Discrete Applied Mathematics最新文献

{"title":"Commutative distance degree-regular graphs","authors":"Cristian M. Conde ,&nbsp;Ezequiel Dratman ,&nbsp;Verónica Moyano ,&nbsp;Adrián Pastine","doi":"10.1016/j.dam.2026.01.026","DOIUrl":"10.1016/j.dam.2026.01.026","url":null,"abstract":"<div><div>This paper introduces and studies a family of graphs characterized by a commutative structure among their distance matrices, with a focus on those exhibiting distance-based regularity via constant row sums. This notion extends classical degree-regularity to a broader context, and, in particular, presents a new generalization of distance-regular graphs. We provide characterizations of this class, examine their relation to classical notions of regularity, and present several examples and constructions that highlight their behavior.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 397-410"},"PeriodicalIF":1.0,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146078351","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":"Aspects of a randomly growing cluster in Rd,d≥2","authors":"Alan Frieze ,&nbsp;Ravi Kannan ,&nbsp;Wesley Pegden","doi":"10.1016/j.dam.2025.12.062","DOIUrl":"10.1016/j.dam.2025.12.062","url":null,"abstract":"<div><div>We consider a simple model of a growing cluster of points in <span><math><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>,</mo><mi>d</mi><mo>≥</mo><mn>2</mn></mrow></math></span>. Beginning with a point <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> located at the origin, we generate a random sequence of points <span><math><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo></mrow></math></span>. To generate <span><math><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>i</mi><mo>≥</mo><mn>2</mn></mrow></math></span> we choose a uniform integer <span><math><mi>j</mi></math></span> in <span><math><mrow><mrow><mo>[</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo>]</mo></mrow><mo>=</mo><mfenced><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>i</mi><mo>−</mo><mn>1</mn></mrow></mfenced></mrow></math></span> and then let <span><math><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>+</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></math></span> where <span><math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mrow><mo>(</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>. Here the <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> are independent copies of the Normal distribution <span><math><mrow><mi>N</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msup><mrow><mi>i</mi></mrow><mrow><mo>−</mo><mi>α</mi></mrow></msup></mrow></math></span> for some <span><math><mrow><mi>α</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>. We prove that for any <span><math><mrow><mi>α</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span> the resulting point set is bounded a.s., and moreover, that the points generated look like samples from a <span><math><mi>β</mi></math></span>-dimensional subset of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> from the standpoint of the minimum lengths of combinatorial structures on the point-sets, where <span><math><mrow><mi>β</mi><mo>=</mo><mo>min</mo><mrow><mo>(</mo><mi>d</mi><mo>,</mo><mn>1</mn><mo>/</mo><mi>α</mi><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 287-292"},"PeriodicalIF":1.0,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980021","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":"Some results on minimum saturated graphs","authors":"Chenke Zhang ,&nbsp;Qing Cui ,&nbsp;Jinze Hu ,&nbsp;Erfei Yue ,&nbsp;Shengjin Ji","doi":"10.1016/j.dam.2025.12.040","DOIUrl":"10.1016/j.dam.2025.12.040","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a graph and <span><math><mi>F</mi></math></span> be a family of graphs. We say a graph <span><math><mi>G</mi></math></span> is <span><math><mi>F</mi></math></span>-saturated if <span><math><mi>G</mi></math></span> does not contain any member in <span><math><mi>F</mi></math></span> and for any <span><math><mrow><mi>e</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mover><mrow><mi>G</mi></mrow><mo>¯</mo></mover><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>G</mi><mo>+</mo><mi>e</mi></mrow></math></span> creates a copy of some member in <span><math><mi>F</mi></math></span>. The saturation number of <span><math><mi>F</mi></math></span> is the minimum number of edges of an <span><math><mi>F</mi></math></span>-saturated graphs of <span><math><mi>n</mi></math></span> vertices, denoted by <span><math><mrow><mi>sat</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span>. If <span><math><mrow><mi>F</mi><mo>=</mo><mrow><mo>{</mo><mi>F</mi><mo>}</mo></mrow></mrow></math></span>, then we write it as <span><math><mrow><mi>sat</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> for short. In