{"title":"Obstructions for local tournament orientation completions with cut-vertices","authors":"Kevin Hsu, Jing Huang","doi":"10.1016/j.dam.2025.07.009","DOIUrl":"10.1016/j.dam.2025.07.009","url":null,"abstract":"<div><div>Local tournaments are oriented graphs that generalize tournaments and their underlying graphs are intimately related to proper circular-arc graphs. According to Skrien, a connected graph can be oriented as a local tournament if and only if it is a proper circular-arc graph. Proper interval graphs are precisely the graphs which can be oriented as acyclic local tournaments.</div><div>The orientation completion problem for local tournaments (respectively, acyclic local tournaments) asks whether a partially oriented graph can be completed to a local tournament (respectively, an acyclic local tournament). It has been proved that the orientation completion problems for local tournaments and acyclic local tournaments are both polynomial time solvable.</div><div>Obstructions for local tournament orientation completions are the minimal partially oriented graphs which cannot be completed to local tournaments. Determining the obstructions for local tournament orientation completions is of algorithmic importance. Recently, the obstructions for acyclic local tournament orientation completions have been determined. It is however much more challenging to determine the obstructions for general local tournament orientation completions. In this paper, we focus on the case when cut-vertices are present and we determine the obstructions for local tournament orientation completions which have cut-vertices.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"378 ","pages":"Pages 174-186"},"PeriodicalIF":1.0,"publicationDate":"2025-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144680460","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jacques Carlier , Antoine Jouglet , Kristina Kumbria , Abderrahim Sahli
{"title":"More powerful energetic reasoning for the cumulative scheduling problem","authors":"Jacques Carlier , Antoine Jouglet , Kristina Kumbria , Abderrahim Sahli","doi":"10.1016/j.dam.2025.07.016","DOIUrl":"10.1016/j.dam.2025.07.016","url":null,"abstract":"<div><div>Energetic reasoning is an efficient filtering technique for the Cumulative Scheduling Problem. In this paper we propose a new definition of the energy balance of intervals, together with a new checker that is more accurate for each interval. Our approach involves solving a tripartition problem. For checking the intervals, we also propose a cubic algorithm leveraging our approach. We report computational results that confirm that it is more efficient than the classical approaches.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"378 ","pages":"Pages 187-209"},"PeriodicalIF":1.0,"publicationDate":"2025-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144680728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Xiaoxiao Qin , Fangyu Zhao , Hong-Jian Lai , Bofeng Huo
{"title":"Supereulerian of regular matroids with cogirth conditions","authors":"Xiaoxiao Qin , Fangyu Zhao , Hong-Jian Lai , Bofeng Huo","doi":"10.1016/j.dam.2025.07.007","DOIUrl":"10.1016/j.dam.2025.07.007","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a 2-connected simple graph on sufficiently large <span><math><mi>n</mi></math></span> vertices. Catlin proved that if <span><math><mrow><mi>δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>5</mn></mrow></mfrac></mrow></math></span>, then <span><math><mi>G</mi></math></span> is supereulerian. Let <span><math><mi>M</mi></math></span> be a connected simple regular matroid. Huo et al. proved that <span><math><mi>M</mi></math></span> is supereulerian if <span><math><mrow><msup><mrow><mi>g</mi></mrow><mrow><mo>∗</mo></mrow></msup><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow><mo>≥</mo><mo>max</mo><mrow><mo>{</mo><mfrac><mrow><mi>r</mi><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow><mrow><mn>10</mn></mrow></mfrac><mo>,</mo><mn>9</mn><mo>}</mo></mrow></mrow></math></span>. In this paper, we prove that a simple connected regular matroid <span><math><mi>M</mi></math></span> is supereulerian if <span><math><mrow><msup><mrow><mi>g</mi></mrow><mrow><mo>∗</mo></mrow></msup><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow><mo>≥</mo><mo>max</mo><mrow><mo>{</mo><mfrac><mrow><mi>r</mi><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow><mrow><mn>10</mn></mrow></mfrac><mo>,</mo><mn>8</mn><mo>}</mo></mrow></mrow></math></span>, or <span><math><mrow><msup><mrow><mi>g</mi></mrow><mrow><mo>∗</mo></mrow></msup><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow><mo>≥</mo><mo>max</mo><mrow><mo>{</mo><mfrac><mrow><mi>r</mi><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow><mrow><mn>15</mn></mrow></mfrac><mo>,</mo><mn>9</mn><mo>}</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 327-334"},"PeriodicalIF":1.0,"publicationDate":"2025-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144679101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Dense subsets of asymptotic bases","authors":"Jin-Hui Fang","doi":"10.1016/j.dam.2025.07.014","DOIUrl":"10.1016/j.dam.2025.07.014","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>h</mi><mo>⩾</mo><mn>2</mn></mrow></math></span> be an integer. A set <span><math><mi>A</mi></math></span> of nonnegative integers is defined as an asymptotic basis of order <span><math><mi>h</mi></math></span> if all sufficiently large integers can be expressed as a sum of <span><math><mi>h</mi></math></span> elements