Discrete Applied Mathematics最新文献

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Pushing Cops and Robber on graphs of maximum degree four 把警察和强盗推到最高四度的图上

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-13 DOI: 10.1016/j.dam.2025.12.067

Harmender Gahlawat

Cops and Robber is a game played on graphs where a set of cops aim to capture the position of a single robber. The main parameter of interest in this game is the cop number, which is the minimum number of cops that are sufficient to guarantee the capture of the robber.

In a directed graph

\vec{G}

, the push operation on a vertex

v

reverses the orientation of all arcs incident to

v

. We consider a variation of the classical Cops and Robber on oriented graphs, where in its turn, each cop can either move to an out-neighbor of its current vertex or push some vertex of the graph, whereas, the robber can move to an out-neighbor in its turn. [Das et al., CALDAM, 2023] introduced this variant and established that if

\vec{G}

is an orientation of a subcubic graph, then one cop with push ability has a winning strategy. We extend these results to establish that if

\vec{G}

is an orientation of a 3-degenerate graph, or of a graph with maximum degree 4, then one cop with push ability has a winning strategy. Moreover, we establish that if

\vec{G}

can be made to be a directed acyclic graph, then one cop with push ability has a winning strategy.

《Cops and robbers》是一款基于图表的游戏，其中一组警察的目标是抓住一名抢劫犯的位置。在这个游戏中最重要的参数是警察数量，即能够保证抓住抢劫犯的最少警察数量。在有向图G - l - l中，顶点v上的推操作反转了所有与v相关的弧线的方向。我们考虑有向图上经典cop和抢劫者的一种变体，其中每个cop可以移动到其当前顶点的外邻居或推图的某个顶点，而抢劫者可以移动到其外邻居。[Das et al.， CALDAM， 2023]引入了这种变体，并建立了如果G / l是亚立方图的一个方向，那么具有推送能力的一方具有获胜策略。我们扩展了这些结果来证明如果G - l是一个3-简并图的一个方向，或者是一个最大度为4的图的一个方向，那么一个具有推能力的cop有一个获胜策略。此外，我们还证明了如果G / l可以构成一个有向无环图，那么一个具有推能力的cop有一个获胜策略。

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引用次数: 0

Non-inclusive g-extra diagnosability of interconnection networks under MM* model MM*模型下互联网络的非包容性g-extra可诊断性

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-13 DOI: 10.1016/j.dam.2025.12.069

Weixing Zheng , Shuming Zhou , Eddie Cheng

With the popularization and deepening of applications on high-performance computing platforms, which are often built upon large-scale multiprocessor systems, system-level diagnosis has become essential to ensure system reliability and fault tolerability. Classical diagnosability and various conditional diagnosabilities are vital metrics to evaluate a system’s capability to accurately identify faulty processors. However, the upper bounds of these metrics commonly rely on the conventional assumptions of inclusiveness between faulty sets—one situation that rarely holds in practical scenarios. To overcome this deficiency, the non-inclusive diagnosability has been introduced and explored under various diagnostic models. Despite significant progress, the study of non-inclusive

g

-extra diagnosability under the MM* model remains up in the air. This work addresses this gap by determining the non-inclusive 2-extra diagnosability of general interconnection networks under the MM* model. As applications, we derive the non-inclusive 2-extra diagnosabilities of several well-known networks, including hypercube,

k

-ary

n

-cube, and bubble sort graph. In addition, we propose a novel diagnosis algorithm, NFDAM, for non-inclusive

g

-extra diagnosis, which runs in polynomial time with a complexity of

O (| V (G) | \cdot {(Δ (G))}^{3})

, where

Δ (G)

denotes the maximum degree of graph

G

. Simulation results demonstrate the effectiveness of the proposed algorithm in fault identification.

随着高性能计算平台应用的普及和深入，系统级诊断对于保证系统的可靠性和容错性至关重要。高性能计算平台通常建立在大型多处理器系统之上。经典可诊断性和各种条件可诊断性是评估系统准确识别故障处理器能力的重要指标。然而，这些指标的上限通常依赖于错误集之间的包容性的传统假设——这种情况在实际场景中很少成立。为了克服这一缺陷，在各种诊断模型下引入了非包容性可诊断性。尽管取得了重大进展，但在MM*模型下对非包容性g-extra可诊断性的研究仍然悬而未决。本工作通过确定MM*模型下一般互连网络的非包容性2-额外可诊断性来解决这一差距。作为应用，我们推导了几个著名网络的非包含2-额外可诊断性，包括超立方体、k-ary n-立方体和冒泡排序图。此外，我们提出了一种新的非包含G -extra诊断算法NFDAM，该算法运行时间为多项式，复杂度为O(|V(G)|⋅（Δ(G)）3)，其中Δ(G)表示图G的最大程度，仿真结果表明了该算法在故障识别中的有效性。

