Discrete Applied Mathematics最新文献

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Covering a supermodular-like function in a mixed hypergraph 覆盖混合超图中的类超模函数

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-22 DOI: 10.1016/j.dam.2026.01.023

Hui Gao

In this paper, we solve a conjecture by Szigeti in [Matroid-rooted packing of arborescences], which characterizes mixed hypergraphs

F = (V, E \cup A)

for which there exists an orientation

\vec{E}

E

such that

e_{\vec{E} \cup A} (P) \geq \sum_{X \in P} h (X) - b (\cup P)

for every subpartition

P

V

, where

h

is an integer-valued, intersecting supermodular function on

V

and

b

a submodular function on

V

. As a corollary, another conjecture in the same paper is confirmed, which characterizes mixed hypergraphs admitting a packing of mixed hyperarborescences such that their roots form a basis in a given matroid, each vertex

v

belongs to exactly

k

of them and is the root of at least

f (v)

and at most

g (v)

of them.

本文用Szigeti在[树形的矩阵根填充]中求解了一个猜想，该猜想刻画了混合超图F=(V,E∪a)，其中对于V的每一子划分P存在一个方向E∈E，使得E∈∪a (P)≥∑X∈Ph(X)−b(∪P)，其中h是V上的一个整数值相交超模函数，b是V上的一个子模函数，作为一个推论，证实了同一论文中的另一个猜想。这是混合超图的特征，它允许混合超树序列的填充，使得它们的根形成给定矩阵中的一组基，每个顶点v恰好属于其中的k个顶点，并且是至少f(v)最多g(v)的根。

引用次数: 0

Fault tolerability of Cayley graphs generated by transposition unicyclic graphs with a triangle 带三角形的转置单环图生成的Cayley图的容错性

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-22 DOI: 10.1016/j.dam.2026.01.018

Weixing Zheng , Shuming Zhou , Lulu Yang

<div><div>The rapid expansion of multiprocessor systems in modern computing platforms has posed new challenges to ensure system reliability and fault resilience under complex and large-scale failure scenarios. Classical connectivity and diagnosability are two key parameters to evaluate the reliability and self-diagnostic capability of multiprocessor systems. As generalizations of traditional connectivity and diagnosability, <math><mi>g</mi></math>-extra connectivity and <math><mi>r</mi></math>-component connectivity, together with their corresponding diagnosabilities, offer a more refined characterization of fault tolerability. In this paper, we determine the fault tolerability of Cayley graphs <math><mrow><mi>U</mi><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math> generated by transposition unicyclic graphs with a triangle. We show that the <math><mi>g</mi></math>-extra connectivity of <math><mrow><mi>U</mi><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math> is <math><mrow><mrow><mo>(</mo><mi>g</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mi>n</mi><mo>−</mo><mfrac><mrow><mrow><mo>(</mo><mi>g</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>g</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math> for <math><mrow><mn>2</mn><mo>≤</mo><mi>g</mi><mo>≤</mo><mo>|</mo><mi>M</mi><mo>|</mo><mo>+</mo><mn>2</mn></mrow></math>, where <math><mi>M</mi></math> is the maximum matching of the generating graph of <math><mrow><mi>U</mi><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math> with the removal of a 3-cycle. Furthermore, we show that the <math><mi>g</mi></math>-extra diagnosability of <math><mrow><mi>U</mi><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math> under both PMC and MM* models is uniformly <math><mrow><mrow><mo>(</mo><mi>g</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mi>n</mi><mo>−</mo><mfrac><mrow><mrow><mo>(</mo><mi>g</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>g</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>g</mi></mrow></math>. In addition, we prove that the <math><mrow><mo>(</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math>-component connectivity of <math><mrow><mi>U</mi><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math> is <math><mrow><mi>r</mi><mi>n</mi><mo>−</mo><mfrac><mrow><mi>r</mi><mrow><mo>(</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math>, and <math><mrow><mo>(</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math>-component diagnosability of <math><mrow><mi>U</mi><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></