from <span><math><mi>A</mi></math></span>. Write <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>L</mi></mrow></msub><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow></mrow></math></span> as the lower asymptotic density of <span><math><mi>A</mi></math></span> and <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>U</mi></mrow></msub><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow></mrow></math></span> as the upper asymptotic density of <span><math><mi>A</mi></math></span>. In 1989, Nathanson and Sárközy proved that, if <span><math><mi>A</mi></math></span> is an asymptotic basis of order <span><math><mi>h</mi></math></span> and <span><math><mi>B</mi></math></span> is a subset of <span><math><mi>A</mi></math></span> with <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>L</mi></mrow></msub><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow><mo>></mo><mn>1</mn><mo>/</mo><mi>h</mi></mrow></math></span>, then there exists a finite subset <span><math><mi>F</mi></math></span> of <span><math><mi>A</mi></math></span> such that <span><math><mrow><mi>F</mi><mo>∪</mo><mi>B</mi></mrow></math></span> is an asymptotic basis of order <span><math><mi>h</mi></math></span>. Recently, Xu and Chen [C. R. Math. Acad. Sci. Paris 362 (2024), 45-49.] further proved that, if <span><math><mi>A</mi></math></span> is a set of nonnegative integers with <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>L</mi></mrow></msub><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>></mo><mn>0</mn></mrow></math></span>, then there exists a subset <span><math><mi>B</mi></math></span> of <span><math><mi>A</mi></math></span> with <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>L</mi></mrow></msub><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow><mo>></mo><mn>0</mn></mrow></math></span> such that <span><math><mrow><mi>F</mi><mo>∪</mo><mi>B</mi></mrow></math></span> is not an asymptotic basis of order <span><math><mi>h</mi></math></span> for any finite set <span><math><mi>F</mi></math></span>. In this paper, we improve the above result, that is: if <span><math><mi>A</mi></math></span> is a set of nonnegative integers with <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>L</mi></mrow></msub><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>></mo><mn>0</mn></mrow></math></span>, then there exists a subset <span><math><mi>B</mi></math></span> of <span><math><mi>A</mi></math></span> such that <span><span><span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>L</mi></mrow></msub><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow><mo>⩾</mo><mfrac><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>L</mi></mrow></msub><mrow><mo>(</mo><m","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"378 ","pages":"Pages 171-173"},"PeriodicalIF":1.0,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144680459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
François Rioult , Amira Mouakher , Abdelkader Ouali
{"title":"Size-optimal Boolean matrix factorization","authors":"François Rioult , Amira Mouakher , Abdelkader Ouali","doi":"10.1016/j.dam.2025.06.027","DOIUrl":"10.1016/j.dam.2025.06.027","url":null,"abstract":"<div><div>The pioneering work of Belohlavek et al. established a compelling connection between Boolean matrix factorization (BMF) and formal concept analysis (FCA), demonstrating that formal concepts serve as optimal factors for decomposing binary matrices. However, identifying the size-optimal decomposition remains an NP-hard problem, posing significant computational challenges. In this paper, we present a novel reformulation of the Boolean rank computation problem using hypergraph theory. Specifically, we show that the Boolean rank of a matrix corresponds to the size of the minimum transversal of the hypergraph constructed from the intervals of its formal concepts. This reformulation provides a theoretical foundation for understanding the structure of optimal factorizations and offers a new perspective on the problem. To validate our approach, we conducted an extensive experimental study to evaluate the characteristics of the solutions computed by our algorithm. The results demonstrated that our method not only achieved optimal factorizations but also exhibited favorable properties in terms of stability and separation.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 314-326"},"PeriodicalIF":1.0,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144679100","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Fault-tolerant dispersion of mobile robots","authors":"Prabhat Kumar Chand , Manish Kumar , Sumathi Sivasubramaniam , Anisur Rahaman Molla","doi":"10.1016/j.dam.2025.06.068","DOIUrl":"10.1016/j.dam.2025.06.068","url":null,"abstract":"<div><div>The <em>dispersion</em> problem, where mobile robots spread evenly across a graph, has recently gained attention for its potential to solve a variety of problems in distributed robotics such as relocating autonomous vehicles to garages or charging stations, covering a region, guarding, load balancing, etc. Dispersion has also been recognized as a foundational primitive for solving a variety of distributed graph problems. Recent studies demonstrate its utility in facilitating computations such as dominating set construction <span><span>[8]</span></span>, maximal independent set (MIS) formation <span><span>[29]</span></span>, leader election & computing a minimum spanning tree (MST) <span><span>[16]</span></span>, triangle counting <span><span>[6]</span></span>, etc. These works assume a robust system in which all robots are fault-free. However, in real-world systems, faults are inevitable, and it is essential to design mechanisms that ensure unexpected crashes do not obstruct the dispersion process, which forms a critical foundation for addressing these problems.