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引用次数: 0

Enumerating minimal defensive alliances 列举最小的防御联盟

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-12 DOI: 10.1016/j.dam.2025.12.066

Zhidan Feng , Henning Fernau , Kevin Mann

In this paper, we study the task of enumerating (and counting) locally and globally minimal defensive alliances in graphs. We consider general graphs as well as special graph classes, like trees, bipartite graphs, and split graphs. From an input-sensitive perspective, our presented algorithms are mostly optimal, meaning that their running times (neglecting polynomial factors) match concrete families of graphs that contain that many minimal alliances.

在本文中，我们研究了在图中枚举（计数）局部和全局最小防御联盟的任务。我们考虑一般图以及特殊的图类，如树、二部图和分裂图。从输入敏感的角度来看，我们提出的算法大多是最优的，这意味着它们的运行时间（忽略多项式因素）匹配包含许多最小联盟的具体图族。

引用次数: 0

Weighted cages 加权的笼子里

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-12 DOI: 10.1016/j.dam.2025.12.047

G. Araujo-Pardo , C. De la Cruz , M. Matamala , M.A. Pizaña

Cages (

r

-regular graphs of girth

g

and minimum order) and their variants have been studied for over seventy years. Here we propose a new variant, weighted cages. We characterize their existence; for cases

g = 3, 4

we determine their order; we give Moore-like bounds and present some computational results.

笼形图（周长g和最小阶的r-正则图）及其变体已经被研究了70多年。在这里，我们提出一种新的变体，加权笼。我们描述它们的存在；对于情况g=3,4，我们确定它们的顺序；给出了类摩尔边界，并给出了一些计算结果。

引用次数: 0

Robust optimization of multi-skill resource-constrained project networks considering staff fatigue under uncertainty 考虑不确定性下员工疲劳的多技能资源约束项目网络鲁棒优化

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-12 DOI: 10.1016/j.dam.2025.12.071

Zihan Ning , Ran Ma , Zhengwen He , Xiaoyan Zhang

In project implementation, uncertainty factors have significant impacts. How to effectively safeguard staff working conditions while ensuring timely project delivery under uncertain environments has emerged as a critical issue urgently requiring resolution in the project management domain. The Multi-Skill Resource-Constrained Project Scheduling Problem (MS-RCPSP), as a typical combinatorial optimization challenge, provides theoretical foundations for addressing the issues. This problem can be abstracted as a complex multi-layer network graph optimization model. This study focuses on robust project scheduling under multi-skill resource constraints, aiming to generate proactive optimization schedules that simultaneously minimize staff maximum fatigue accumulation and maximize robustness. To accurately reflect realistic execution environments, this research incorporates activity uncertainty into staff fatigue levels considerations within the model construction, thereby better characterizing the actual impact of uncertain factors on staff fatigue levels. In terms of algorithmic design, this paper introduces priority selection strategies during the schedule generation process to enhance solution quality, and develops an Improved Non-dominated Sorting Genetic Algorithm II (INSGA-II) featuring dual-mode adaptive selection mechanisms and diversity-driven elite retention adjustments. Experimental results demonstrate that through multi-metric comparisons, the proposed comprehensive algorithm significantly outperforms traditional methods. Furthermore, this study conducts in-depth experimental analysis and interpretation regarding solution robustness and the improved fatigue function, validating the effectiveness and practicality of the proposed methodology.

在项目实施过程中，不确定性因素对项目的影响是显著的。如何在不确定的环境下有效保障员工的工作条件，同时保证项目的及时交付，已成为项目管理领域亟待解决的关键问题。多技能资源约束项目调度问题（MS-RCPSP）作为典型的组合优化问题，为解决这一问题提供了理论基础。该问题可以抽象为一个复杂的多层网络图优化模型。本研究关注多技能资源约束下的鲁棒项目调度，旨在生成同时最小化人员最大疲劳积累和最大化鲁棒性的主动优化调度。为了准确反映现实的执行环境，本研究在模型构建中将活动的不确定性纳入到员工疲劳水平的考虑中，从而更好地表征不确定性因素对员工疲劳水平的实际影响。在算法设计方面，本文在调度生成过程中引入优先级选择策略以提高求解质量，并开发了一种具有双模式自适应选择机制和多样性驱动的精英保留调整的改进非支配排序遗传算法II （INSGA-II）。实验结果表明，通过多指标比较，本文提出的综合算法明显优于传统方法。此外，本文还对解的鲁棒性和改进的疲劳函数进行了深入的实验分析和解释，验证了所提出方法的有效性和实用性。