现代计算平台中多处理器系统的快速发展对复杂大规模故障场景下系统的可靠性和故障恢复能力提出了新的挑战。经典连通性和可诊断性是评价多处理器系统可靠性和自诊断能力的两个关键参数。作为传统连接性和可诊断性的概括，g-extra连接性和r-component连接性及其相应的可诊断性提供了更精确的容错性表征。本文确定了由带三角形的转置单环图生成的Cayley图UGn的容错性。我们证明了当2≤g≤|M|+2时，UGn的g-额外连通性为（g+1）n−(g+1)(g+2)2，其中M为UGn的生成图与去掉一个3环的最大匹配。进一步证明了在PMC和MM*模型下，UGn的g-extra可诊断性一致为（g+1）n−(g+1)(g+2)2+g。此外，我们证明了UGn的（r+1）分量连通性为rn−r(r+1)2，对于2≤r≤|M|+2，在PMC和MM*模型下，UGn的（r+1）分量可诊断性一致为（r+1）n−(r+1)(r+2)2。

引用次数: 0

Globally minimal defensive alliances: A parameterized perspective 全球最小防御联盟：参数化视角

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-22 DOI: 10.1016/j.dam.2026.01.022

Ajinkya Gaikwad, Soumen Maity

A defensive alliance in an undirected graph

G = (V, E)

is a non-empty set

S \subseteq V

such that every vertex

v \in S

has at least as many neighbours (including itself) in

S

as it has in

V ∖ S

. In this paper, we consider the notion of global minimality. A defensive alliance

S

is called a globally minimal defensive alliance if no proper subset of

S

is a defensive alliance. Given an undirected graph

G

and a positive integer

k

, we study Globally Minimal Defensive Alliance, where the goal is to check whether

G

has a globally minimal defensive alliance of size at least

k

. This problem is NP-hard but its parameterized complexity has remained open until now. The goal of this paper is to provide new insight into the complexity of Globally Minimal Defensive Alliance, parameterized by the structure of the input graph. We show that the problem is fixed-parameter tractable (FPT) when parameterized by the neighbourhood diversity of the input graph. The result for neighbourhood diversity implies that the problem is FPT parameterized by vertex cover number also. We prove that the problem parameterized by the vertex cover number of the input graph does not admit a polynomial compression unless coNP

\subseteq

NP/poly. Furthermore, we show that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, pathwidth, treewidth and treedepth. Finally, we prove that, given a vertex

r \in V (G)

, deciding whether

G

has a globally minimal defensive alliance of any size that contains

r

is NP-complete.

无向图G=（V，E）中的防御联盟是一个非空集S⊥V，使得每个顶点V∈S在S中的邻居（包括其自身）至少与在V∈S中的邻居一样多。在本文中，我们考虑了全局极小性的概念。如果防御联盟S没有适当的子集是防御联盟，则称为全局最小防御联盟。给定一个无向图G和一个正整数k，我们研究全局最小防御联盟，其目标是检验G是否有一个规模至少为k的全局最小防御联盟。这个问题是np困难的，但其参数化复杂度至今仍然是开放的。本文的目标是通过输入图的结构参数化，对全局最小防御联盟的复杂性提供新的见解。我们证明了当输入图的邻域多样性参数化时，问题是固定参数可处理的（FPT）。邻域多样性的结果表明问题也是由顶点覆盖数参数化的FPT。证明了输入图的顶点覆盖数参数化的问题不允许多项式压缩，除非coNP≠NP/poly。此外，我们证明了问题是W[1]-硬参数化的相当有限的结构参数，如反馈顶点集数，路径宽度，树宽和树深。最后，我们证明了给定顶点r∈V(G)，决定G是否存在一个包含r的任意大小的全局最小防御联盟是np完全的。

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引用次数: 0

Semidefinite programming bounds and a Branch-and-bound algorithm for the Chordless Cycle Problem 无弦循环问题的半定规划界及分支定界算法

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-22 DOI: 10.1016/j.dam.2026.01.015

Dilson Lucas Pereira , Dilson Almeida Guimarães , Alexandre Salles da Cunha , Abilio Lucena

In this paper, we investigate the Chordless Cycle Problem (CCP), that asks for a maximum cardinality set of vertices whose induced subgraph is a cycle for a given connected undirected graph. In order to solve the CCP exactly, we propose an enhanced Lagrangian Relaxation Algorithm (LRA) and a Branch-and-bound algorithm, BBLAGSDP, that relies on the LRA. Enhancements come mostly from the fact that the matrix

Λ

of Lagrangian multipliers attached to the semidefinite programming (SDP) constraint involved in our relaxation for the CCP is not only positive semidefinite, but also symmetric. For this reason, we are allowed to replace

Λ

by its factorization

Λ = Γ Γ^{T}

, reformulate its accompanying Lagrangian Dual Problem (LDP) in terms of

Γ

and, finally, avoid the need for computing an eigendecomposition of

Λ

, that would otherwise be the most expensive step required for solving the LDP. On the one hand, our LRA approximates very well the exact SDP bounds, computed by a convex optimization solver from the literature. And it does so in significantly smaller computational times. On the other hand, our SDP bounds are also much stronger than the previously available Linear Programming (LP) bounds. Computational experiments conduced with 458 instances indicate that BBLAGSDP is by far the best performing algorithm, among the five exact methods we compare here, as the density of the input graph increases.