</div><div>In this paper, we consider the mobile robot dispersion problem in the presence of crash faults. Mobile robot dispersion consists of <span><math><mrow><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></math></span> robots operating in an <span><math><mi>n</mi></math></span>-node anonymous graph. The goal is to ensure that, regardless of the initial placement of the robots across the nodes, the final configuration places at most one robot per node. In a crash-fault setting, up to <span><math><mrow><mi>f</mi><mo>≤</mo><mi>k</mi></mrow></math></span> robots may fail arbitrarily by crashing, thereby losing all stored information and becoming incapable of communication. In this paper, we solve the dispersion problem under crash-fault conditions by analysing two different initial configurations: (i) the rooted configuration, and (ii) the arbitrary configuration. In the rooted case, all robots start at a single node. In contrast, the arbitrary configuration allows the robots to be initially distributed across the graph in <span><math><mrow><mi>l</mi><mo><</mo><mi>k</mi></mrow></math></span> arbitrary clusters. For the rooted case, we design an algorithm that achieves dispersion in the presence of faulty robots in <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> rounds, improving upon the prior results by <span><span>[30]</span></span>, <span><span>[31]</span></span>. For the arbitrary configuration, we present an algorithm that solves dispersion in <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mrow><mo>(</mo><mi>f</mi><mo>+</mo><mi>l</mi><mo>)</mo></mrow><mi>⋅</mi><mo>min</mo><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mi>k</mi><mi>Δ</mi><mo>,</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> rounds, assuming the number of edges <","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 299-313"},"PeriodicalIF":1.0,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144670273","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"{1,2}-good-neighbor conditional diagnosability of Cayley graphs generated by k-trees","authors":"Shu-Li Zhao , Bao-Cheng Zhang , Jou-Ming Chang","doi":"10.1016/j.dam.2025.07.010","DOIUrl":"10.1016/j.dam.2025.07.010","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> be a connected graph representing a computing system. A subset <span><math><mrow><mi>F</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is called a <span><math><mi>g</mi></math></span>-good-neighbor faulty set if any vertex in <span><math><mrow><mi>G</mi><mo>−</mo><mi>F</mi></mrow></math></span> (the graph obtained by removing <span><math><mi>F</mi></math></span> from <span><math><mi>G</mi></math></span>) has at least <span><math><mi>g</mi></math></span> neighbors in <span><math><mrow><mi>G</mi><mo>−</mo><mi>F</mi></mrow></math></span>. The <span><math><mi>g</mi></math></span>-good-neighbor diagnosability, denoted as <span><math><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>g</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is defined as the maximum size of a <span><math><mi>g</mi></math></span>-good-neighbor faulty set that the system can reliably identify. Previous research has explored the problem of determining <span><math><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>g</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> for specific types of graphs, particularly Cayley graphs generated by trees or complete graphs. This paper extends these findings by addressing the same problem for Cayley graphs generated by <span><math><mi>k</mi></math></span>-trees, denoted as <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>. Specifically, we establish that <span><math><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mn>2</mn><mrow><mo>(</mo><mi>k</mi><mi>n</mi><mo>−</mo><mfrac><mrow><mi>k</mi><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mn>4</mn><mrow><mo>(</mo><mi>k</mi><mi>n</mi><mo>−</mo><mfrac><mrow><mi>k</mi><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mo>−</mo><mn>7</mn></mrow></math></span> under both the PMC model and the MM* model for <span><math><mrow><mi>n</mi><mo>≥</mo><mn>5</mn></mrow></math></span> and <span><math><mrow><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 289-298"},"PeriodicalIF":1.0,"publicationDate":"2025-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144653257","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Component edge connectivity of folded hypercube-like networks","authors":"Litao Guo , Xiangyan Kong","doi":"10.1016/j.dam.2025.07.011","DOIUrl":"10.1016/j.dam.2025.07.011","url":null,"abstract":"<div><div>Connectivity is a critical parameter which can measure the reliability of networks. Suppose <span><math><mrow><mi>ω</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is the number of components of <span><math><mi>G</mi></math></span> and <span><math><mrow><mi>W</mi><mo>⊆</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, if <span><math><mrow><mi>ω</mi><mrow><mo>(</mo><mi>G</mi><mo>−</mo><mi>W</mi><mo>)</mo></mrow><mo>≥</mo><mi>t</mi></mrow></math></span>, then <span><math><mi>W</mi></math></span> is a <span><math><mi>t</mi></math></span>-component edge cut of <span><math><mi>G</mi></math></span>. The