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引用次数: 0

A note on robber locating game 关于抢劫犯定位游戏的说明

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-12 DOI: 10.1016/j.dam.2025.12.061

Rahul Das, Ushnish Sarkar

The robber locating game on a simple connected graph is a variant of pursuit evasion game where a cop wants to locate a mobile, invisible and omniscient robber hiding in the vertices of the graph using distance queries. This game has two versions depending on whether the robber is allowed to backtrack or not. The minimum numbers of rounds required to locate the robber, against any strategy he adopts, in the two versions of the game are referred to as backtrack and non-backtrack location numbers of the graph respectively. However, if the robber is immobile, then the minimum numbers of rounds required to locate the invisible robber is called the sequential metric dimension of the graph. This article gives a lower bound of the backtrack and non-backtrack location numbers in terms of the metric dimension and diameter. This has been done by establishing a lower bound of the sequential metric dimension. A hideout is a subgraph

G

of a graph

H

such that the mobile, invisible and omniscient robber can win by remaining on the vertices of

G

. We prove sufficient conditions for a graph to be a hideout for both backtracking and non-backtracking scenarios.

简单连通图上的抢劫犯定位游戏是追捕逃避游戏的一种变体，其中警察希望利用距离查询来定位隐藏在图顶点中的移动的、不可见的、无所不知的抢劫犯。这个游戏有两个版本，取决于强盗是否被允许返回。在两个版本的博弈中，针对抢劫者采取的任何策略，定位抢劫者所需的最小回合数分别称为图的回溯和非回溯位置数。然而，如果抢劫者是固定不动的，那么定位看不见的抢劫者所需的最小回合数称为图的顺序度量维度。本文根据公制尺寸和直径给出了回溯和非回溯位置数的下界。这是通过建立顺序度量维度的下界来实现的。藏身处是图H的子图G，使得移动的、不可见的、无所不知的强盗可以通过留在G的顶点上而获胜。我们证明了图在回溯和非回溯两种情况下都是藏身处的充分条件。

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引用次数: 0

Aspects of a randomly growing cluster in Rd,d≥2 随机生长的聚类在Rd，d≥2的方面

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-12 DOI: 10.1016/j.dam.2025.12.062

Alan Frieze , Ravi Kannan , Wesley Pegden

We consider a simple model of a growing cluster of points in

R^{d}, d \geq 2

. Beginning with a point

X_{1}

located at the origin, we generate a random sequence of points

X_{1}, X_{2}, \dots, X_{i}, \dots,

. To generate

X_{i}, i \geq 2

we choose a uniform integer

j

[i - 1] = (1, 2, \dots, i - 1)

and then let

X_{i} = X_{j} + D_{i}

where

D_{i} = (δ_{1}, \dots, δ_{d})

. Here the

δ_{j}

are independent copies of the Normal distribution

N (0, σ_{i})

, where

σ_{i} = i^{- α}

for some

α > 0

. We prove that for any

α > 0

the resulting point set is bounded a.s., and moreover, that the points generated look like samples from a

β

-dimensional subset of

R^{d}

from the standpoint of the minimum lengths of combinatorial structures on the point-sets, where

β = min (d, 1 / α)

我们考虑一个简单的模型，在Rd，d≥2点的增长簇。从原点的点X1开始，我们生成点X1，X2，…，Xi，…，的随机序列。为了生成Xi，i≥2，我们在[i−1]=1,2，…，i−1中选择一个均匀整数j，然后令Xi=Xj+Di，其中Di=（δ1，…，δd）。这里的δj是正态分布N（0,σi）的独立副本，其中σi=i - α，对于某些α>；0。我们证明了对于任意α>；0生成的点集是有界的，并且从点集上组合结构的最小长度的角度来看，生成的点看起来像来自Rd的β维子集的样本，其中β=min（d,1/α）。

引用次数: 0

On 2-distance (2Δ+6)-coloring of planar graphs 平面图形的2-距离（2Δ+6）着色

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-09 DOI: 10.1016/j.dam.2025.12.063

Jiahao Yu, Min Chen

Given a graph

G = (V, E)

, a 2-distance

k

-coloring is a mapping

π : V (G) \to {1, 2, \dots, k}

such that any two vertices within distance at most two receive distinct colors. A graph admitting such a coloring is called 2-distance. It is known that every planar graph is 2-distance

(2 Δ + 16)

-colorable if the maximum degree

Δ \geq 8

(Song and Lai, 2018) and 2-distance

(2 Δ + 7)

-colorable if

Δ \geq 9

(Bousquet et al., 2023). In this paper, we strengthen these bounds by proving that every planar graph with

Δ \geq 8

is 2-distance

(2 Δ + 6)

-colorable, thereby advancing progress toward Wegner’s conjecture.

