{"title":"An introduction to the deduction number","authors":"Andrea Burgess , Danny Dyer , Mozhgan Farahani","doi":"10.1016/j.dam.2025.02.024","DOIUrl":"10.1016/j.dam.2025.02.024","url":null,"abstract":"<div><div>The deduction game is a variation of the game of cops and robber on graphs in which searchers must capture an invisible evader in at most one move. Searchers know each others’ initial locations, but can only communicate if they are on the same vertex. Thus, searchers must deduce other searchers’ movement and move accordingly. We introduce the deduction number and study it for various classes of graphs. We provide upper bounds for the deduction number of the Cartesian product of graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"369 ","pages":"Pages 20-27"},"PeriodicalIF":1.0,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143561881","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Characterization of trees with second minimum eccentricity energy","authors":"Iswar Mahato","doi":"10.1016/j.dam.2025.02.036","DOIUrl":"10.1016/j.dam.2025.02.036","url":null,"abstract":"<div><div>The eccentricity matrix of a connected graph <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is obtained from the distance matrix of <span><math><mi>G</mi></math></span> by keeping the largest entries in each row and each column, and putting the remaining entries as zero. The eigenvalues of <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> are the <span><math><mi>E</mi></math></span>-eigenvalues of <span><math><mi>G</mi></math></span>. The eccentricity energy (or the <span><math><mi>E</mi></math></span>-energy) of <span><math><mi>G</mi></math></span> is the sum of the absolute values of all <span><math><mi>E</mi></math></span>-eigenvalues of <span><math><mi>G</mi></math></span>. In this article, we characterize the trees with second minimum <span><math><mi>E</mi></math></span>-energy among all trees on <span><math><mrow><mi>n</mi><mo>≥</mo><mn>5</mn></mrow></math></span> vertices.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"369 ","pages":"Pages 78-87"},"PeriodicalIF":1.0,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143561797","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":"Output-sensitive enumeration of maximal cliques in temporal graphs","authors":"Filippo Brunelli , Alessio Conte , Roberto Grossi , Andrea Marino","doi":"10.1016/j.dam.2025.02.025","DOIUrl":"10.1016/j.dam.2025.02.025","url":null,"abstract":"<div><div>Community detection is one of the fundamental tasks in graph analysis, and cliques, representing a fully interconnected subset of nodes, are one of the archetypal models of community, typically focusing on those that are maximal under inclusion. Recent years witnessed an increase in prominence of temporal graphs, i.e., graphs where edges may freely appear and disappear. This scenario caused a wave of research devoted to understand the temporal structure, and how to adapt classical graph methods to this more complex model. Several adaptations have been proposed for cliques, as well as algorithms for finding maximal cliques. These, however, do not have output-sensitive guarantees, meaning considerable time could be spent to find a small number of solutions.</div><div>In this paper, we show how two different proposed models of temporal graphs, and the corresponding definitions of clique, are essentially equivalent, allowing us to consider a single model of clique without losing generality; furthermore, we develop an output-sensitive algorithm for finding all maximal cliques in the restricted scenario where each edge is present for a continuous interval: the algorithm runs in polynomial total time using polynomial space, or in incremental polynomial time if we allow exponential space usage.</div><div>While the restricted scenario limits the generality of the result, no output-sensitive algorithm for the problem exists as of yet, making this the first such result, and a first step into output-sensitive community detection in temporal graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"369 ","pages":"Pages 66-77"},"PeriodicalIF":1.0,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552891","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":"Unary NP-hardness of transportation and batching scheduling to minimize the total weighted completion time","authors":"Hongjun Wei, Yuan Gao, Jinjiang Yuan","doi":"10.1016/j.dam.2025.02.023","DOIUrl":"10.1016/j.dam.2025.02.023","url":null,"abstract":"<div><div>In this paper, we study the coordination of transportation and batching scheduling with one single vehicle to minimize the total weighted completion time. When the batch capacity is 2, the computational complexity of this problem has been reported open in the literature. We show in this paper that the problem is unary NP-hard.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"369 ","pages":"Pages 45-52"},"PeriodicalIF":1.0,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552967","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}
E.M.M. Coelho , H. Coelho , L. Faria , M.P. Ferreira , S. Klein
