the non-inclusive g-good-neighbor diagnosability tNg(G) of a multiprocessor system G, which requires every pair of g-good-neighbor faulty sets is non-inclusive. The Rg-conditional diagnosability tRg(G) of a system G is a generalization of conditional diagnosability, which requires at least g fault-free neighbors for each node. In this paper, we explore the relationships between the non-inclusive g-good-neighbor diagnosability and the Rg-conditional diagnosability of G under the PMC and MM* models. We first show tNg(G)≤tRg(G) for g≥1, and also give some conditions for equality. Next, we discuss the non-inclusive g-good-neighbor diagnosability of hypercubes, (n,k)-star graphs and (n,k)-bubble-sort graphs. We show that the non-inclusive g-good-neighbor diagnosability of n-dimensional hypercubes is less that its Rg-conditional diagnosability for 2≤g≤n−22, and determine the non-inclusive g-good-neighbor diagnosability of (n,k)-star graphs and (n,k)-bubble-sort graphs. Finally, we plot and compare the non-inclusive g-good-neighbor diagnosability and the g-good-neighbor diagnosability of (n,k)-star graphs and (n,k)-bubble-sort graphs under the PMC and MM* models, respectively. It can be seen that their non-inclusive g-good-neighbor diagnosability is significantly larger than their g-good-neighbor diagnosability.

Hypercube embeddings and Cayley graphs generated by transpositions

A graph is called a partial cube if it can be embedded into a hypercube isometrically. In this paper, we study a class of Cayley graphs—Cayley graphs generated by transpositions—and show that a Cayley graph generated by transpositions is a partial cube if and only if is a bubble sort graph. This result enhances a result of Alahmadi et al. in 2016: is a partial cube. As a corollary, we give the analytical expressions of the Wiener indices of bubble sort graphs.

The Rg-conditional diagnosability of international networks

The Rg-conditional diagnosability of a multiprocessor system modeled by a graph G, denoted by tRg(G), is a generalization of conditional diagnosability, which restricts every vertex contains at least g fault-free neighbors. Particularly, the R1-conditional diagnosability is the conditional diagnosability. The Rg-conditional connectivity of a graph G, denoted by κRg(G), is the minimum number of vertices, whose deletion will disconnect the graph and every vertex of G has at least g neighbors in the remaining subgraphs. In this paper, the relationships between the Rg-conditional connectivity of a graph G and its Rg-conditional diagnosability under the PMC and MM* models are explored. We establish the Rg-conditional diagnosability tRg(G) equals κR2g+1(G)+g under some reasonable conditions, except the R1-conditional diagnosability of G under the MM* model. Moreover, we show under the MM* model, tR1(G)=κR2(G) with similar conditions. Applying our results, the Rg-conditional diagnosability of the (n,k)-star graphs and the (n,k)-bubble-sort graphs are determined.

Fault-Tolerant Strongly Hamiltonian Laceability and Hyper-Hamiltonian Laceability of Cayley Graphs Generated by Transposition Trees

AbstractA bipartite graph is Hamiltonian laceable if any two of its vertices in different partite sets are connected by a Hamiltonian path. A Hamiltonian laceable graph $G$ is called strongly Hamiltonian laceable if any two of its vertices in the same partite set are connected by a path of length $|V(G)|-2$. A Hamiltonian laceable graph $G$ (with two partite sets $V_0, V_1$) is called hyper-Hamiltonian laceable, if for any vertex $v \in V_{i}$ for $i \in \{0,1\}$, there is a Hamiltonian path of $G-\{v\}$ between any two vertices in $V_{1-i}$. In this paper, we focus on the edge-fault-tolerant strongly Hamiltonian laceability and hyper-Hamiltonian laceability on the class of Cayley graphs generated by transposition trees, which are a generalization of star graph and bubble-sort graph. For every $n$-dimensional Cayley graph generated by a transposition tree $\Gamma _n$, we show that $\Gamma _{n}-F$ is strongly Hamiltonian laceable for any $F \subseteq E(\Gamma _{n})$ with $|F|\leq n-3$, which generalizes results in [ 1, 11], and show that $\Gamma _{n}-F$ is hyper-Hamiltonian laceable for any $F \subseteq E(\Gamma _{n})$ with $|F|\leq n-4$.

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

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.

Subnetwork reliability analysis of bubble-sort graph networks

The bubble-sort graph network Bn is recognized as an attractive interconnection network topology for building multiprocessor computer systems. In this paper, the subnetwork reliability of Bn is analyzed in the presence of node failures. An upper bound and a lower bound on the Bn−1 subnetwork reliability of Bn are established under the probability fault model, and the theoretical results are proved to be in accordance with and near to the simulation results. Moreover, under the node fault model, the mean time to failure to maintain the fault-free status of different number of disjoint Bn−1 subnetworks in Bn is evaluated by using the fixed partitioning and the flexible partitioning, respectively, and it is shown that Bn has better robustness under the flexible partitioning pattern when disjoint Bn−1 subnetworks need to be used.

A Novel Interconnection Network with Improved Network Cost Through Shuffle-Exchange Permutation Graph

The interconnection network represents an interconnected structure of processors that strongly determines the performance quality of a parallel processing system. The shuffle-exchange permutation (SEP) network with three degrees has high fault tolerance and can be efficiently simulated through star, bubble-sort, and pancake graphs. This study proposes a new interconnection network: the new SEP (NSEP), which improves the diameter and reduces network cost by adding one edge to the SEP network, and presents its graph properties and routing algorithms. The NSEP network, with a degree of connectivity of four, demonstrated maximum fault tolerance and Hamiltonian cycle. Furthermore, the diameter was seen to be improved by 40% or more and the network cost by 20% or more.