number of edges in the least <span><math><mi>t</mi></math></span>-component edge cut is the <span><math><mi>t</mi></math></span>-component edge connectivity <span><math><mrow><mi>c</mi><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mi>G</mi></math></span>. In this note, we study the folded hypercube-like networks and obtain its the <span><math><mi>t</mi></math></span>-component edge connectivity of folded hypercube-like networks for some <span><math><mi>t</mi></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"378 ","pages":"Pages 162-170"},"PeriodicalIF":1.0,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144654704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Optimality for the restricted edge connectivity in exchanged ternary n-cubes","authors":"Yuxing Yang, Xiaohui Hua","doi":"10.1016/j.dam.2025.07.006","DOIUrl":"10.1016/j.dam.2025.07.006","url":null,"abstract":"<div><div>The <em>restricted edge connectivity</em> <span><math><mrow><msup><mrow><mi>λ</mi></mrow><mrow><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of a connected graph <span><math><mi>G</mi></math></span> is the minimum number of edges whose removal results in a disconnected graph without a singleton. <span><math><mi>G</mi></math></span> is <span><math><msup><mrow><mi>λ</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span><em>-optimal</em> if <span><math><mrow><msup><mrow><mi>λ</mi></mrow><mrow><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> equals to the minimum edge degree of <span><math><mi>G</mi></math></span>. Let <span><math><mi>r</mi></math></span>, <span><math><mi>s</mi></math></span> and <span><math><mi>t</mi></math></span> be three non-negative integers, and let <span><math><mrow><mi>n</mi><mo>=</mo><mi>r</mi><mo>+</mo><mi>s</mi><mo>+</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow></math></span>. The exchanged ternary <span><math><mi>n</mi></math></span>-cube, denoted by <span><math><mrow><mi>E</mi><mn>3</mn><mi>C</mi><mrow><mo>(</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span>, is a variant of the ternary <span><math><mi>n</mi></math></span>-cube. In this paper, we loosen the restriction <span><math><mrow><mo>min</mo><mrow><mo>{</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>}</mo></mrow><mo>≥</mo><mn>1</mn></mrow></math></span> in the original definition of <span><math><mrow><mi>E</mi><mn>3</mn><mi>C</mi><mrow><mo>(</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> to <span><math><mrow><mo>min</mo><mrow><mo>{</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>}</mo></mrow><mo>≥</mo><mn>0</mn></mrow></math></span>. Let <span><math><mrow><mi>x</mi><mi>y</mi><mi>z</mi></mrow></math></span> be the ascending order permutation of <span><math><mi>r</mi></math></span>, <span><math><mi>s</mi></math></span> and <span><math><mi>t</mi></math></span>. For <span><math><mrow><mi>z</mi><mo>≥</mo><mn>1</mn></mrow></math></span>, we prove that <span><math><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow></math></span>\u0000 <span><math><mrow><mi>E</mi><mn>3</mn><mi>C</mi><mrow><mo>(</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> is <span><math><msup><mrow><mi>λ</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>-optimal, and <span><math><mrow><mo>(</mo><mi>i</mi><mi>i</mi><mo>)</mo></mrow></math></span>\u0000 <span><math><mrow><msup><mrow><mi>λ</mi></mrow><mrow><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>E</mi><mn>3</mn><mi>C</mi><mrow><mo>(</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>2</mn></mrow></math></span> if <span><math><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span>, and <span><math><mrow><msup><mrow><mi>λ</mi></mrow><mrow><mo>′<","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"378 ","pages":"Pages 151-161"},"PeriodicalIF":1.0,"publicationDate":"2025-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144633481","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Related characterizations of the Shapley value and the weighted Shapley values via relaxations of differential marginality","authors":"André Casajus","doi":"10.1016/j.dam.2025.06.069","DOIUrl":"10.1016/j.dam.2025.06.069","url":null,"abstract":"<div><div>Casajus (2018) [7] provides a characterization of the class of positively weighted Shapley values for finite games from an infinite universe of players via three properties: efficiency, the null player out property, and superweak differential marginality. The latter requires two players’ payoffs to change in the same direction whenever only their joint productivity changes, that is, their individual productivities stay the same. Strengthening this property into (weak) differential marginality yields a characterization of the Shapley value. We suggest a relaxation of superweak differential marginality into two subproperties: (i) hyperweak differential marginality and (ii) superweak differential marginality for infinite subdomains. The former (i) only rules out changes in the opposite direction. The latter (ii) requires changes in the same direction for players within certain infinite subuniverses. Together with efficiency and the null player out property, these properties characterize the class of weighted Shapley values.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"378 ","pages":"Pages 136-150"},"PeriodicalIF":1.0,"publicationDate":"2025-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144633361","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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