本文研究了给定连通无向图的无弦环问题，该问题要求其诱导子图为一个环的顶点的最大基数集。为了准确地解决CCP问题，我们提出了一种增强的拉格朗日松弛算法（LRA）和基于LRA的分支定界算法BBLAGSDP。增强主要来自于这样一个事实，即拉格朗日乘子的矩阵Λ附加到我们对CCP的松弛所涉及的半定规划（SDP）约束上，不仅是正半定的，而且是对称的。由于这个原因，我们可以用它的因式分解Λ=ΓΓT来替换Λ，用Γ重新表达它的拉格朗日对偶问题（LDP），最后，避免计算Λ的特征分解，否则这将是求解LDP所需的最昂贵的步骤。一方面，我们的LRA非常接近精确的SDP边界，由文献中的凸优化求解器计算。而且它的计算时间大大缩短了。另一方面，我们的SDP界也比以前可用的线性规划（LP）界强得多。458个实例的计算实验表明，随着输入图密度的增加，BBLAGSDP是目前为止我们比较的五种精确方法中性能最好的算法。

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引用次数: 0

Neighbor connectivity of bubble-sort star graphs 泡状排序星图的邻居连通性

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-22 DOI: 10.1016/j.dam.2026.01.027

Liying Zhao , Shumin Zhang , Bo Zhu , Jou-Ming Chang

In a spy network, when a particular spy is threatened, all associated partners in the surrounding vicinity may also be at risk. To better evaluate the reliability and security of the spy network, particularly in the context of the underground resistance movement, some early scholars introduced the concept of neighbor connectivity. This notion has recently been extended to assess the stability of interconnection networks, particularly in light of multi-processor or distributed computing systems. The topology of a network is typically modeled as a simple undirected graph. The neighbor connectivity of a graph

G

, denoted as

κ_{NB} (G)

(resp. the edge neighbor connectivity, denoted as

λ_{NB} (G)

), is defined as the minimum number of vertices (resp. edges) whose removal, along with their closed neighborhood, results in

G

becoming disconnected, empty, or complete (resp. trivial). Bubble-sort star graphs possess several favorable properties, including bipartiteness, fault tolerance, and vertex transitivity, making them attractive as network structures. In this paper, we establish that

κ_{NB} (B S_{n}) = ⌊ \frac{3 n - 2}{4} ⌋

for

n \geq 4

and

λ_{NB} (B S_{n}) = 2 n - 3

for

n \geq 2

在一个间谍网络中，当一个特定的间谍受到威胁时，周围的所有相关伙伴也可能处于危险之中。为了更好地评估间谍网络的可靠性和安全性，特别是在地下抵抗运动的背景下，一些早期学者引入了邻居连通性的概念。这个概念最近被扩展到评估互连网络的稳定性，特别是在多处理器或分布式计算系统的情况下。网络的拓扑结构通常被建模为一个简单的无向图。图G的邻居连通性，记为κNB(G)。边缘邻居连通性，记为λNB(G))，定义为最小顶点数(p。边)，它的移除，以及它们的封闭邻域，导致G变得不连接，空的，或完全（见第2章）。琐碎的)。气泡排序星图具有双分性、容错性和顶点传递性等优点，使其成为具有吸引力的网络结构。本文建立了当n≥4时κNB(BSn)=⌊3n−24⌋，当n≥2时λNB(BSn)=2n−3。

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引用次数: 0

Maximum subgraph and wirelength analysis of extended Sierpiński networks in parallel computing 并行计算中扩展Sierpiński网络的最大子图和带宽分析

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-21 DOI: 10.1016/j.dam.2026.01.013

P. Leo Joshwa , R. Sundara Rajan , T.M. Rajalaxmi

The Sierpiński networks,

S (p, q)