{"title":"On the absolute and relative oriented clique problems’ time complexity","authors":"E.M.M. Coelho , H. Coelho , L. Faria , M.P. Ferreira , S. Klein","doi":"10.1016/j.dam.2025.02.039","DOIUrl":"10.1016/j.dam.2025.02.039","url":null,"abstract":"<div><div>Given a graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span>, the size of the largest clique <span><math><mrow><mi>ω</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is always less than or equal to the chromatic number <span><math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mi>G</mi></math></span>. The oriented coloring of an oriented graph <span><math><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover></math></span> assigns colors to the vertices of <span><math><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover></math></span>, such that the arcs connecting vertices in different color classes always have the same direction and the smallest number <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mrow><mo>(</mo><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover><mo>)</mo></mrow></mrow></math></span> of colors in an oriented coloring is the oriented chromatic number of <span><math><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover></math></span>. Oriented colorings have fundamental implications for homomorphisms of oriented graphs and significant applications in distributed processing and task scheduling. In 2004, Klostermeyer and MacGillivray defined the concept of an “analogue of clique” for oriented coloring in which a subgraph <span><math><mover><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mi>o</mi></mrow></msub></mrow><mo>⃗</mo></mover></math></span> of <span><math><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover></math></span> is an absolute oriented clique if the oriented distance between a pair of vertices of <span><math><mover><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mi>o</mi></mrow></msub></mrow><mo>⃗</mo></mover></math></span> in <span><math><mover><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mi>o</mi></mrow></msub></mrow><mo>⃗</mo></mover></math></span> is at most 2. The authors defined the absolute oriented clique number of <span><math><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover></math></span> as the number of vertices <span><math><mrow><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mover><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mi>o</mi></mrow></msub></mrow><mo>⃗</mo></mover><mo>)</mo></mrow><mo>|</mo></mrow><mo>=</mo><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mrow><mo>(</mo><mover><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mi>o</mi></mrow></msub></mrow><mo>⃗</mo></mover><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>a</mi><mi>o</mi></mrow></msub><mrow><mo>(</mo><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover><mo>)</mo></mrow></mrow></math></span> in a maximum absolute oriented clique <span><math><mover><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mi>o</mi></mrow></msub></mrow><mo>⃗</mo></move","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"369 ","pages":"Pages 53-65"},"PeriodicalIF":1.0,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552968","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":"Connectivity and super connectivity of enhanced folded hypercube-like networks","authors":"Litao Guo , Wantao Ning","doi":"10.1016/j.dam.2025.02.034","DOIUrl":"10.1016/j.dam.2025.02.034","url":null,"abstract":"<div><div>We define a new class of graphs called enhanced folded hypercube-like networks <span><math><mrow><mi>E</mi><mi>F</mi><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span>. We also investigate the reliability of this class of graphs in terms of the (edge) connectivity and super (edge) connectivity.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"369 ","pages":"Pages 14-19"},"PeriodicalIF":1.0,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143534673","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":"Hamilton cycles in vertex-transitive graphs of order 6p","authors":"Shaofei Du, Tianlei Zhou","doi":"10.1016/j.dam.2025.02.033","DOIUrl":"10.1016/j.dam.2025.02.033","url":null,"abstract":"<div><div>It was shown by Kutnar and Šparl in 2009 that every connected vertex-transitive graph of order <span><math><mrow><mn>6</mn><mi>p</mi></mrow></math></span>, where <span><math><mi>p</mi></math></span> is a prime, contains a Hamilton path. In this paper, it will be shown that every such graph contains a Hamilton cycle, except for the Petersen graph by replacing each vertex by a triangle.