Pub Date : 2026-01-05DOI: 10.1016/j.dam.2025.12.049
Yifan Li , Shuming Zhou , Qifan Zhang
For a network , the subversion at the vertex set (resp., edge set) of is defined as the removal of the closed neighborhood of the vertex set (resp., all end vertices of the edge set) from , where the vertex set (resp., edge set) is referred as subverted vertices (resp., edges). Neighbor connectivity and edge neighbor connectivity serve as key indicators for assessing the subversion of spy networks and network disruptions throughout the deletion of closed neighborhood. The neighbor connectivity (resp., edge neighbor connectivity ) of a network is defined as the minimum number of subverted vertices (resp., edges) required to disconnect it, make it empty or complete (resp., trivial). Gu et al. (IEEE Trans. Netw. Sci. Eng. 11 (5) (2024) 1-13) conjectured that whether holds for all compound graphs constructed by the underlying block . In this paper, we solve this conjecture and determine the (edge) neighbor connectivity of a class of hypercube-based compound network, including half hypercube, hierarchical hypercube, hierarchical cubic network and dual-cube-like network. In addition, we present network vulnerability analysis algorithms based on neighborhood fault pattern. To evaluate their effectiveness, taking the half hypercube, hierarchical cubic network and real-world network dwt-918 as examples, we perform experimental simulations to analyze both the cardinality distribution of subverted vertices and topological configurations of survival graph.
对于网络G，在顶点集(p。，边集)定义为顶点集（resp.）的闭邻域的移除。，边集的所有端点)来自G，其中顶点集(resp。，边集)被称为颠覆顶点(如。,边缘)。邻居连通性和边缘邻居连通性是评估间谍网络颠覆和网络中断的关键指标。邻居连通性κNB(G)。，网络G的边缘邻居连通性λNB(G)定义为颠覆顶点的最小个数(p。需要断开它，使其为空或完整(参见。琐碎的)。Gu et al. （IEEE译）Netw。科学。Eng. 11（5)(2024) 1-13）推测κNB(G)=δ(G)−12+1是否对所有由底层块Qn构造的复合图G成立。本文解决了这一猜想，并确定了一类基于超立方体的复合网络（包括半超立方体、分层超立方体、分层立方网络和双类立方体网络）的（边）邻居连通性。此外，提出了基于邻域故障模式的网络漏洞分析算法。为了评估它们的有效性，我们以半超立方体、分层立方网络和现实世界网络dwt-918为例，进行了实验模拟，分析了颠覆顶点的基数分布和生存图的拓扑构型。
{"title":"Neighbor connectivity of hypercube-based compound network","authors":"Yifan Li , Shuming Zhou , Qifan Zhang","doi":"10.1016/j.dam.2025.12.049","DOIUrl":"10.1016/j.dam.2025.12.049","url":null,"abstract":"<div><div>For a network <span><math><mi>G</mi></math></span>, the subversion at the vertex set (resp., edge set) of <span><math><mi>G</mi></math></span> is defined as the removal of the closed neighborhood of the vertex set (resp., all end vertices of the edge set) from <span><math><mi>G</mi></math></span>, where the vertex set (resp., edge set) is referred as subverted vertices (resp., edges). Neighbor connectivity and edge neighbor connectivity serve as key indicators for assessing the subversion of spy networks and network disruptions throughout the deletion of closed neighborhood. The neighbor connectivity <span><math><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mi>N</mi><mi>B</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> (resp., edge neighbor connectivity <span><math><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>N</mi><mi>B</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>) of a network <span><math><mi>G</mi></math></span> is defined as the minimum number of subverted vertices (resp., edges) required to disconnect it, make it empty or complete (resp., trivial). Gu et al. (IEEE Trans. Netw. Sci. Eng. 11 (5) (2024) 1-13) conjectured that whether <span><math><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mi>NB</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mfenced><mrow><mfrac><mrow><mi>δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mo>+</mo><mn>1</mn></mrow></math></span> holds for all compound graphs <span><math><mi>G</mi></math></span> constructed by the underlying block <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. In this paper, we solve this conjecture and determine the (edge) neighbor connectivity of a class of hypercube-based compound network, including half hypercube, hierarchical hypercube, hierarchical cubic network and dual-cube-like network. In addition, we present network vulnerability analysis algorithms based on neighborhood fault pattern. To evaluate their effectiveness, taking the half hypercube, hierarchical cubic network and real-world network dwt-918 as examples, we perform experimental simulations to analyze both the cardinality distribution of subverted vertices and topological configurations of survival graph.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 1-15"},"PeriodicalIF":1.0,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145898206","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 : 2026-01-05DOI: 10.1016/j.dam.2025.12.044
Bilal Ahmad Rather , Hilal Ahmad Ganie , Jainfeng Wang
In a connected graph , the distance signless Laplacian is defined as , where is the diagonal matrix of vertex transmissions and is the distance matrix indexed by the vertices of , such that , where represents the distance between the vertices and . Motivated by the Laplacian and signless Laplacian matrices of , Aouchiche and Hensen (2013) developed the idea of distance (signless) Laplacian matrix, which has attracted the interest among numerous spectral graph theory researchers in the field of algebraic graph theory. The spectral investigation of resulted in numerous articles. In this paper, we present a review of research on the distance signless Laplacian of connected graphs.
