Pub Date : 2026-01-19DOI: 10.1016/j.dam.2026.01.007
Xiaohui Bei , Alexander Lam , Xinhang Lu , Warut Suksompong
We study the allocation of indivisible items that form an undirected graph and investigate the worst-case welfare loss when requiring that each agent must receive a connected subgraph. Our focus is on both egalitarian and utilitarian welfare. Specifically, we introduce the concept of egalitarian (resp., utilitarian) price of connectivity, which captures the worst-case ratio between the optimal egalitarian (resp., utilitarian) welfare among all allocations and that among connected allocations. We provide tight or asymptotically tight bounds on the price of connectivity for several large classes of graphs in the case of two agents—including graphs with vertex connectivity 1 or 2 and complete bipartite graphs—as well as for paths, stars, and cycles in the general case where the number of agents can be arbitrary.
{"title":"Welfare loss in connected resource allocation","authors":"Xiaohui Bei , Alexander Lam , Xinhang Lu , Warut Suksompong","doi":"10.1016/j.dam.2026.01.007","DOIUrl":"10.1016/j.dam.2026.01.007","url":null,"abstract":"<div><div>We study the allocation of indivisible items that form an undirected graph and investigate the worst-case welfare loss when requiring that each agent must receive a connected subgraph. Our focus is on both egalitarian and utilitarian welfare. Specifically, we introduce the concept of <em>egalitarian (resp., utilitarian) price of connectivity</em>, which captures the worst-case ratio between the optimal egalitarian (resp., utilitarian) welfare among all allocations and that among connected allocations. We provide tight or asymptotically tight bounds on the price of connectivity for several large classes of graphs in the case of two agents—including graphs with vertex connectivity 1 or 2 and complete bipartite graphs—as well as for paths, stars, and cycles in the general case where the number of agents can be arbitrary.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"385 ","pages":"Pages 1-23"},"PeriodicalIF":1.0,"publicationDate":"2026-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145993603","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}
Pub Date : 2026-01-19DOI: 10.1016/j.dam.2026.01.012
Ruiqing Feng, Qi Yan, Xuan Zheng
The partial Petrial polynomial was first introduced by Gross, Mansour, and Tucker as a generating function that enumerates the Euler genera of all possible partial Petrials on a ribbon graph. Yan and Li later extended this polynomial invariant to circle graphs by utilizing the correspondence between circle graphs and bouquets. Their explicit computation demonstrated that paths produce binomial polynomials, specifically those containing exactly two non-zero terms. This discovery led them to pose a fundamental characterization problem: identify all connected circle graphs whose partial Petrial polynomial is binomial. In this paper, we solve this open problem in terms of local complementation and prove that for connected circle graphs, the binomial property holds precisely when the graph is a path.
{"title":"Characterizing circle graphs with binomial partial Petrial polynomials","authors":"Ruiqing Feng, Qi Yan, Xuan Zheng","doi":"10.1016/j.dam.2026.01.012","DOIUrl":"10.1016/j.dam.2026.01.012","url":null,"abstract":"<div><div>The partial Petrial polynomial was first introduced by Gross, Mansour, and Tucker as a generating function that enumerates the Euler genera of all possible partial Petrials on a ribbon graph. Yan and Li later extended this polynomial invariant to circle graphs by utilizing the correspondence between circle graphs and bouquets. Their explicit computation demonstrated that paths produce binomial polynomials, specifically those containing exactly two non-zero terms. This discovery led them to pose a fundamental characterization problem: identify all connected circle graphs whose partial Petrial polynomial is binomial. In this paper, we solve this open problem in terms of local complementation and prove that for connected circle graphs, the binomial property holds precisely when the graph is a path.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 411-416"},"PeriodicalIF":1.0,"publicationDate":"2026-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023054","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}
Pub Date : 2026-01-17DOI: 10.1016/j.dam.2026.01.003
Alberto José Ferrari , Valeria Leoni , Graciela Nasini , Gabriel Valiente
In computational biology and bioinformatics, hypergraphs model metabolic pathways and networks representing compounds as vertices and reactions as hyperedges. In a previous work we considered the problem of assigning a direction to the hyperedges of a hypergraph minimizing the number of source and sink vertices. We proved that this problem is NP-hard and that it is polynomial-time solvable on graphs.