Small cycles, generalized prisms and Hamiltonian cycles in the Bubble-sort graph

The Bubble-sort graph BSn,n⩾2, is a Cayley graph over the symmetric group Symn generated by transpositions from the set {(12),(23),…,(n−1n)}. It is a bipartite graph containing all even cycles of length ℓ, where 4⩽ℓ⩽n!. We give an explicit combinatorial characterization of all its 4- and 6-cycles. Based on this characterization, we define generalized prisms in BSn,n⩾5, and present a new approach to construct a Hamiltonian cycle based on these generalized prisms.

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

<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>

The 3-Good-Neighbor Connectivity of Modified Bubble-Sort Graphs

Let G = V G , E G be a connected graph. A subset F ⊆ V G is called a g -good-neighbor cut if G − F is disconnected and each vertex of G − F has at least g neighbors. The g -good-neighbor connectivity of G is the minimum cardinality of g -good-neighbor cuts. The n -dimensional modified bubble-sort graph MB n is a special Cayley graph. It has many good properties. In this paper, we prove that the 3-good-neighbor connectivity of MB n is 8 n − 24 for n ≥ 6 .

Linearly many faults in Cayley graphs generated by transposition triangle free unicyclic graphs

We consider a kind of Cayley graphs, including the modified bubble-sort graphs MBn, the Cayley graphs generated by lollipop graphs Hn,n−1 and the other Cayley graphs generated by unicyclic triangle free graphs. We get that if we delete linearly many vertices in Cayley graphs generated by transposition unicyclic triangle free graphs, then the resulting graphs has a large connected component that comprises of nearly all residual vertices.

Reliability assessment of the Cayley graph generated by trees

The r-component connectivity cκr(G) of a non-complete graph G is the minimum number of vertices whose removal results in a disconnected graph with at least r components. This parameter generalizes the classical connectivity. Hence, it is an important parameter to evaluate the reliability and fault tolerance of a network. Let Γn be the n-dimensional Cayley graph generated by transposition trees, and g(Γn) denote the girth of Γn (i.e., the length of a shortest cycle of Γn). It has been shown that either g(Γn)=4 or g(Γn)=6. In this paper, we obtain that cκ3(Γn)=2n−4 if g(Γn)=4 and cκ3(Γn)=2n−3 if g(Γn)=6 for n≥4. As corollaries, the 3-component connectivity of the star graph Sn and the bubble-sort graph Bn can be obtained directly. Also, the above consequences explicitly point out that there is a flaw in the recent result of Xu et al., (2020).

The fault tolerance of [formula omitted]-bubble-sort networks

The h-good neighbor connectivity κh(G) is an important parameter to evaluate the fault tolerance of an interconnection network G. So far, almost all known results of κh(G) are about small h′s except the hypercubes, the star graphs, the k-ary n-cubes and hierarchical hypercube network HHCn et al. In this paper, we focus on κh(Bn,k) for the (n,k)-bubble-sort network Bn,k, which is a generalization of the bubble-sort graph Bn as it is a special case of (n,k)-bubble-sort networks for k=n−1. We show that κh(Bn,k)=n+h(k−2)−1 for 2≤k≤⌈n+32⌉ and 0≤h≤⌊n−32⌋, which implies that at least n+h(k−2)−1 vertices of Bn,k have to be deleted to get a disconnected graph without vertices of degree less than h.

Conditional diagnosability of component-composition graphs under the PMC model

Diagnosability is an important metric for measuring the reliability of multiprocessor systems. The conditional diagnosability, which is more general than the classical diagnosability, is to measure the diagnosability of a multiprocessor system under the assumption that all of the neighbors of any node in the system cannot fail at the same time. This paper adopts the PMC model and outlines the common properties of a wide class of interconnection networks, called component-composition graphs (CCGs), to determine their conditional diagnosability by using the derived properties. As applications, the diagnosability of hypercube-like networks (including hypercubes, crossed cubes, Möbius cubes, twisted cubes, locally twisted cubes, generalized twisted cubes, and recursive circulants), and bubble-sort graphs, all of which belong to CCGs can be readily obtained.

Constructing dual-CISTs of pancake graphs and performance assessment of protection routings on some Cayley networks

For a connected graph $$G=(V,E)$$ , two spanning trees $$T_1$$ and $$T_2$$ of G are said to be a pair of completely independent spanning trees (or a dual-CIST for short) if for any two vertices $$u,v\in V$$ , the paths joining u and v in the two trees have no common vertex except for u and v. Although the existence of a dual-CIST in the underlying graph of a network has the practical application of protection routing on fault-tolerance, it has been proved that determining whether a graph G admits a dual-CIST is NP-complete. As we know that Cayley graphs are a large family of graphs, some of its subclasses have been attracted and thus graphs in these subclasses have been adopted as the topologies of interconnection networks, such as the n-dimensional star graphs $$S_n$$ , bubble sort graphs $$BS_n$$ , pancake graph $$P_n$$ , alternating group networks $$AGN_n$$ and so on. Pai and Chang (IEEE/ACM Trans Netw 27(3): 1112–1123, 2019) recently showed that there exist dual-CISTs in $$S_n$$ , $$BS_n$$ , $$AGN_n$$ for $$n\geqslant 5$$ and provided their corresponding protection routings. So far, the problem of constructing dual-CISTs on $$P_n$$ has not been dealt with yet. In this sequel, we continue the investigation of the construction of dual-CISTs in pancake graphs as a complementary result. Since $$P_n$$ , $$S_n$$ , and $$BS_n$$ are with the same scale, we experimentally assess the performance of protection routing through simulation results for comparing them when $$n=5,6,7$$ .