{"title":"Distance signless Laplacian spectra of graphs: A survey","authors":"Bilal Ahmad Rather , Hilal Ahmad Ganie , Jainfeng Wang","doi":"10.1016/j.dam.2025.12.044","DOIUrl":"10.1016/j.dam.2025.12.044","url":null,"abstract":"<div><div>In a connected graph <span><math><mi>G</mi></math></span>, the distance signless Laplacian is defined as <span><math><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>Q</mi></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>D</mi><mi>i</mi><mi>a</mi><mi>g</mi><mrow><mo>(</mo><mi>T</mi><mi>r</mi><mo>)</mo></mrow><mo>+</mo><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>D</mi><mi>i</mi><mi>a</mi><mi>g</mi><mrow><mo>(</mo><mi>T</mi><mi>r</mi><mo>)</mo></mrow></mrow></math></span> is the diagonal matrix of vertex transmissions and <span><math><mrow><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mrow><mo>(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub></mrow></math></span> is the distance matrix indexed by the vertices of <span><math><mi>G</mi></math></span>, such that <span><math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>=</mo><mi>d</mi><mrow><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>d</mi><mrow><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> represents the distance between the vertices <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>. Motivated by the Laplacian and signless Laplacian matrices of <span><math><mi>G</mi></math></span>, Aouchiche and Hensen (2013) developed the idea of distance (signless) Laplacian matrix, which has attracted the interest among numerous spectral graph theory researchers in the field of algebraic graph theory. The spectral investigation of <span><math><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>Q</mi></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> resulted in numerous articles. In this paper, we present a review of research on the distance signless Laplacian of connected graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 41-138"},"PeriodicalIF":1.0,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145898207","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 : 2026-01-05DOI: 10.1016/j.dam.2025.12.050
Gideon Amir , Yaakov Malinovsky
In this note, we extend a recent result on the uniqueness of the maximum score in a classical round-robin tournament to general round-robin tournament models with equally strong players, where the scores take values in .
{"title":"Uniqueness of maximum scores in countable-outcome round-robin tournaments","authors":"Gideon Amir , Yaakov Malinovsky","doi":"10.1016/j.dam.2025.12.050","DOIUrl":"10.1016/j.dam.2025.12.050","url":null,"abstract":"<div><div>In this note, we extend a recent result on the uniqueness of the maximum score in a classical round-robin tournament to general round-robin tournament models with equally strong players, where the scores take values in <span><math><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mspace></mspace><mn>1</mn><mo>]</mo></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 16-22"},"PeriodicalIF":1.0,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145898208","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 : 2026-01-05DOI: 10.1016/j.dam.2025.12.040
Chenke Zhang , Qing Cui , Jinze Hu , Erfei Yue , Shengjin Ji
Let be a graph and be a family of graphs. We say a graph is -saturated if does not contain any member in and for any , creates a copy of some member in . The saturation number of is the minimum number of edges of an -saturated graphs of vertices, denoted by . If , then we write it as for short. In this paper, we determine the exact value of , and as its application, we obtain two bounds on for and sufficiently large . Furthermore, is determined, where is a linear forest without isolated vertices.