In a more general setting, a compound can be a source or a sink in a particular metabolic pathway but, in the context of a metabolic network, it may become both a sink of one pathway and a source of another pathway (an internal vertex). Therefore, in the present work we address a more general form of the hypergraph orientation problem in which some vertices are constrained to be a source, a sink, or an internal vertex. We prove that it remains polynomial-time solvable on graphs by giving a linear-time algorithm. We propose a polynomial-size ILP formulation of the problem, which, applied to the biochemical reactions stored in the Kyoto Encyclopedia of Genes and Genomes (KEGG) database, shows that metabolic pathways and networks, and random hypergraphs with thousands of vertices and hyperedges, can be oriented in a few seconds on a personal computer.
{"title":"The hypergraph orientation problem with vertex constraints","authors":"Alberto José Ferrari , Valeria Leoni , Graciela Nasini , Gabriel Valiente","doi":"10.1016/j.dam.2026.01.003","DOIUrl":"10.1016/j.dam.2026.01.003","url":null,"abstract":"<div><div>In computational biology and bioinformatics, hypergraphs model metabolic pathways and networks representing compounds as vertices and reactions as hyperedges. In a previous work we considered the problem of assigning a direction to the hyperedges of a hypergraph minimizing the number of source and sink vertices. We proved that this problem is NP-hard and that it is polynomial-time solvable on graphs.</div><div>In a more general setting, a compound can be a source or a sink in a particular metabolic pathway but, in the context of a metabolic network, it may become both a sink of one pathway and a source of another pathway (an internal vertex). Therefore, in the present work we address a more general form of the hypergraph orientation problem in which some vertices are constrained to be a source, a sink, or an internal vertex. We prove that it remains polynomial-time solvable on graphs by giving a linear-time algorithm. We propose a polynomial-size ILP formulation of the problem, which, applied to the biochemical reactions stored in the Kyoto Encyclopedia of Genes and Genomes (KEGG) database, shows that metabolic pathways and networks, and random hypergraphs with thousands of vertices and hyperedges, can be oriented in a few seconds on a personal computer.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 355-366"},"PeriodicalIF":1.0,"publicationDate":"2026-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977709","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}
Pub Date : 2026-01-16DOI: 10.1016/j.dam.2026.01.009
Jesse Geneson , Shen-Fu Tsai
<div><div>Hernando et al. (2008) introduced the fault-tolerant metric dimension <span><math><mrow><mtext>ftdim</mtext><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, which is the size of the smallest resolving set <span><math><mi>S</mi></math></span> of a graph <span><math><mi>G</mi></math></span> such that <span><math><mrow><mi>S</mi><mo>−</mo><mfenced><mrow><mi>s</mi></mrow></mfenced></mrow></math></span> is also a resolving set of <span><math><mi>G</mi></math></span> for every <span><math><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></math></span>. They found an upper bound <span><math><mrow><mtext>ftdim</mtext><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mo>dim</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mn>2</mn><mi>⋅</mi><msup><mrow><mn>5</mn></mrow><mrow><mo>dim</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mo>dim</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> denotes the standard metric dimension of <span><math><mi>G</mi></math></span>. It was unknown whether there exists a family of graphs where <span><math><mrow><mtext>ftdim</mtext><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> grows exponentially in terms of <span><math><mrow><mo>dim</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, until recently when Knor et al. (2024) found a family with <span><math><mrow><mtext>ftdim</mtext><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mo>dim</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><msup><mrow><mn>2</mn></mrow><mrow><mo>dim</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span> for any possible value of <span><math><mrow><mo>dim</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. We improve the upper bound on fault-tolerant metric dimension by showing that <span><math><mrow><mtext>ftdim</mtext><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mo>dim</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msup><mrow><mn>3</mn></mrow><mrow><mo>dim</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> for every connected graph <span><math><mi>G</mi></math></span>. Moreover, we find an infinite family of connected graphs <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> such that <span><math><mrow><mo>dim</mo><mrow><mo>(</mo><msub><mrow><mi>J</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mi>k</mi></mrow></math></span> and <span><math><mrow><mtext>ftdim</mtext><mrow><mo>(</mo><msub><mrow><mi>J</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow><mo>≥</mo><msup><mrow><mn>3</mn></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mi>k</mi><mo>−</mo><mn>1</mn></mrow></mat
