Pub Date : 2026-05-31Epub Date: 2026-01-22DOI: 10.1016/j.dam.2026.01.022
Ajinkya Gaikwad, Soumen Maity
A defensive alliance in an undirected graph is a non-empty set such that every vertex has at least as many neighbours (including itself) in as it has in . In this paper, we consider the notion of global minimality. A defensive alliance is called a globally minimal defensive alliance if no proper subset of is a defensive alliance. Given an undirected graph and a positive integer , we study Globally Minimal Defensive Alliance, where the goal is to check whether has a globally minimal defensive alliance of size at least . This problem is NP-hard but its parameterized complexity has remained open until now. The goal of this paper is to provide new insight into the complexity of Globally Minimal Defensive Alliance, parameterized by the structure of the input graph. We show that the problem is fixed-parameter tractable (FPT) when parameterized by the neighbourhood diversity of the input graph. The result for neighbourhood diversity implies that the problem is FPT parameterized by vertex cover number also. We prove that the problem parameterized by the vertex cover number of the input graph does not admit a polynomial compression unless coNP NP/poly. Furthermore, we show that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, pathwidth, treewidth and treedepth. Finally, we prove that, given a vertex , deciding whether has a globally minimal defensive alliance of any size that contains is NP-complete.
{"title":"Globally minimal defensive alliances: A parameterized perspective","authors":"Ajinkya Gaikwad, Soumen Maity","doi":"10.1016/j.dam.2026.01.022","DOIUrl":"10.1016/j.dam.2026.01.022","url":null,"abstract":"<div><div>A <em>defensive alliance</em> in an undirected graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> is a non-empty set <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi></mrow></math></span> such that every vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>S</mi></mrow></math></span> has at least as many neighbours (including itself) in <span><math><mi>S</mi></math></span> as it has in <span><math><mrow><mi>V</mi><mo>∖</mo><mi>S</mi></mrow></math></span>. In this paper, we consider the notion of global minimality. A defensive alliance <span><math><mi>S</mi></math></span> is called a <em>globally minimal defensive alliance</em> if no proper subset of <span><math><mi>S</mi></math></span> is a defensive alliance. Given an undirected graph <span><math><mi>G</mi></math></span> and a positive integer <span><math><mi>k</mi></math></span>, we study <span>Globally Minimal Defensive Alliance</span>, where the goal is to check whether <span><math><mi>G</mi></math></span> has a globally minimal defensive alliance of size at least <span><math><mi>k</mi></math></span>. This problem is NP-hard but its parameterized complexity has remained open until now. The goal of this paper is to provide new insight into the complexity of <span>Globally Minimal Defensive Alliance</span>, parameterized by the structure of the input graph. We show that the problem is fixed-parameter tractable (FPT) when parameterized by the neighbourhood diversity of the input graph. The result for neighbourhood diversity implies that the problem is FPT parameterized by vertex cover number also. We prove that the problem parameterized by the vertex cover number of the input graph does not admit a polynomial compression unless coNP <span><math><mo>⊆</mo></math></span> NP/poly. Furthermore, we show that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, pathwidth, treewidth and treedepth. Finally, we prove that, given a vertex <span><math><mrow><mi>r</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, deciding whether <span><math><mi>G</mi></math></span> has a globally minimal defensive alliance of any size that contains <span><math><mi>r</mi></math></span> is NP-complete.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"385 ","pages":"Pages 86-99"},"PeriodicalIF":1.0,"publicationDate":"2026-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039498","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 : 2026-05-31Epub 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":"2026-05-31","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 : 2026-05-31Epub Date: 2026-01-30DOI: 10.1016/j.dam.2026.01.028
Vladimir Gurvich , Mariya Naumova
In several recent papers some concepts of convex analysis were extended to discrete sets. The present paper is one more step in this direction. It is well known that a local minimum of a convex function is always its global minimum. We study some discrete objects that share this property and provide several examples of convex families related to graphs and to two-person games in normal form.
{"title":"More on discrete convexity","authors":"Vladimir Gurvich , Mariya Naumova","doi":"10.1016/j.dam.2026.01.028","DOIUrl":"10.1016/j.dam.2026.01.028","url":null,"abstract":"<div><div>In several recent papers some concepts of convex analysis were extended to discrete sets. The present paper is one more step in this direction. It is well known that a local minimum of a convex function is always its global minimum. We study some discrete objects that share this property and provide several examples of convex families related to graphs and to two-person games in normal form.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"385 ","pages":"Pages 146-164"},"PeriodicalIF":1.0,"publicationDate":"2026-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146090578","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 : 2026-05-31Epub Date: 2026-02-13DOI: 10.1016/j.dam.2026.02.009
Hanna Furmańczyk , Vahan Mkrtchyan
In this paper, we consider some general properties of block graphs as well as the equitable coloring problem in this class of graphs. In the first part we establish the relation between two structural parameters for general block graphs which can be valuable for studying the parameterized computational complexity of many problems for block graphs. We also give complete characterization of block graphs with given value of parameter . In the next part of the paper we confirm the hypothesis for some subclass of block graphs in which the problem of EQUITABLE COLORING is unlikely to be polynomial time solvable.