{"title":"Some results on minimum saturated graphs","authors":"Chenke Zhang , Qing Cui , Jinze Hu , Erfei Yue , Shengjin Ji","doi":"10.1016/j.dam.2025.12.040","DOIUrl":"10.1016/j.dam.2025.12.040","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a graph and <span><math><mi>F</mi></math></span> be a family of graphs. We say a graph <span><math><mi>G</mi></math></span> is <span><math><mi>F</mi></math></span>-saturated if <span><math><mi>G</mi></math></span> does not contain any member in <span><math><mi>F</mi></math></span> and for any <span><math><mrow><mi>e</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mover><mrow><mi>G</mi></mrow><mo>¯</mo></mover><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>G</mi><mo>+</mo><mi>e</mi></mrow></math></span> creates a copy of some member in <span><math><mi>F</mi></math></span>. The saturation number of <span><math><mi>F</mi></math></span> is the minimum number of edges of an <span><math><mi>F</mi></math></span>-saturated graphs of <span><math><mi>n</mi></math></span> vertices, denoted by <span><math><mrow><mi>sat</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span>. If <span><math><mrow><mi>F</mi><mo>=</mo><mrow><mo>{</mo><mi>F</mi><mo>}</mo></mrow></mrow></math></span>, then we write it as <span><math><mrow><mi>sat</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> for short. In this paper, we determine the exact value of <span><math><mrow><mi>sat</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mrow><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span>, and as its application, we obtain two bounds on <span><math><mrow><mi>sat</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>∪</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>k</mi><mo>≥</mo><mn>10</mn></mrow></math></span> and sufficiently large <span><math><mi>n</mi></math></span>. Furthermore, <span><math><mrow><mi>sat</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∨</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> is determined, where <span><math><mi>F</mi></math></span> is a linear forest without isolated vertices.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 23-33"},"PeriodicalIF":1.0,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145898209","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-29DOI: 10.1016/j.dam.2025.12.043
Yong-De Feng, Yawen Chen, Baoyindureng Wu
An odd graph is a graph for which every vertex satisfies . An odd spanning tree of is a spanning tree such that for all . It is known that for any complete graph of even order has an odd spanning tree. In this paper, we establish the exact number of labeled odd spanning trees in . By employing the classical Prüfer sequence and constructing the corresponding generating function, we prove that the number of labeled odd spanning trees in is given by (where is even).
{"title":"The number of odd spanning trees in the complete graphs","authors":"Yong-De Feng, Yawen Chen, Baoyindureng Wu","doi":"10.1016/j.dam.2025.12.043","DOIUrl":"10.1016/j.dam.2025.12.043","url":null,"abstract":"<div><div>An odd graph is a graph <span><math><mi>G</mi></math></span> for which every vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> satisfies <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>≡</mo><mn>1</mn><mspace></mspace><mrow><mo>(</mo><mo>mod</mo><mspace></mspace><mn>2</mn><mo>)</mo></mrow></mrow></math></span>. An odd spanning tree <span><math><mi>T</mi></math></span> of <span><math><mi>G</mi></math></span> is a spanning tree such that <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>T</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>≡</mo><mn>1</mn><mspace></mspace><mrow><mo>(</mo><mo>mod</mo><mspace></mspace><mn>2</mn><mo>)</mo></mrow></mrow></math></span> for all <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow></mrow></math></span>. It is known that for any complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of even order has an odd spanning tree. In this paper, we establish the exact number of labeled odd spanning trees in <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. By employing the classical Prüfer sequence and constructing the corresponding generating function, we prove that the number of labeled odd spanning trees in <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is given by <span><math><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></mfrac><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mfenced><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></mfenced><msup><mrow><mrow><mo>(</mo><mn>2</mn><mi>k</mi><mo>−</mo><mi>n</mi><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup></mrow></math></span>\u0000 <!--> <!-->(where <span><math><mi>n</mi></math></span> is even).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 295-299"},"PeriodicalIF":1.0,"publicationDate":"2025-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145885019","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 : 2025-12-26DOI: 10.1016/j.dam.2025.12.022
Louis Deaett , Derek Young
For a simple graph, the minimum rank problem is to determine the smallest rank among the symmetric matrices whose off-diagonal nonzero entries occur in positions corresponding to the edges of the graph. Bounds on this minimum rank (and on an equivalent value, the maximum nullity) are given by various graph parameters, most notably the zero forcing number and its variants. For a matrix, replacing each nonzero entry with the symbol gives its zero-nonzero pattern. The associated minimum rank problem is to determine, given only this pattern, the smallest possible rank of the matrix. The most fundamental lower bound on this minimum rank is the triangle number of the pattern. A cobipartite graph is the complement of a bipartite graph; its vertices can be partitioned into two cliques. Such a graph corresponds to a zero-nonzero pattern in a natural way. Over an infinite field, the maximum nullity of the graph and the minimum rank of the pattern obey a simple relationship. We show that the zero forcing number of the graph and the triangle number of the pattern follow this same relationship. This has implications for the relationship between the two minimum rank problems. We also explore how, for cobipartite graphs, variants of the zero forcing number and other parameters important to the minimum rank problem are related, as well as how, for graphs in general, these parameters can be interpreted in terms of the zero-nonzero patterns of the symmetric matrices associated with the graph.