Hernando et al.（2008）引入了容错度量维度ftdim(G)，它是图G的最小解析集S的大小，使得S−S也是每个S∈S的G的解析集。他们发现了一个上界ftdim(G)≤dim(G)(1+2·5dim(G)−1)，其中dim(G)表示G的标准度量维度。不知道是否存在一类图，其中ftdim(G)以dim(G)为指数增长，直到最近Knor等人（2024）发现ftdim(G)=dim(G)+2dim(G)−1对于任何可能的dim(G)值。通过证明对于每一个连通图G， ftdim(G)≤dim(G)(1+3dim(G)−1)，我们改进了容错度量维的上界，并且我们找到了一个无限族的连通图Jk，使得对于每一个正整数k， dim(Jk)=k和ftdim(Jk)≥3k−1−k−1。我们的结果表明limk→∞maxG:dim(G)=klog3(ftdim(G))k=1。此外，我们考虑容错边缘度量维数ftedim(G)，并将其与边缘度量维数edim(G)进行定界，表明limk→∞maxG:edim(G)=klog2(ftedim(G))k=1。我们还得到了邻接维数和k截断度量维数容错的尖锐极值界。此外，我们还得到了其他一些关于度量维数及其变体的极值问题的尖锐界。特别地，我们证明了关于边度量维的极值问题与极值集理论中Erdős和Kleitman（1974）的开放问题之间的等价性。
{"title":"Fault tolerance for metric dimension and its variants","authors":"Jesse Geneson , Shen-Fu Tsai","doi":"10.1016/j.dam.2026.01.009","DOIUrl":"10.1016/j.dam.2026.01.009","url":null,"abstract":"<div><div>Hernando et al. (2008) introduced the fault-tolerant metric dimension <span><math><mrow><mtext>ftdim</mtext><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, which is the size of the smallest resolving set <span><math><mi>S</mi></math></span> of a graph <span><math><mi>G</mi></math></span> such that <span><math><mrow><mi>S</mi><mo>−</mo><mfenced><mrow><mi>s</mi></mrow></mfenced></mrow></math></span> is also a resolving set of <span><math><mi>G</mi></math></span> for every <span><math><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></math></span>. They found an upper bound <span><math><mrow><mtext>ftdim</mtext><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mo>dim</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mn>2</mn><mi>⋅</mi><msup><mrow><mn>5</mn></mrow><mrow><mo>dim</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mo>dim</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> denotes the standard metric dimension of <span><math><mi>G</mi></math></span>. It was unknown whether there exists a family of graphs where <span><math><mrow><mtext>ftdim</mtext><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> grows exponentially in terms of <span><math><mrow><mo>dim</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, until recently when Knor et al. (2024) found a family with <span><math><mrow><mtext>ftdim</mtext><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mo>dim</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><msup><mrow><mn>2</mn></mrow><mrow><mo>dim</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span> for any possible value of <span><math><mrow><mo>dim</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. We improve the upper bound on fault-tolerant metric dimension by showing that <span><math><mrow><mtext>ftdim</mtext><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mo>dim</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msup><mrow><mn>3</mn></mrow><mrow><mo>dim</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> for every connected graph <span><math><mi>G</mi></math></span>. Moreover, we find an infinite family of connected graphs <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> such that <span><math><mrow><mo>dim</mo><mrow><mo>(</mo><msub><mrow><mi>J</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mi>k</mi></mrow></math></span> and <span><math><mrow><mtext>ftdim</mtext><mrow><mo>(</mo><msub><mrow><mi>J</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow><mo>≥</mo><msup><mrow><mn>3</mn></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mi>k</mi><mo>−</mo><mn>1</mn></mrow></mat","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 339-354"},"PeriodicalIF":1.0,"publicationDate":"2026-01-16","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}
Pub Date : 2026-01-16DOI: 10.1016/j.dam.2026.01.005
Shu-Li Zhao, Bao-Cheng Zhang
<div><div>Let <span><math><mi>G</mi></math></span> be a connected graph, <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow><mo>≥</mo><mn>2</mn></mrow></math></span>, a tree <span><math><mi>T</mi></math></span> in <span><math><mi>G</mi></math></span> is called a pendant <span><math><mi>S</mi></math></span>-tree if <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow></mrow></math></span> and the degree of each vertex in <span><math><mi>S</mi></math></span> is equal to one. Two pendant <span><math><mi>S</mi></math></span>-trees <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are called internally disjoint if <span><math><mrow><mi>E</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mo>∩</mo><mi>E</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mo>=</mo><mo>0̸</mo></mrow></math></span> and <span><math><mrow><mi>V</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mo>∩</mo><mi>V</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mo>=</mo><mi>S</mi></mrow></math></span>. For an integer <span><math><mi>k</mi></math></span> with <span><math><mrow><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></math></span>, the pendant-tree <span><math><mi>k</mi></math></span>-connectivity of a graph <span><math><mi>G</mi></math></span> is defined as <span><math><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>m</mi><mi>i</mi><mi>n</mi><mrow><mo>{</mo></mrow><msub><mrow><mi>τ</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow><mo>|</mo><mi>S</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mo>|</mo><mi>S</mi><mo>|</mo><mo>=</mo><mi>k</mi><mrow><mo>}</mo></mrow></mrow></math></span>, where <span><math><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow></mrow></math></span> denotes the maximum number <span><math><mi>r</mi></math></span> of internally disjoint pendant <span><math><mi>S</mi></math></span>-trees in <span><math><mi>G</mi></math></span>. The pendant-tree <span><math><mi>k</mi></math></span>-connectivity is a generalization of traditional connectivity. In this paper, we mainly investigate the pendant-tree 4-connectivity of the regular graph with given properties, denoted by <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, which was introduced in Zhao and Hao (2019). As applications of the main result, the pendant-tree 4
{"title":"The pendant-tree connectivity of some regular graphs","authors":"Shu-Li Zhao, Bao-Cheng Zhang","doi":"10.1016/j.dam.2026.01.005","DOIUrl":"10.1016/j.dam.2026.01.005","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a connected graph, <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow><mo>≥</mo><mn>2</mn></mrow></math></span>, a tree <span><math><mi>T</mi></math></span> in <span><math><mi>G</mi></math></span> is called a pendant <span><math><mi>S</mi></math></span>-tree if <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow></mrow></math></span> and the degree of each vertex in <span><math><mi>S</mi></math></span> is equal to one. Two pendant <span><math><mi>S</mi></math></span>-trees <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are called internally disjoint if <span><math><mrow><mi>E</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mo>∩</mo><mi>E</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mo>=</mo><mo>0̸</mo></mrow></math></span> and <span><math><mrow><mi>V</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mo>∩</mo><mi>V</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mo>=</mo><mi>S</mi></mrow></math></span>. For an integer <span><math><mi>k</mi></math></span> with <span><math><mrow><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></math></span>, the pendant-tree <span><math><mi>k</mi></math></span>-connectivity of a graph <span><math><mi>G</mi></math></span> is defined as <span><math><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>m</mi><mi>i</mi><mi>n</mi><mrow><mo>{</mo></mrow><msub><mrow><mi>τ</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow><mo>|</mo><mi>S</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mo>|</mo><mi>S</mi><mo>|</mo><mo>=</mo><mi>k</mi><mrow><mo>}</mo></mrow></mrow></math></span>, where <span><math><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow></mrow></math></span> denotes the maximum number <span><math><mi>r</mi></math></span> of internally disjoint pendant <span><math><mi>S</mi></math></span>-trees in <span><math><mi>G</mi></math></span>. The pendant-tree <span><math><mi>k</mi></math></span>-connectivity is a generalization of traditional connectivity. In this paper, we mainly investigate the pendant-tree 4-connectivity of the regular graph with given properties, denoted by <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, which was introduced in Zhao and Hao (2019). As applications of the main result, the pendant-tree 4","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 352-360"},"PeriodicalIF":1.0,"publicationDate":"2026-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980016","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}
Pub Date : 2026-01-15DOI: 10.1016/j.dam.2026.01.002
Yushuang Mou , Qiang Sun , Chao Zhang
Weighted signed networks capture both positive and negative relationships between individuals, with link weights representing the intensity of these relationships. We model cooperation in such networks as a cooperative game restricted by a weighted signed network. To address the distribution problem in these games, we introduce the weighted signed Myerson value (WS-Myerson value), which is grounded in structural balance theory and incorporates the minimum cost required to achieve balance within the network. We prove that the WS-Myerson value is uniquely determined by the axioms of component efficiency, fairness for conflict players, and marginality.