We also give an equitable -algorithm for all block graphs from . As a by-product we prove that the equitable chromatic spectrum for the subclass of block graphs from is gap-free.
{"title":"Block graphs — Some general results and their equitable colorings","authors":"Hanna Furmańczyk , Vahan Mkrtchyan","doi":"10.1016/j.dam.2026.02.009","DOIUrl":"10.1016/j.dam.2026.02.009","url":null,"abstract":"<div><div>In this paper, we consider some general properties of block graphs as well as the equitable coloring problem in this class of graphs. In the first part we establish the relation between two structural parameters for general block graphs which can be valuable for studying the parameterized computational complexity of many problems for block graphs. We also give complete characterization of block graphs with given value of parameter <span><math><msub><mrow><mi>α</mi></mrow><mrow><mo>min</mo></mrow></msub></math></span>. In the next part of the paper we confirm the hypothesis for some subclass of block graphs in which the problem of EQUITABLE COLORING is unlikely to be polynomial time solvable.</div><div>We also give an equitable <span><math><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow></math></span>-algorithm for all block graphs <span><math><mrow><mi>G</mi><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>;</mo><mi>B</mi><mo>)</mo></mrow></mrow></math></span> from <span><math><mi>GLS</mi></math></span>. As a by-product we prove that the equitable chromatic spectrum for the subclass of block graphs from <span><math><mi>GLS</mi></math></span> is gap-free.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"385 ","pages":"Pages 178-193"},"PeriodicalIF":1.0,"publicationDate":"2026-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190103","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 : 2026-05-31Epub 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":"2026-05-31","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 : 2026-05-15Epub Date: 2026-01-07DOI: 10.1016/j.dam.2025.12.045
Pingshan Li, Ke Yang, Wei Jin
A subset of vertices is called a dissociation set if it induces a subgraph with vertex degree at most one. Recently, Yuan et al. established the upper bound of the maximum number of dissociation sets among all connected graphs of order and characterized the corresponding extremal graphs. They also proposed a question regarding the second-largest number of dissociation sets among all connected graphs of order and the corresponding extremal graphs. In this paper, we give a positive answer to this question.
{"title":"A note on the second-largest number of dissociation sets in connected graphs","authors":"Pingshan Li, Ke Yang, Wei Jin","doi":"10.1016/j.dam.2025.12.045","DOIUrl":"10.1016/j.dam.2025.12.045","url":null,"abstract":"<div><div>A subset of vertices is called a dissociation set if it induces a subgraph with vertex degree at most one. Recently, Yuan et al. established the upper bound of the maximum number of dissociation sets among all connected graphs of order <span><math><mi>n</mi></math></span> and characterized the corresponding extremal graphs. They also proposed a question regarding the second-largest number of dissociation sets among all connected graphs of order <span><math><mi>n</mi></math></span> and the corresponding extremal graphs. In this paper, we give a positive answer to this question.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 34-40"},"PeriodicalIF":1.0,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928977","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 : 2026-05-15Epub Date: 2026-01-09DOI: 10.1016/j.dam.2025.12.063
Jiahao Yu, Min Chen
Given a graph , a 2-distance -coloring is a mapping such that any two vertices within distance at most two receive distinct colors. A graph admitting such a coloring is called 2-distance. It is known that every planar graph is 2-distance -colorable if the maximum degree (Song and Lai, 2018) and 2-distance -colorable if (Bousquet et al., 2023). In this paper, we strengthen these bounds by proving that every planar graph with is 2-distance -colorable, thereby advancing progress toward Wegner’s conjecture.