给定一个图G=(V，E)，一个2-distance k-coloring是一个映射π:V(G)→{1,2，…，k}，使得距离不超过两个的任意两个顶点得到不同的颜色。允许这种着色的图称为2-距离。已知，如果最大度Δ≥8 (Song and Lai, 2018)，则每个平面图为2-distance （2Δ+16）可色(2-distance （2Δ+7）可色（Δ≥9）（Bousquet et al., 2023）。在本文中，我们通过证明Δ≥8的每个平面图都是2-距离（2Δ+6）可色来加强这些界限，从而推进了Wegner猜想的进展。

引用次数: 0

Construction, extension and paths of near-homogeneous tournaments 构建、扩展和路径近乎同质化的比赛

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-09 DOI: 10.1016/j.dam.2025.12.059

Rongxia Tang , Zhaojun Chen , Zan-Bo Zhang

A homogeneous tournament is a tournament with

4 t + 3

vertices such that every arc is contained in exactly

t + 1

cycles of length 3. Homogeneous tournaments are the first class of tournaments that are proved to be path extendable, which means that every nonhamiltonian path

P

in such a tournament

T

can be extended to a path

P^{'}

with the same initial and terminal vertex and

V (P^{'}) = V (P) \cup {u}

for a certain vertex

u \in V (T) ∖ V (P)

. In order to find more path extendable tournaments we study the generalization of homogeneous tournaments called near-homogeneous tournaments, in which every arc is contained in

t

t + 1

cycles of length 3. Near-homogeneity has been defined in tournaments with

4 t + 1

vertices. In this paper, we raise a new method to construct near-homogeneous tournaments with

4 t + 1

vertices. We then show that the definition of near-homogeneous tournament can be extended to tournaments with an even number of vertices. Finally we verify path extendability of near-homogeneous tournaments, thus expand the class of path extendable tournaments.

齐次比赛是指有4t+3个顶点的比赛，这样每个弧都包含在长度为3的t+1个循环中。齐次竞赛是第一类被证明是路径可扩展的竞赛，这意味着在这样一个竞赛T中的每一个非哈密顿路径P都可以被扩展到具有相同初始顶点和终点顶点的路径P '，并且对于某个顶点u∈V(T)∈V(P)∈V(P)∈V(P)≠V(P)∪{u}。为了找到更多路径可扩展的比赛，我们研究了齐次比赛的推广，称为近齐次比赛，其中每个弧都包含在t或t+1个长度为3的循环中。近同质性在4t+1个顶点的比赛中被定义。本文提出了一种构造具有4t+1个顶点的近齐次竞赛的新方法。然后，我们证明了近齐次比赛的定义可以扩展到具有偶数个顶点的比赛。最后，我们验证了近似同构竞赛的路径可扩展性，从而扩展了路径可扩展性竞赛的类别。

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引用次数: 0

Extremal distance spectra of graphs and essential connectivity 图的极值距离谱和基本连通性

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-09 DOI: 10.1016/j.dam.2025.12.070

Daoxia Zhang, Dan Li, Wenxiu Ding

A graph is trivial if it contains only one vertex. The essential connectivity of

G

, denoted by

κ^{'} (G)

, is the minimum number of vertices of

G

whose removal produces a disconnected graph with at least two non-trivial components. In this paper, we determine the

n

-vertex graph of given essential connectivity with minimum distance spectral radius. We also characterize the extremal graphs attaining the minimum distance spectral radius among all connected graphs with fixed essential connectivity and minimum degree. Furthermore, we characterize the extremal digraphs with minimum distance spectral radius among the strongly connected digraphs with given essential connectivity.

如果一个图只包含一个顶点，那么它是平凡的。G的基本连通性，用κ ' (G)表示，是G的最小顶点数，其移除会产生至少有两个非平凡分量的断开图。在本文中，我们确定了具有最小距离谱半径的给定基本连通的n顶点图。我们还描述了在所有具有固定基本连通性和最小度的连通图中获得最小距离谱半径的极值图。进一步，我们刻画了具有给定基本连通性的强连通有向图中具有最小距离谱半径的极值有向图。

引用次数: 0

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