this paper, we determine the exact value of <span><math><mrow><mi>sat</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mrow><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span>, and as its application, we obtain two bounds on <span><math><mrow><mi>sat</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>∪</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>k</mi><mo>≥</mo><mn>10</mn></mrow></math></span> and sufficiently large <span><math><mi>n</mi></math></span>. Furthermore, <span><math><mrow><mi>sat</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∨</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> is determined, where <span><math><mi>F</mi></math></span> is a linear forest without isolated vertices.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 23-33"},"PeriodicalIF":1.0,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145898209","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":"Some properties of outerplanar graphs and the least eigenvalue","authors":"Guanglong Yu ,&nbsp;Lin Sun ,&nbsp;Xin Li","doi":"10.1016/j.dam.2026.01.010","DOIUrl":"10.1016/j.dam.2026.01.010","url":null,"abstract":"<div><div>Some structural properties of the (edge) maximal bipartite outerplanar graphs are represented in this paper. As well, among all outerplanar graphs of order <span><math><mrow><mi>n</mi><mo>≥</mo><mn>18</mn></mrow></math></span>, the minimum least eigenvalue is completely determined.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 409-417"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146077876","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":"Cliques and high odd holes in graphs with chromatic number equal to maximum degree","authors":"Rachel Galindo,&nbsp;Jessica McDonald ,&nbsp;Songling Shan","doi":"10.1016/j.dam.2026.01.014","DOIUrl":"10.1016/j.dam.2026.01.014","url":null,"abstract":"<div><div>We give a uniform and self-contained proof that if <span><math><mi>G</mi></math></span> is a connected graph with <span><math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>G</mi><mo>≠</mo><mover><mrow><msub><mrow><mi>C</mi></mrow><mrow><mn>7</mn></mrow></msub></mrow><mo>¯</mo></mover></mrow></math></span>, then <span><math><mi>G</mi></math></span> contains either <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></msub></math></span> or an odd hole where every vertex has degree at least <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow></math></span> in <span><math><mi>G</mi></math></span>. This was previously proved in series of two papers by Chen, Lan, Lin, and Zhou, who used the Strong Perfect Graph Theorem for the cases <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>6</mn></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 383-386"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023128","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":"Maximum subgraph and wirelength analysis of extended Sierpiński networks in parallel computing","authors":"P. Leo Joshwa ,&nbsp;R. Sundara Rajan ,&nbsp;T.M. Rajalaxmi","doi":"10.1016/j.dam.2026.01.013","DOIUrl":"10.1016/j.dam.2026.01.013","url":null,"abstract":"<div><div>The Sierpiński networks, <span><math><mrow><mi>S</mi><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span>, serves as an important model for multiprocessor interconnection systems due to its hierarchical and self-similar properties. Its extended variant, <span><math><mrow><msup><mrow><mi>S</mi></mrow><mrow><mo>+</mo><mo>+</mo></mrow></msup><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span>, 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 <span><math><mrow><msup><mrow><mi>S</mi></mrow><mrow><mo>+</mo><mo>+</mo></mrow></msup><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>p</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, and <span><math><mrow><mi>q</mi><mo>≥</mo><mn>3</mn></mrow></math></span> 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 <span><math><mi>r</mi></math></span> vertices in <span><math><mrow><msup><mrow><mi>S</mi></mrow><mrow><mo>+</mo><mo>+</mo></mrow></msup><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span> where <span><math><mrow><mn>1</mn><mo>≤</mo><mi>r</mi><mo>≤</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span>. Moreover, we investigate the minimum wirelength embedding of <span><math><mrow><msup><mrow><mi>S</mi></mrow><mrow><mo>+</mo><mo>+</mo></mrow></msup><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span> into structures such as paths, caterpillars, and 1-hierarchical caterpillars, contributing to the study of efficient graph embeddings in hierarchical systems.