Pub Date : 2026-01-21DOI: 10.1016/j.dam.2026.01.029
Kun Cheng
A well-known result of Chvátal and Erdős from 1972 states that a graph with connectivity not less than its independence number plus one is hamiltonian-connected. A graph is called an -graph if any induced subgraph of of order has size at least We prove that every -connected -graph is hamiltonian-connected except where and is an arbitrary graph of order . This generalizes the Chvátal–Erdős theorem.
{"title":"A generalization of the Chvátal–Erdős theorem","authors":"Kun Cheng","doi":"10.1016/j.dam.2026.01.029","DOIUrl":"10.1016/j.dam.2026.01.029","url":null,"abstract":"<div><div>A well-known result of Chvátal and Erdős from 1972 states that a graph with connectivity not less than its independence number plus one is hamiltonian-connected. A graph <span><math><mi>G</mi></math></span> is called an <span><math><mrow><mo>[</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>]</mo></mrow></math></span>-graph if any induced subgraph of <span><math><mi>G</mi></math></span> of order <span><math><mi>s</mi></math></span> has size at least <span><math><mrow><mi>t</mi><mo>.</mo></mrow></math></span> We prove that every <span><math><mi>k</mi></math></span>-connected <span><math><mrow><mo>[</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>]</mo></mrow></math></span>-graph is hamiltonian-connected except <span><math><mrow><mi>k</mi><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∨</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo></mrow></math></span> where <span><math><mrow><mi>k</mi><mo>≥</mo><mn>2</mn></mrow></math></span> and <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is an arbitrary graph of order <span><math><mi>k</mi></math></span>. This generalizes the Chvátal–Erdős theorem.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"385 ","pages":"Pages 65-71"},"PeriodicalIF":1.0,"publicationDate":"2026-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039495","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.2025.12.056
Michael A. Henning , Paras Vinubhai Maniya , Dinabandhu Pradhan
A disjunctive dominating set of a graph is a set such that every vertex in has a neighbor in or has at least two vertices in at distance 2 from it. The disjunctive domination number of , denoted by , is the minimum cardinality among all disjunctive dominating sets of . In this paper, we show that if is a maximal outerplanar graph of order with vertices of degree 2, then , and this bound is sharp.
{"title":"Disjunctive domination in maximal outerplanar graphs","authors":"Michael A. Henning , Paras Vinubhai Maniya , Dinabandhu Pradhan","doi":"10.1016/j.dam.2025.12.056","DOIUrl":"10.1016/j.dam.2025.12.056","url":null,"abstract":"<div><div>A disjunctive dominating set of a graph <span><math><mi>G</mi></math></span> is a set <span><math><mrow><mi>D</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> such that every vertex in <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>∖</mo><mi>D</mi></mrow></math></span> has a neighbor in <span><math><mi>D</mi></math></span> or has at least two vertices in <span><math><mi>D</mi></math></span> at distance 2 from it. The disjunctive domination number of <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><msubsup><mrow><mi>γ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is the minimum cardinality among all disjunctive dominating sets of <span><math><mi>G</mi></math></span>. In this paper, we show that if <span><math><mi>G</mi></math></span> is a maximal outerplanar graph of order <span><math><mrow><mi>n</mi><mo>≥</mo><mn>7</mn></mrow></math></span> with <span><math><mi>k</mi></math></span> vertices of degree 2, then <span><math><mrow><msubsup><mrow><mi>γ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mrow><mo>⌊</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>9</mn></mrow></mfrac><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo>)</mo></mrow><mo>⌋</mo></mrow></mrow></math></span>, and this bound is sharp.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"385 ","pages":"Pages 24-61"},"PeriodicalIF":1.0,"publicationDate":"2026-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039493","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.2026.01.004
Chuanye Zheng, Liqiong Xu
<div><div>As multiprocessor systems scale up in size and complexity to meet increasing computational demands, link or processor failures are inevitable. Thus reliability of multiprocessor systems needs to be considered. Restricting the surviving components within multiprocessor systems can enhance the evaluation of their reliability. Recently, Yang et al. introduced a new parameter called <span><math><mi>h</mi></math></span>-extra <span><math><mi>r</mi></math></span>-component edge-connectivity, which requires that for a connected graph <span><math><mi>G</mi></math></span> and an edge-cut <span><math><mi>F</mi></math></span> of <span><math><mi>G</mi></math></span>, there exist at least <span><math><mi>r</mi></math></span> components surviving in <span><math><mrow><mi>G</mi><mo>−</mo><mi>F</mi></mrow></math></span> and the order of each component is not less than <span><math><mi>h</mi></math></span>. In this paper, we consider the <span><math><mi>h</mi></math></span>-extra <span><math><mi>r</mi></math></span>-component edge-connectivity of the 3-ary <span><math><mi>n</mi></math></span>-cube <span><math><msubsup><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msubsup></math></span> and determine that <span><math><mrow><mi>c</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn>4</mn></mrow><mrow><mi>h</mi></mrow></msubsup><mrow><mo>(</mo><msubsup><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>)</mo></mrow><mo>=</mo><mn>6</mn><mi>n</mi><mi>h</mi><mo>−</mo><mn>3</mn><mi>e</mi><msub><mrow><mi>x</mi></mrow><mrow><mi>h</mi></mrow></msub><mrow><mo>(</mo><msubsup><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>)</mo></mrow><mo>−</mo><mn>3</mn><mi>h</mi></mrow></math></span> for <span><math><mrow><mi>n</mi><mo>≥</mo><mn>5</mn></mrow></math></span> and <span><math><mrow><mn>1</mn><mo>≤</mo><mi>h</mi><mo>≤</mo><mi>δ</mi><mi>⋅</mi><msup><mrow><mn>3</mn></mrow><mrow><mrow><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow></math></span> where <span><math><mrow><mi>δ</mi><mo>=</mo><mn>1</mn></mrow></math></span> if <span><math><mi>n</mi></math></span> is odd and <span><math><mrow><mi>δ</mi><mo>=</mo><mn>2</mn></mrow></math></span> if <span><math><mi>n</mi></math></span> is even, <span><math><mrow><mi>c</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>r</mi></mrow><mrow><msup><mrow><mn>3</mn></mrow><mrow><mi>k</mi></mrow></msup></mrow></msubsup><mrow><mo>(</mo><msubsup><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn><mi>k</mi><mo>)</mo></mrow><msup><mrow><mn>3</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>e</mi><msub><mrow><mi>x</mi></mrow><m
随着多处理器系统的规模和复杂性不断扩大，以满足不断增长的计算需求，链路或处理器故障是不可避免的。因此，需要考虑多处理器系统的可靠性。限制多处理机系统中幸存部件的数量可以提高对系统可靠性的评估。最近,杨等人提出了一个新的参数称为h-extra r-component edge,这要求一个连通图G和edge-cut F (G,存在至少r G−F幸存的组件,每个组件的顺序不小于h。在本文中,我们考虑h-extra r-component edge的3-ary n立方体Qn3和确定cλ4 h (Qn3) = 6 nh−3 exh (Qn3)−3 h n≥5和1 h≤≤δ⋅3⌈n2⌉−−1δ= 1如果n是奇怪和δ= 2如果n是偶数,cλr3k (Qn3) = (r−1)(2 n−2 k) 3 k 12 exr−−1 (Qn3)⋅3 k 1≤(r−1)3 k≤δ⋅3⌈n2⌉−1和cλ43 k (Qn3) = (2 n−2 k−1)3 k + 1 0≤k≤n−2。
{"title":"Assessing reliability of 3-ary n-cubes based on the h-extra r-component edge-connectivity","authors":"Chuanye Zheng, Liqiong Xu","doi":"10.1016/j.dam.2026.01.004","DOIUrl":"10.1016/j.dam.2026.01.004","url":null,"abstract":"<div><div>As multiprocessor systems scale up in size and complexity to meet increasing computational demands, link or processor failures are inevitable. Thus reliability of multiprocessor systems needs to be considered. Restricting the surviving components within multiprocessor systems can enhance the evaluation of their reliability. Recently, Yang et al. introduced a new parameter called <span><math><mi>h</mi></math></span>-extra <span><math><mi>r</mi></math></span>-component edge-connectivity, which requires that for a connected graph <span><math><mi>G</mi></math></span> and an edge-cut <span><math><mi>F</mi></math></span> of <span><math><mi>G</mi></math></span>, there exist at least <span><math><mi>r</mi></math></span> components surviving in <span><math><mrow><mi>G</mi><mo>−</mo><mi>F</mi></mrow></math></span> and the order of each component is not less than <span><math><mi>h</mi></math></span>. In this paper, we consider the <span><math><mi>h</mi></math></span>-extra <span><math><mi>r</mi></math></span>-component edge-connectivity of the 3-ary <span><math><mi>n</mi></math></span>-cube <span><math><msubsup><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msubsup></math></span> and determine that <span><math><mrow><mi>c</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn>4</mn></mrow><mrow><mi>h</mi></mrow></msubsup><mrow><mo>(</mo><msubsup><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>)</mo></mrow><mo>=</mo><mn>6</mn><mi>n</mi><mi>h</mi><mo>−</mo><mn>3</mn><mi>e</mi><msub><mrow><mi>x</mi></mrow><mrow><mi>h</mi></mrow></msub><mrow><mo>(</mo><msubsup><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>)</mo></mrow><mo>−</mo><mn>3</mn><mi>h</mi></mrow></math></span> for <span><math><mrow><mi>n</mi><mo>≥</mo><mn>5</mn></mrow></math></span> and <span><math><mrow><mn>1</mn><mo>≤</mo><mi>h</mi><mo>≤</mo><mi>δ</mi><mi>⋅</mi><msup><mrow><mn>3</mn></mrow><mrow><mrow><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow></math></span> where <span><math><mrow><mi>δ</mi><mo>=</mo><mn>1</mn></mrow></math></span> if <span><math><mi>n</mi></math></span> is odd and <span><math><mrow><mi>δ</mi><mo>=</mo><mn>2</mn></mrow></math></span> if <span><math><mi>n</mi></math></span> is even, <span><math><mrow><mi>c</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>r</mi></mrow><mrow><msup><mrow><mn>3</mn></mrow><mrow><mi>k</mi></mrow></msup></mrow></msubsup><mrow><mo>(</mo><msubsup><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn><mi>k</mi><mo>)</mo></mrow><msup><mrow><mn>3</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>e</mi><msub><mrow><mi>x</mi></mrow><m","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 361-371"},"PeriodicalIF":1.0,"publicationDate":"2026-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146024235","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.2026.01.007
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}
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