Discrete Applied Mathematics最新文献

{"title":"Polarity on H-split graphs","authors":"Fernando Esteban Contreras-Mendoza,&nbsp;César Hernández-Cruz","doi":"10.1016/j.dam.2025.12.065","DOIUrl":"10.1016/j.dam.2025.12.065","url":null,"abstract":"<div><div>A partition <span><math><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></mrow></math></span> of the vertex set of a graph <span><math><mi>G</mi></math></span> is called a polar partition of <span><math><mi>G</mi></math></span> when <span><math><mrow><mi>G</mi><mrow><mo>[</mo><mi>A</mi><mo>]</mo></mrow></mrow></math></span> and <span><math><mover><mrow><mi>G</mi><mrow><mo>[</mo><mi>B</mi><mo>]</mo></mrow></mrow><mo>¯</mo></mover></math></span> are complete multipartite graphs. If <span><math><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></mrow></math></span> is a polar partition of <span><math><mi>G</mi></math></span> in which <span><math><mrow><mi>G</mi><mrow><mo>[</mo><mi>A</mi><mo>]</mo></mrow></mrow></math></span> and <span><math><mover><mrow><mi>G</mi><mrow><mo>[</mo><mi>B</mi><mo>]</mo></mrow></mrow><mo>¯</mo></mover></math></span> have at most <span><math><mi>s</mi></math></span> and <span><math><mi>k</mi></math></span> parts, respectively, then <span><math><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></mrow></math></span> is an <span><math><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span>-polar partition of <span><math><mi>G</mi></math></span>, and <span><math><mi>G</mi></math></span> is said to be an <span><math><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span>-polar graph. A graph is said to be unipolar (monopolar) if its vertex set admits a polar partition <span><math><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></mrow></math></span> such that <span><math><mi>A</mi></math></span> is a clique (an independent set, respectively). A graph admitting a <span><math><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span>-polar partition is usually called a split graph.</div><div>Naturally, most problems related to polar partitions are trivial on split graphs, even when some of them are very hard in general. In this work, we present some results related to polar partitions on two graph classes generalizing split graphs. Our main results include efficient algorithms to decide whether graphs in these classes admit such partitions. We also establish upper bounds on the order of minimal <span><math><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span>-polar obstructions for these families, for any <span><math><mi>s</mi></math></span> and <span><math><mi>k</mi></math></span> (possibly <span><math><mrow><mi>s</mi><mo>=</mo><mi>∞</mi></mrow></math></span> or <span><math><mrow><mi>k</mi><mo>=</mo><mi>∞</mi></mrow></math></span>).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 229-242"},"PeriodicalIF":1.0,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Construction, extension and paths of near-homogeneous tournaments","authors":"Rongxia Tang ,&nbsp;Zhaojun Chen ,&nbsp;Zan-Bo Zhang","doi":"10.1016/j.dam.2025.12.059","DOIUrl":"10.1016/j.dam.2025.12.059","url":null,"abstract":"<div><div>A homogeneous tournament is a tournament with <span><math><mrow><mn>4</mn><mi>t</mi><mo>+</mo><mn>3</mn></mrow></math></span> vertices such that every arc is contained in exactly <span><math><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></math></span> cycles of length 3. Homogeneous tournaments are the first class of tournaments that are proved to be path extendable, which means that every nonhamiltonian path <span><math><mi>P</mi></math></span> in such a tournament <span><math><mi>T</mi></math></span> can be extended to a path <span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> with the same initial and terminal vertex and <span><math><mrow><mi>V</mi><mrow><mo>(</mo><msup><mrow><mi>P</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></mrow><mo>=</mo><mi>V</mi><mrow><mo>(</mo><mi>P</mi><mo>)</mo></mrow><mo>∪</mo><mrow><mo>{</mo><mi>u</mi><mo>}</mo></mrow></mrow></math></span> for a certain vertex <span><math><mrow><mi>u</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>∖</mo><mi>V</mi><mrow><mo>(</mo><mi>P</mi><mo>)</mo></mrow></mrow></math></span>. In order to find more path extendable tournaments we study the generalization of homogeneous tournaments called near-homogeneous tournaments, in which every arc is contained in <span><math><mi>t</mi></math></span> or <span><math><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></math></span> cycles of length 3. Near-homogeneity has been defined in tournaments with <span><math><mrow><mn>4</mn><mi>t</mi><mo>+</mo><mn>1</mn></mrow></math></span> vertices. In this paper, we raise a new method to construct near-homogeneous tournaments with <span><math><mrow><mn>4</mn><mi>t</mi><mo>+</mo><mn>1</mn></mrow></math></span> vertices. We then show that the definition of near-homogeneous tournament can be extended to tournaments with an even number of vertices. Finally we verify path extendability of near-homogeneous tournaments, thus expand the class of path extendable tournaments.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 268-280"},"PeriodicalIF":1.0,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928981","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Extremal distance spectra of graphs and essential connectivity","authors":"Daoxia Zhang,&nbsp;Dan Li,&nbsp;Wenxiu Ding","doi":"10.1016/j.dam.2025.12.070","DOIUrl":"10.1016/j.dam.2025.12.070","url":null,"abstract":"<div><div>A graph is trivial if it contains only one vertex. The essential connectivity of <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><msup><mrow><mi>κ</mi></mrow><mrow><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is the minimum number of vertices of <span><math><mi>G</mi></math></span> whose removal produces a disconnected graph with at least two non-trivial components. In this paper, we determine the <span><math><mi>n</mi></math></span>-vertex graph of given essential connectivity with minimum distance spectral radius. We also characterize the extremal graphs attaining the minimum distance spectral radius among all connected graphs with fixed essential connectivity and minimum degree. Furthermore, we characterize the extremal digraphs with minimum distance spectral radius among the strongly connected digraphs with given essential connectivity.