, serves as an important model for multiprocessor interconnection systems due to its hierarchical and self-similar properties. Its extended variant,

S^{+ +} (p, q)

, further enhances these characteristics, making it highly relevant for applications in parallel computing and VLSI design. In this work, we address one of the open problems called the Maximum Subgraph Problem (MSP) for extended Sierpiński networks

S^{+ +} (p, q)

p \geq 2

, and

q \geq 3

posed by Joshwa et al. (2025). In addition, we use lexicographic ordering and propose a computational approach implemented in Sage to determine the maximum number of edges formed by

r

vertices in

S^{+ +} (p, q),

where

1 \leq r \leq q^{p} + q^{p - 1}

. Moreover, we investigate the minimum wirelength embedding of

S^{+ +} (p, q)

into structures such as paths, caterpillars, and 1-hierarchical caterpillars, contributing to the study of efficient graph embeddings in hierarchical systems.

Sierpiński网络S（p,q）由于其分层和自相似的特性而成为多处理器互连系统的重要模型。其扩展版本s++ (p,q)进一步增强了这些特性，使其与并行计算和VLSI设计中的应用高度相关。在这项工作中，我们解决了由Joshwa等人（2025）提出的扩展Sierpiński网络S++(p,q)， p≥2和q≥3的最大子图问题（MSP）的一个开放问题。此外，我们使用字典排序，并提出了一种在Sage中实现的计算方法来确定S++(p,q)中r个顶点形成的最大边数，其中1≤r≤qp+qp−1。此外，我们还研究了s++ (p,q)在路径、毛虫和1-分层毛虫等结构中的最小无线嵌入，为分层系统中高效图嵌入的研究做出了贡献。

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引用次数: 0

Refinements of Combinatorial Nullstellensatz via polynomial supports 基于多项式支持的组合零态的改进

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-21 DOI: 10.1016/j.dam.2026.01.025

Wei Cao

Nica introduced the notion of nullity for finite sets and utilized it to refine the Combinatorial Nullstellensatz over structured grids. Inspired by Nica’s approach, we replace the concept of nullity with the support of polynomials, thereby obtaining further refinements of the Combinatorial Nullstellensatz.

Nica引入了有限集的零性概念，并利用它来改进结构化网格上的组合零性。受Nica方法的启发，我们用多项式的支持取代了零的概念，从而得到了组合零的进一步改进。

引用次数: 0

A generalization of the Chvátal–Erdős theorem Chvátal-Erdős定理的推广

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-21 DOI: 10.1016/j.dam.2026.01.029

Kun Cheng

A well-known result of Chvátal and Erdős from 1972 states that a graph with connectivity not less than its independence number plus one is hamiltonian-connected. A graph

G

is called an

[s, t]

-graph if any induced subgraph of

G

of order

s

has size at least

t .

We prove that every

k

-connected

[k + 1, 2]

-graph is hamiltonian-connected except

k K_{1} \lor G_{k},

where

k \geq 2

and

G_{k}

is an arbitrary graph of order

k

. This generalizes the Chvátal–Erdős theorem.

1972年Chvátal和Erdős的一个著名结果表明，连通性不小于其独立数加1的图是哈密顿连通的。如果G的s阶诱导子图的大小至少为t，则图G称为[s，t]-图。我们证明除kK1≥2且Gk是k阶任意图外，所有k连通的[k+1,2]-图都是哈密顿连通的，这推广了Chvátal-Erdős定理。

引用次数: 0

Disjunctive domination in maximal outerplanar graphs 极大外平面图的析取支配

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-19 DOI: 10.1016/j.dam.2025.12.056

Michael A. Henning , Paras Vinubhai Maniya , Dinabandhu Pradhan

A disjunctive dominating set of a graph

G

is a set

D \subseteq V (G)

such that every vertex in

V (G) ∖ D

has a neighbor in

D

or has at least two vertices in

D

at distance 2 from it. The disjunctive domination number of

G

, denoted by

γ_{2}^{d} (G)

, is the minimum cardinality among all disjunctive dominating sets of

G

. In this paper, we show that if

G

is a maximal outerplanar graph of order

n \geq 7

with

k

vertices of degree 2, then

γ_{2}^{d} (G) \leq ⌊ \frac{2}{9} (n + k) ⌋

, and this bound is sharp.