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 367-382"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023129","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":"Equitable greedy scalar equilibria for coalitional semi-cooperative games","authors":"Davoud Foroutannia ,&nbsp;Davoud Mahmoudinia","doi":"10.1016/j.dam.2025.11.036","DOIUrl":"10.1016/j.dam.2025.11.036","url":null,"abstract":"<div><div>In this paper, we examine coalitional semi-cooperative games in strategic form, where players within each coalition adhere to the equitable preference relation. Equitable preference is a refinement of the Pareto preference relation. It adds to the reflexivity, strict monotonicity, and transitivity of the Pareto preference order the requirements of impartiality and the Pigou–Dalton principle of transfers. We introduce the concept of <span><math><mi>C</mi></math></span>-equitable, in which <span><math><mi>C</mi></math></span> is a complete set of coalitions. By generating a finite subset of Pareto optimal solutions, this concept can help decision-makers select the most suitable solution, especially when there is a large number of Pareto equilibria in some real situations.</div><div>A scalar equilibrium achieved by maximizing an appropriate utility function over the acceptable joint actions may not be <span><math><mi>C</mi></math></span>-equitable in many instances. Another goal of this paper is to generate the <span><math><mi>C</mi></math></span>-equitable equilibria using utility functions. To achieve this, we require the aggregate utility functions to be Schur-convex, in addition to assuming strict increases, to ensure that each scalar equilibrium is a <span><math><mi>C</mi></math></span>-equitable equilibrium.</div><div>The greedy scalar equilibrium is one of the well-known scalar equilibria that assigns actions to players that yield the largest individual payoffs jointly possible. Our final concept is <span><math><mi>C</mi></math></span>-equitable greedy scalar equilibrium, which incorporates the notions of equitability and greed to select the optimal set of coalitions for a semi-cooperative game.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 52-64"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792200","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":"The structural properties of {(2,3),6}-spheres","authors":"Rui Yang,&nbsp;Chengli Liu","doi":"10.1016/j.dam.2025.12.018","DOIUrl":"10.1016/j.dam.2025.12.018","url":null,"abstract":"<div><div>A <span><math><mrow><mrow><mo>{</mo><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo></mrow><mo>,</mo><mn>6</mn><mo>}</mo></mrow><mtext>-</mtext></mrow></math></span> sphere <span><math><mi>S</mi></math></span> is a <span><math><mrow><mn>6</mn><mtext>-</mtext></mrow></math></span>regular planar graph whose faces are only <span><math><mrow><mn>2</mn><mtext>-</mtext></mrow></math></span>length and <span><math><mrow><mn>3</mn><mtext>-</mtext></mrow></math></span>length. Furthermore, it consists of exactly six <span><math><mrow><mn>2</mn><mtext>-</mtext></mrow></math></span>length faces and even number of <span><math><mrow><mn>3</mn><mtext>-</mtext></mrow></math></span>length faces by Euler’s formula. A graph <span><math><mi>G</mi></math></span> is called <em>cyclically</em> <span><math><mrow><mi>k</mi><mtext>-</mtext></mrow></math></span><em>edge-connected</em>, if it cannot be separated into at least two components such that each contains cycles by deleting fewer than <span><math><mi>k</mi></math></span> edges. We denote the maximum value of <span><math><mrow><mi>k</mi><mo>∈</mo><mi>N</mi></mrow></math></span> such that <span><math><mi>G</mi></math></span> is cyclically <span><math><mrow><mi>k</mi><mtext>-</mtext></mrow></math></span>edge-connected by <span><math><mrow><mi>c</mi><mi>λ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, then the positive integer <span><math><mrow><mi>c</mi><mi>λ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is referred to as <em>cyclical edge-connectivity</em> of <span><math><mi>G</mi></math></span>. In this paper, we prove that the cyclical edge-connectivity of <span><math><mrow><mrow><mo>{</mo><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo></mrow><mo>,</mo><mn>6</mn><mo>}</mo></mrow><mtext>-</mtext></mrow></math></span>sphere is 4 or 6 or 8, and we give the structures of <span><math><mrow><mrow><mo>{</mo><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo></mrow><mo>,</mo><mn>6</mn><mo>}</mo></mrow><mtext>-</mtext></mrow></math></span>spheres with cyclical edge-connectivities 4 and 6, respectively. A face of <span><math><mi>G</mi></math></span> is called <em>resonant</em> if its boundary is <span><math><mrow><mi>M</mi><mtext>-</mtext></mrow></math></span>alternating cycle, where <span><math><mi>M</mi></math></span> is a perfect matching of <span><math><mi>G</mi></math></span>. Furthermore, in this paper, we show that every <span><math><mrow><mn>2</mn><mtext>-</mtext></mrow></math></span>length face of a <span><math><mrow><mrow><mo>{</mo><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo></mrow><mo>,</mo><mn>6</mn><mo>}</mo></mrow><mtext>-</mtext></mrow></math></span>sphere is resonant.