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"368 ","pages":"Pages 165-175"},"PeriodicalIF":1.0,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552118","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":"Structure of (bull, diamond)-free graphs and its applications","authors":"Suchismita Mishra","doi":"10.1016/j.dam.2025.02.026","DOIUrl":"10.1016/j.dam.2025.02.026","url":null,"abstract":"<div><div>In this paper, we discuss the complete structure of the (bull, diamond)-free graphs. As an application of that, we give the characterization of the partitionable (bull, diamond)-free graphs. Moreover, we show that such a partition for a partitionable (bull, diamond)-free graph can be found in polynomial time. Additionally, we show that the cop number of a (bull, diamond)-free graph containing a triangle is at most two less than its diameter. Furthermore, the cop number of a connected (<span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, bull, diamond)-free graph with a triangle, is at most <span><math><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></math></span>, for any natural number <span><math><mrow><mi>n</mi><mo>></mo><mn>3</mn></mrow></math></span>. We also discuss a couple of applications of the structural theorem of the (bull, diamond)-free graphs in the conclusions.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"368 ","pages":"Pages 176-183"},"PeriodicalIF":1.0,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552119","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":"Tight bounds on odd chromatic number of some standard graph products","authors":"Priyamvada","doi":"10.1016/j.dam.2025.02.041","DOIUrl":"10.1016/j.dam.2025.02.041","url":null,"abstract":"<div><div>An <em>odd coloring</em> of a graph <span><math><mi>G</mi></math></span> is an assignment <span><math><mrow><mi>f</mi><mo>:</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>→</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></mrow></math></span> of colors to the vertices of <span><math><mi>G</mi></math></span> such that <span><math><mi>f</mi></math></span> is a proper vertex coloring and for every non-isolated vertex <span><math><mi>v</mi></math></span>, there is a color that occurs an odd number of times within its open neighborhood. The minimum number of colors required by any odd coloring of <span><math><mi>G</mi></math></span> is called the <em>odd chromatic number</em> of <span><math><mi>G</mi></math></span> and is denoted by <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. In this paper, we give tight upper bounds on the odd chromatic number of various standard graph products and operations, including the lexicographic product, corona product, edge corona product and Mycielskian of a graph. Moreover, we give tight lower bounds on the odd chromatic number of corona product and edge corona product of graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"369 ","pages":"Pages 1-13"},"PeriodicalIF":1.0,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143534446","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":"A note on secure domination number in 2K2-free graphs","authors":"Xiaodong Chen, Tianhao Li, Jiayuan Zhang","doi":"10.1016/j.dam.2025.02.019","DOIUrl":"10.1016/j.dam.2025.02.019","url":null,"abstract":"<div><div>A dominating set <span><math><mi>D</mi></math></span> of a graph <span><math><mi>G</mi></math></span> is secure if for each vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>−</mo><mi>D</mi><mo>,</mo></mrow></math></span>\u0000 <span><math><mi>D</mi></math></span> contains a neighbor <span><math><mi>u</mi></math></span> of <span><math><mi>v</mi></math></span> such that <span><math><mrow><mrow><mo>(</mo><mi>D</mi><mo>−</mo><mrow><mo>{</mo><mi>u</mi><mo>}</mo></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>{</mo><mi>v</mi><mo>}</mo></mrow></mrow></math></span> is a dominating set of <span><math><mi>G</mi></math></span>. The minimum cardinality of a secure dominating set in <span><math><mi>G</mi></math></span> is the secure domination number of <span><math><mi>G</mi></math></span> and denoted by <span><math><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>s</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>.</mo></mrow></math></span> A graph is <span><math><mrow><mn>2</mn><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>-free if it does not contain two independent edges as an induced subgraph. Let <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> denote the independence number of <span><math><mrow><mi>G</mi><mo>.</mo></mrow></math></span> Several results gave the upper bound of <span><math><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>s</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> by a function of <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>.</mo></mrow></math></span> In this note, we shows that <span><math><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>s</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow></math></span> for every <span><math><mrow><mn>2</mn><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>-free graph <span><math><mrow><mi>G</mi><mo>;</mo></mrow></math></span> moreover, we give an example to show the bound in our result is best possible.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"368 ","pages":"Pages 162-164"},"PeriodicalIF":1.0,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552120","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}
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