{"title":"Relationships between minimum rank problem parameters for cobipartite graphs","authors":"Louis Deaett , Derek Young","doi":"10.1016/j.dam.2025.12.022","DOIUrl":"10.1016/j.dam.2025.12.022","url":null,"abstract":"<div><div>For a simple graph, the minimum rank problem is to determine the smallest rank among the symmetric matrices whose off-diagonal nonzero entries occur in positions corresponding to the edges of the graph. Bounds on this minimum rank (and on an equivalent value, the maximum nullity) are given by various graph parameters, most notably the zero forcing number and its variants. For a matrix, replacing each nonzero entry with the symbol <span><math><mo>∗</mo></math></span> gives its zero-nonzero pattern. The associated minimum rank problem is to determine, given only this pattern, the smallest possible rank of the matrix. The most fundamental lower bound on this minimum rank is the triangle number of the pattern. A cobipartite graph is the complement of a bipartite graph; its vertices can be partitioned into two cliques. Such a graph corresponds to a zero-nonzero pattern in a natural way. Over an infinite field, the maximum nullity of the graph and the minimum rank of the pattern obey a simple relationship. We show that the zero forcing number of the graph and the triangle number of the pattern follow this same relationship. This has implications for the relationship between the two minimum rank problems. We also explore how, for cobipartite graphs, variants of the zero forcing number and other parameters important to the minimum rank problem are related, as well as how, for graphs in general, these parameters can be interpreted in terms of the zero-nonzero patterns of the symmetric matrices associated with the graph.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"381 ","pages":"Pages 378-397"},"PeriodicalIF":1.0,"publicationDate":"2025-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145883704","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 : 2025-12-24DOI: 10.1016/j.dam.2025.12.035
Ambroise Baril , Miguel Couceiro , Victor Lagerkvist
Semiring algebras have been shown to provide a suitable language to formalize many noteworthy combinatorial problems. For instance, the Shortest-Path problem can be seen as a special case of the Algebraic-Path problem when applied to the tropical semiring. The application of semirings typically makes it possible to solve extended problems without increasing the computational complexity. In this article we further exploit the idea of using semiring algebras to address and tackle several extensions of classical computational problems by dynamic programming.
We consider a general approach which allows us to define a semiring extension of any problem with a reasonable notion of a certificate (e.g., an NP problem). This allows us to consider cost variants of these combinatorial problems, as well as their counting extensions where the goal is to determine how many solutions a given problem admits. The approach makes no particular assumptions (such as idempotence) on the semiring structure. We also propose a new associative algebraic operation on semirings, called -product, which enables our dynamic programming algorithms to count the number of solutions of minimal costs. We illustrate the advantages of our framework on two well-known but computationally very different NP-hard problems, namely, Connected-Dominating-Set problems and finite-domain Constraint Satisfaction Problems (Csps). In particular, we prove fixed parameter tractability (FPT) with respect to clique-width and tree-width of the input. This also allows us to count solutions of minimal cost, which is an overlooked problem in the literature.
{"title":"New perspectives on semiring applications to dynamic programming","authors":"Ambroise Baril , Miguel Couceiro , Victor Lagerkvist","doi":"10.1016/j.dam.2025.12.035","DOIUrl":"10.1016/j.dam.2025.12.035","url":null,"abstract":"<div><div>Semiring algebras have been shown to provide a suitable language to formalize many noteworthy combinatorial problems. For instance, the <span>Shortest-Path</span> problem can be seen as a special case of the <span>Algebraic-Path</span> problem when applied to the tropical semiring. The application of semirings typically makes it possible to solve extended problems without increasing the computational complexity. In this article we further exploit the idea of using semiring algebras to address and tackle several extensions of classical computational problems by dynamic programming.</div><div>We consider a general approach which allows us to define a semiring extension of <em>any</em> problem with a reasonable notion of a certificate (e.g., an <span>NP</span> problem). This allows us to consider cost variants of these combinatorial problems, as well as their counting extensions where the goal is to determine how many solutions a given problem admits. The approach makes no particular assumptions (such as idempotence) on the semiring structure. We also propose a new associative algebraic operation on semirings, called <span><math><mi>Δ</mi></math></span>-product, which enables our dynamic programming algorithms to count the number of solutions of minimal costs. We illustrate the advantages of our framework on two well-known but computationally very different <span>NP</span>-hard problems, namely, <span>Connected-Dominating-Set</span> problems and finite-domain <span>Constraint Satisfaction Problems</span> (<span>Csp</span>s). In particular, we prove fixed parameter tractability (<span>FPT</span>) with respect to clique-width and tree-width of the input. This also allows us to count solutions of minimal cost, which is an overlooked problem in the literature.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 243-279"},"PeriodicalIF":1.0,"publicationDate":"2025-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842076","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}
In this work, assuming that preferences are asymmetric, complete and transitive, we aim to prove the equivalence, in bilateral conflicts, between the concepts of Maximin stability and Generalized Metarationality within the graph model for conflict resolution (GMCR). This result is surprising, as these concepts are apparently distinct, due to the movements of the opponent of the focal DM, in the Generalized Metarationality, be limited by the use of a policy. To achieve our goal, we define a Maximin tree to build maximin policies from the choices made by the focal DM opponent in this tree. We show that two kinds of maximin policies are necessary: one for conflict analysis with odd horizon and another for even horizon conflict analysis. This result, in addition to improving our understanding of these concepts, facilitates the determination of states that satisfy generalized metarationality, since maximin states can be determined by means of a backward induction procedure.