{"title":"The Myerson value for games with weighted signed networks","authors":"Yushuang Mou , Qiang Sun , Chao Zhang","doi":"10.1016/j.dam.2026.01.002","DOIUrl":"10.1016/j.dam.2026.01.002","url":null,"abstract":"<div><div>Weighted signed networks capture both positive and negative relationships between individuals, with link weights representing the intensity of these relationships. We model cooperation in such networks as a cooperative game restricted by a weighted signed network. To address the distribution problem in these games, we introduce the weighted signed Myerson value (WS-Myerson value), which is grounded in structural balance theory and incorporates the minimum cost required to achieve balance within the network. We prove that the WS-Myerson value is uniquely determined by the axioms of component efficiency, fairness for conflict players, and marginality.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 400-410"},"PeriodicalIF":1.0,"publicationDate":"2026-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145976747","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}
Pub Date : 2026-01-14DOI: 10.1016/j.dam.2025.12.068
Bernard De Baets , Emilio De Santis
A set of candidates is presented to a commission. At every round, each member of this commission votes by pairwise comparison, and one-half of the candidates is deleted from the tournament, the remaining ones proceeding to the next round until the th round (the final one) in which the final winner is declared. The candidates are arranged on a board in a given order, which is maintained among the remaining candidates at all rounds. A study of the size of the commission is carried out in order to obtain the desired result of any candidate being a possible winner. For candidates with , we identify a voting profile with voters such that any candidate could win simply by choosing a proper initial order of the candidates. Moreover, in the setting of a random number of voters, we obtain the same results, with high probability, when the expected number of voters is large.
{"title":"Voting profiles admitting all candidates as knockout winners","authors":"Bernard De Baets , Emilio De Santis","doi":"10.1016/j.dam.2025.12.068","DOIUrl":"10.1016/j.dam.2025.12.068","url":null,"abstract":"<div><div>A set of <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span> candidates is presented to a commission. At every round, each member of this commission votes by pairwise comparison, and one-half of the candidates is deleted from the tournament, the remaining ones proceeding to the next round until the <span><math><mi>n</mi></math></span>th round (the final one) in which the final winner is declared. The candidates are arranged on a board in a given order, which is maintained among the remaining candidates at all rounds. A study of the size of the commission is carried out in order to obtain the desired result of any candidate being a possible winner. For <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span> candidates with <span><math><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, we identify a voting profile with <span><math><mrow><mn>4</mn><mi>n</mi><mo>−</mo><mn>3</mn></mrow></math></span> voters such that any candidate could win simply by choosing a proper initial order of the candidates. Moreover, in the setting of a random number of voters, we obtain the same results, with high probability, when the expected number of voters is large.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 340-351"},"PeriodicalIF":1.0,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980019","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}
Pub Date : 2026-01-13DOI: 10.1016/j.dam.2025.12.054
Deqing Xu , Bo Deng , HongJian Lai
The Tutte polynomial of a graph is a fundamental invariant that effectively reflects certain characteristics and properties of the graph, which is also a very important tool for studying other graph parameters. By assigning values to the two variables in the Tutte polynomial, one can obtain combinatorial interpretations of various graph parameters such as the number of spanning trees. It is well known that the computation of the Tutte polynomial of a graph is NP-hard. In this research, a matrix–vector multiplication algorithm with the computational complexity is given to compute the Tutte polynomial for a pericondensed system with hexagons. And the matrix–vector multiplication is used to derive the precise formulation of the Tutte polynomials for single interior pericondensed hexagonal systems, which are applied to explore their spanning trees, chromatic polynomial and flow polynomial.