图G的一个析取支配集是一个集D⊥V(G)，使得V(G) × D中的每个顶点在D中有一个邻居，或者在距离为2的距离上至少有两个顶点在D中。G的析取支配数，用γ2d(G)表示，是G的所有析取支配集中最小的cardinality。本文证明了如果G是一个n≥7阶的极大外平面图，有k个顶点为2度，则γ2d(G)≤⌊29（n+k）⌋，且该界是尖锐的。

引用次数: 0

Assessing reliability of 3-ary n-cubes based on the h-extra r-component edge-connectivity 基于h-额外r分量边连通性的3元n-立方体可靠性评估

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-19 DOI: 10.1016/j.dam.2026.01.004

Chuanye Zheng, Liqiong Xu

<div><div>As multiprocessor systems scale up in size and complexity to meet increasing computational demands, link or processor failures are inevitable. Thus reliability of multiprocessor systems needs to be considered. Restricting the surviving components within multiprocessor systems can enhance the evaluation of their reliability. Recently, Yang et al. introduced a new parameter called <math><mi>h</mi></math>-extra <math><mi>r</mi></math>-component edge-connectivity, which requires that for a connected graph <math><mi>G</mi></math> and an edge-cut <math><mi>F</mi></math> of <math><mi>G</mi></math>, there exist at least <math><mi>r</mi></math> components surviving in <math><mrow><mi>G</mi><mo>−</mo><mi>F</mi></mrow></math> and the order of each component is not less than <math><mi>h</mi></math>. In this paper, we consider the <math><mi>h</mi></math>-extra <math><mi>r</mi></math>-component edge-connectivity of the 3-ary <math><mi>n</mi></math>-cube <math><msubsup><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msubsup></math> and determine that <math><mrow><mi>c</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn>4</mn></mrow><mrow><mi>h</mi></mrow></msubsup><mrow><mo>(</mo><msubsup><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>)</mo></mrow><mo>=</mo><mn>6</mn><mi>n</mi><mi>h</mi><mo>−</mo><mn>3</mn><mi>e</mi><msub><mrow><mi>x</mi></mrow><mrow><mi>h</mi></mrow></msub><mrow><mo>(</mo><msubsup><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>)</mo></mrow><mo>−</mo><mn>3</mn><mi>h</mi></mrow></math> for <math><mrow><mi>n</mi><mo>≥</mo><mn>5</mn></mrow></math> and <math><mrow><mn>1</mn><mo>≤</mo><mi>h</mi><mo>≤</mo><mi>δ</mi><mi>⋅</mi><msup><mrow><mn>3</mn></mrow><mrow><mrow><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow></math> where <math><mrow><mi>δ</mi><mo>=</mo><mn>1</mn></mrow></math> if <math><mi>n</mi></math> is odd and <math><mrow><mi>δ</mi><mo>=</mo><mn>2</mn></mrow></math> if <math><mi>n</mi></math> is even, <math><mrow><mi>c</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>r</mi></mrow><mrow><msup><mrow><mn>3</mn></mrow><mrow><mi>k</mi></mrow></msup></mrow></msubsup><mrow><mo>(</mo><msubsup><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn><mi>k</mi><mo>)</mo></mrow><msup><mrow><mn>3</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>e</mi><msub><mrow><mi>x</mi></mrow><m

随着多处理器系统的规模和复杂性不断扩大，以满足不断增长的计算需求，链路或处理器故障是不可避免的。因此，需要考虑多处理器系统的可靠性。限制多处理机系统中幸存部件的数量可以提高对系统可靠性的评估。最近,杨等人提出了一个新的参数称为h-extra r-component edge,这要求一个连通图G和edge-cut F (G,存在至少r G−F幸存的组件,每个组件的顺序不小于h。在本文中,我们考虑h-extra r-component edge的3-ary n立方体Qn3和确定cλ4 h (Qn3) = 6 nh−3 exh (Qn3)−3 h n≥5和1 h≤≤δ⋅3⌈n2⌉−−1δ= 1如果n是奇怪和δ= 2如果n是偶数,cλr3k (Qn3) = (r−1)(2 n−2 k) 3 k 12 exr−−1 (Qn3)⋅3 k 1≤(r−1)3 k≤δ⋅3⌈n2⌉−1和cλ43 k (Qn3) = (2 n−2 k−1)3 k + 1 0≤k≤n−2。

引用次数: 0

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