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 15-25"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145748179","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":"An approximation algorithm for zero forcing","authors":"Ben Cameron ,&nbsp;Jeannette Janssen ,&nbsp;Rogers Mathew ,&nbsp;Zhiyuan Zhang","doi":"10.1016/j.dam.2025.12.013","DOIUrl":"10.1016/j.dam.2025.12.013","url":null,"abstract":"<div><div>We give an algorithm that finds a zero forcing set which approximates the optimal size by a factor of <span><math><mrow><mi>pw</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow></math></span>, where <span><math><mrow><mi>pw</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is the pathwidth of <span><math><mi>G</mi></math></span>. The algorithm requires a path decomposition of <span><math><mi>G</mi></math></span>, and given this it runs in <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>n</mi><mi>m</mi><mo>)</mo></mrow></mrow></math></span> time, where <span><math><mi>n</mi></math></span> and <span><math><mi>m</mi></math></span> are the order and size of the graph, respectively. This is the first zero forcing algorithm with a guarantee on both the approximation ratio and on the run-time. As a corollary, we obtain a new upper bound on the zero forcing number in terms of the fort number and the pathwidth. The algorithm is based on a correspondence between zero forcing sets and forcing arc sets. This correspondence leads to a new bound on the zero forcing number in terms of vertex cuts, and to new, short proofs for known bounds on the zero forcing number.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 227-242"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842075","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 tolerance for metric dimension and its variants","authors":"Jesse Geneson ,&nbsp;Shen-Fu Tsai","doi":"10.1016/j.dam.2026.01.009","DOIUrl":"10.1016/j.dam.2026.01.009","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Hernando et al. (2008) introduced the fault-tolerant metric dimension &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mtext&gt;ftdim&lt;/mtext&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, which is the size of the smallest resolving set &lt;span&gt;&lt;math&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; of a graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; such that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is also a resolving set of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; for every &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. They found an upper bound &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mtext&gt;ftdim&lt;/mtext&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mo&gt;dim&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;⋅&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;dim&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;dim&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; denotes the standard metric dimension of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. It was unknown whether there exists a family of graphs where &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mtext&gt;ftdim&lt;/mtext&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; grows exponentially in terms of &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;dim&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, until recently when Knor et al. (2024) found a family with &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mtext&gt;ftdim&lt;/mtext&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;dim&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;dim&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; for any possible value of &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;dim&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. We improve the upper bound on fault-tolerant metric dimension by showing that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mtext&gt;ftdim&lt;/mtext&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mo&gt;dim&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;dim&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; for every connected graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. Moreover, we find an infinite family of connected graphs &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; such that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;dim&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mtext&gt;ftdim&lt;/mtext&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mat","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 339-354"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977711","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}