给定一个图G=(V，E)，一个2-distance k-coloring是一个映射π:V(G)→{1,2，…，k}，使得距离不超过两个的任意两个顶点得到不同的颜色。允许这种着色的图称为2-距离。已知，如果最大度Δ≥8 (Song and Lai, 2018)，则每个平面图为2-distance （2Δ+16）可色(2-distance （2Δ+7）可色（Δ≥9）（Bousquet et al., 2023）。在本文中，我们通过证明Δ≥8的每个平面图都是2-距离（2Δ+6）可色来加强这些界限，从而推进了Wegner猜想的进展。
{"title":"On 2-distance (2Δ+6)-coloring of planar graphs","authors":"Jiahao Yu, Min Chen","doi":"10.1016/j.dam.2025.12.063","DOIUrl":"10.1016/j.dam.2025.12.063","url":null,"abstract":"<div><div>Given a graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span>, a 2-distance <span><math><mi>k</mi></math></span>-coloring is a mapping <span><math><mrow><mi>π</mi><mo>:</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>→</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></mrow></math></span> such that any two vertices within distance at most two receive distinct colors. A graph admitting such a coloring is called 2-distance. It is known that every planar graph is 2-distance <span><math><mrow><mo>(</mo><mn>2</mn><mi>Δ</mi><mo>+</mo><mn>16</mn><mo>)</mo></mrow></math></span>-colorable if the maximum degree <span><math><mrow><mi>Δ</mi><mo>≥</mo><mn>8</mn></mrow></math></span> (Song and Lai, 2018) and 2-distance <span><math><mrow><mo>(</mo><mn>2</mn><mi>Δ</mi><mo>+</mo><mn>7</mn><mo>)</mo></mrow></math></span>-colorable if <span><math><mrow><mi>Δ</mi><mo>≥</mo><mn>9</mn></mrow></math></span> (Bousquet et al., 2023). In this paper, we strengthen these bounds by proving that every planar graph with <span><math><mrow><mi>Δ</mi><mo>≥</mo><mn>8</mn></mrow></math></span> is 2-distance <span><math><mrow><mo>(</mo><mn>2</mn><mi>Δ</mi><mo>+</mo><mn>6</mn><mo>)</mo></mrow></math></span>-colorable, thereby advancing progress toward Wegner’s conjecture.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 256-267"},"PeriodicalIF":1.0,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928980","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 : 2026-05-15Epub Date: 2026-01-07DOI: 10.1016/j.dam.2025.12.046
Zi-Ming Zhou , Zhi-Bin Du , Chang-Xiang He
For a graph , let be the sum of the first two largest signless Laplacian eigenvalues of , and . Very recently, Zhou et al. (2024) proved that (the star graph with an additional edge) is the unique graph with minimum value of among the graphs on vertices. In this paper, we prove that the vertex-disjoint union of and possibly some isolated vertices is the unique graph with minimum value of among the graphs with edges.
{"title":"Extremal graphs for the sum of the first two largest signless Laplacian eigenvalues","authors":"Zi-Ming Zhou , Zhi-Bin Du , Chang-Xiang He","doi":"10.1016/j.dam.2025.12.046","DOIUrl":"10.1016/j.dam.2025.12.046","url":null,"abstract":"<div><div>For a graph <span><math><mi>G</mi></math></span>, let <span><math><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the sum of the first two largest signless Laplacian eigenvalues of <span><math><mi>G</mi></math></span>, and <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mn>3</mn><mo>−</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Very recently, Zhou et al. (2024) proved that <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> (the star graph with an additional edge) is the unique graph with minimum value of <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> among the graphs on <span><math><mi>n</mi></math></span> vertices. In this paper, we prove that the vertex-disjoint union of <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> and possibly some isolated vertices is the unique graph with minimum value of <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> among the graphs with <span><math><mrow><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> edges.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 139-144"},"PeriodicalIF":1.0,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928987","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 : 2026-05-15Epub 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":"2026-05-15","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 : 2026-05-15Epub Date: 2026-01-12DOI: 10.1016/j.dam.2025.12.066
Zhidan Feng , Henning Fernau , Kevin Mann
In this paper, we study the task of enumerating (and counting) locally and globally minimal defensive alliances in graphs. We consider general graphs as well as special graph classes, like trees, bipartite graphs, and split graphs. From an input-sensitive perspective, our presented algorithms are mostly optimal, meaning that their running times (neglecting polynomial factors) match concrete families of graphs that contain that many minimal alliances.
{"title":"Enumerating minimal defensive alliances","authors":"Zhidan Feng , Henning Fernau , Kevin Mann","doi":"10.1016/j.dam.2025.12.066","DOIUrl":"10.1016/j.dam.2025.12.066","url":null,"abstract":"<div><div>In this paper, we study the task of enumerating (and counting) locally and globally minimal defensive alliances in graphs. We consider general graphs as well as special graph classes, like trees, bipartite graphs, and split graphs. From an input-sensitive perspective, our presented algorithms are mostly optimal, meaning that their running times (neglecting polynomial factors) match concrete families of graphs that contain that many minimal alliances.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 309-325"},"PeriodicalIF":1.0,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980078","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号