{"title":"Tutte polynomials of single interior pericondensed hexagonal systems","authors":"Deqing Xu , Bo Deng , HongJian Lai","doi":"10.1016/j.dam.2025.12.054","DOIUrl":"10.1016/j.dam.2025.12.054","url":null,"abstract":"<div><div>The Tutte polynomial of a graph is a fundamental invariant that effectively reflects certain characteristics and properties of the graph, which is also a very important tool for studying other graph parameters. By assigning values to the two variables in the Tutte polynomial, one can obtain combinatorial interpretations of various graph parameters such as the number of spanning trees. It is well known that the computation of the Tutte polynomial of a graph is NP-hard. In this research, a matrix–vector multiplication algorithm with the computational complexity <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mo>log</mo><mi>N</mi><mo>)</mo></mrow></mrow></math></span> is given to compute the Tutte polynomial for a pericondensed system with <span><math><mi>N</mi></math></span> hexagons. And the matrix–vector multiplication is used to derive the precise formulation of the Tutte polynomials for single interior pericondensed hexagonal systems, which are applied to explore their spanning trees, chromatic polynomial and flow polynomial.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 326-339"},"PeriodicalIF":1.0,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980017","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}
Pub Date : 2026-01-13DOI: 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 , the push operation on a vertex reverses the orientation of all arcs incident to . 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 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 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 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。
{"title":"Pushing Cops and Robber on graphs of maximum degree four","authors":"Harmender Gahlawat","doi":"10.1016/j.dam.2025.12.067","DOIUrl":"10.1016/j.dam.2025.12.067","url":null,"abstract":"<div><div><span>Cops and Robber</span> is a game played on graphs where a set of <em>cops</em> aim to <em>capture</em> the position of a single <em>robber</em>. The main parameter of interest in this game is the <em>cop number</em>, which is the minimum number of cops that are sufficient to guarantee the capture of the robber.</div><div>In a directed graph <span><math><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover></math></span>, the <em>push</em> operation on a vertex <span><math><mi>v</mi></math></span> reverses the orientation of all arcs incident to <span><math><mi>v</mi></math></span>. We consider a variation of the classical <span>Cops and Robber</span> 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 <span><math><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover></math></span> 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 <span><math><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover></math></span> 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 <span><math><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover></math></span> can be made to be a directed acyclic graph, then one cop with push ability has a winning strategy.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 327-338"},"PeriodicalIF":1.0,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977708","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}
Pub Date : 2026-01-13DOI: 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 -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, -ary -cube, and bubble sort graph. In addition, we propose a novel diagnosis algorithm, NFDAM, for non-inclusive -extra diagnosis, which runs in polynomial time with a complexity of , where denotes the maximum degree of graph . Simulation results demonstrate the effectiveness of the proposed algorithm in fault identification.
{"title":"Non-inclusive g-extra diagnosability of interconnection networks under MM* model","authors":"Weixing Zheng , Shuming Zhou , Eddie Cheng","doi":"10.1016/j.dam.2025.12.069","DOIUrl":"10.1016/j.dam.2025.12.069","url":null,"abstract":"<div><div>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 <span><math><mi>g</mi></math></span>-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, <span><math><mi>k</mi></math></span>-ary <span><math><mi>n</mi></math></span>-cube, and bubble sort graph. In addition, we propose a novel diagnosis algorithm, NFDAM, for non-inclusive <span><math><mi>g</mi></math></span>-extra diagnosis, which runs in polynomial time with a complexity of <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo><mi>⋅</mi><msup><mrow><mrow><mo>(</mo><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> denotes the maximum degree of graph <span><math><mi>G</mi></math></span>. Simulation results demonstrate the effectiveness of the proposed algorithm in fault identification.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 315-326"},"PeriodicalIF":1.0,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977710","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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