{"title":"On the relation between Maximinh stability and Generalized Metarationality in bilateral conflicts","authors":"Alecio Soares Silva , Giannini Italino Alves Vieira , Leandro Chaves Rêgo","doi":"10.1016/j.dam.2025.12.041","DOIUrl":"10.1016/j.dam.2025.12.041","url":null,"abstract":"<div><div>In this work, assuming that preferences are asymmetric, complete and transitive, we aim to prove the equivalence, in bilateral conflicts, between the concepts of Maximin<span><math><msub><mrow></mrow><mrow><mi>h</mi></mrow></msub></math></span> stability and Generalized Metarationality within the graph model for conflict resolution (GMCR). This result is surprising, as these concepts are apparently distinct, due to the movements of the opponent of the focal DM, in the Generalized Metarationality, be limited by the use of a policy. To achieve our goal, we define a Maximin<span><math><msub><mrow></mrow><mrow><mi>h</mi></mrow></msub></math></span> tree to build maximin policies from the choices made by the focal DM opponent in this tree. We show that two kinds of maximin policies are necessary: one for conflict analysis with odd horizon and another for even horizon conflict analysis. This result, in addition to improving our understanding of these concepts, facilitates the determination of states that satisfy generalized metarationality, since maximin states can be determined by means of a backward induction procedure.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 280-294"},"PeriodicalIF":1.0,"publicationDate":"2025-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842077","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}
We give an algorithm that finds a zero forcing set which approximates the optimal size by a factor of , where is the pathwidth of . The algorithm requires a path decomposition of , and given this it runs in time, where and are the order and size of the graph, respectively. This is the first zero forcing algorithm with a guarantee on both the approximation ratio and on the run-time. As a corollary, we obtain a new upper bound on the zero forcing number in terms of the fort number and the pathwidth. The algorithm is based on a correspondence between zero forcing sets and forcing arc sets. This correspondence leads to a new bound on the zero forcing number in terms of vertex cuts, and to new, short proofs for known bounds on the zero forcing number.
{"title":"An approximation algorithm for zero forcing","authors":"Ben Cameron , Jeannette Janssen , Rogers Mathew , Zhiyuan Zhang","doi":"10.1016/j.dam.2025.12.013","DOIUrl":"10.1016/j.dam.2025.12.013","url":null,"abstract":"<div><div>We give an algorithm that finds a zero forcing set which approximates the optimal size by a factor of <span><math><mrow><mi>pw</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow></math></span>, where <span><math><mrow><mi>pw</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is the pathwidth of <span><math><mi>G</mi></math></span>. The algorithm requires a path decomposition of <span><math><mi>G</mi></math></span>, and given this it runs in <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>n</mi><mi>m</mi><mo>)</mo></mrow></mrow></math></span> time, where <span><math><mi>n</mi></math></span> and <span><math><mi>m</mi></math></span> are the order and size of the graph, respectively. This is the first zero forcing algorithm with a guarantee on both the approximation ratio and on the run-time. As a corollary, we obtain a new upper bound on the zero forcing number in terms of the fort number and the pathwidth. The algorithm is based on a correspondence between zero forcing sets and forcing arc sets. This correspondence leads to a new bound on the zero forcing number in terms of vertex cuts, and to new, short proofs for known bounds on the zero forcing number.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 227-242"},"PeriodicalIF":1.0,"publicationDate":"2025-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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