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 243-255"},"PeriodicalIF":1.0,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928986","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the (total) domination in subdivision graphs","authors":"Esteban De Jesús Sánchez ,&nbsp;Abel Cabrera-Martínez ,&nbsp;Ismael Rios-Villamar ,&nbsp;José M. Sigarreta","doi":"10.1016/j.dam.2025.12.055","DOIUrl":"10.1016/j.dam.2025.12.055","url":null,"abstract":"<div><div>In this article, we study the total domination number and the <span><math><mi>k</mi></math></span>-domination number of the subdivision graph operator <span><math><mrow><mstyle><mi>S</mi></mstyle><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> for simple graphs <span><math><mi>G</mi></math></span>. First, we obtain a closed formula for the total domination number of this well-known graph operator. Additionally, for <span><math><mrow><mi>k</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, we obtain the exact value of the <span><math><mi>k</mi></math></span>-domination number of <span><math><mrow><mstyle><mi>S</mi></mstyle><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. The picture is notably different when considering the case <span><math><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow></math></span>, i.e., the domination number. In this case, we obtain some general bounds on the domination number of <span><math><mrow><mstyle><mi>S</mi></mstyle><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, which we further improve when <span><math><mi>G</mi></math></span> is a tree. Finally, we provide closed formulas for the domination number of the subdivision graph of some well-known composite graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 201-210"},"PeriodicalIF":1.0,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928979","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On some critical Ramsey numbers involving paths","authors":"Yahui Zhang,&nbsp;Yan Li,&nbsp;Changxiang He","doi":"10.1016/j.dam.2025.12.048","DOIUrl":"10.1016/j.dam.2025.12.048","url":null,"abstract":"<div><div>For graphs <span><math><mi>F</mi></math></span>, <span><math><mi>G</mi></math></span> and <span><math><mi>H</mi></math></span>, let <span><math><mrow><mi>F</mi><mo>→</mo><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> signify that any red-blue edge-coloring of <span><math><mi>F</mi></math></span> contains either a red <span><math><mi>G</mi></math></span> or a blue <span><math><mi>H</mi></math></span>. The Ramsey number <span><math><mrow><mi>R</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> is defined as <span><math><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mrow><mo>{</mo><mi>r</mi><mspace></mspace><mo>|</mo><mspace></mspace><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>→</mo><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow><mo>}</mo></mrow></mrow></math></span>. In this note, we show that <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>∖</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>→</mo><mrow><mo>(</mo><mi>G</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>∖</mo><msub><mrow><mi>K</mi></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><mo>,</mo><mrow><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>→</mo><mrow><mo>(</mo><mi>G</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> under some bounds on <span><math><mi>n</mi></math></span>, where <span><math><mrow><mi>r</mi><mo>=</mo><mi>R</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> and <span><math><mi>G</mi></math></span> is a given graph with minimum color class of size 1. Our construction of edge-deleted balanced complete bipartite graphs removes more edges than earlier results concerning the critical Ramsey number of paths.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 145-153"},"PeriodicalIF":1.0,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928982","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Pair state transfer in tensor product and double cover","authors":"Ming Jiang ,&nbsp;Xiaogang Liu ,&nbsp;Jing Wang","doi":"10.1016/j.dam.2025.12.053","DOIUrl":"10.1016/j.dam.2025.12.053","url":null,"abstract":"<div><div>Quantum state transfer, first introduced by Bose in 2003, is an important physical phenomenon in quantum networks, which plays a vital role in quantum communication and quantum computing. In 2004, Christandl et al. proposed the concept of perfect state transfer on graphs by modeling the quantum network using graphs, and unveiled the feasibility of applying graph theory to quantum state transfer. In 2018, Chen and Godsil proposed the definition of Laplacian perfect pair state transfer on graphs, which is a brilliant generalization of perfect state transfer. In this paper, we investigate the existence of Laplacian perfect pair state transfer in tensor product and double cover of two regular graphs, respectively, and reveal fundamental connections between perfect state transfer and Laplacian perfect pair state transfer. We give necessary and sufficient conditions for the tensor product of two regular graphs to admit Laplacian perfect pair state transfer, where one of the two regular graphs admits perfect state transfer or Laplacian perfect pair state transfer. Additionally, we characterize the existence of Laplacian perfect pair state transfer in the double cover of two regular graphs. By our results, a variety of families of graphs admitting Laplacian perfect pair state transfer can be constructed.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 165-176"},"PeriodicalIF":1.0,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Vertex evaluation of multiplex graphs using Forman curvature","authors":"Taiki Yamada","doi":"10.1016/j.dam.2025.12.058","DOIUrl":"10.1016/j.dam.2025.12.058","url":null,"abstract":"<div><div>The identification of vertices that play a central role in network analysis is a fundamental challenge. Although traditional centrality measures have been extensively employed for this purpose, the increasing complexity of modern networks necessitates the use of sophisticated metrics. The concept of Forman curvature has recently garnered significant attention as a promising approach. We define the Forman curvature for multiplex graphs, which are a category of complex networks characterized by multiple layers of connections between nodes. We then prove the key properties of the Forman curvature in the context of multiplex graphs and show its usefulness in identifying vertices occupying central positions within these networks. Moreover, through a series of comparative experiments with traditional graph features and graph kernels, we demonstrate that the Forman curvature can function as an effective metric for classifying the overall structure of networks.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 211-228"},"PeriodicalIF":1.0,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A note on robber locating game","authors":"Rahul Das,&nbsp;Ushnish Sarkar","doi":"10.1016/j.dam.2025.12.061","DOIUrl":"10.1016/j.dam.2025.12.061","url":null,"abstract":"<div><div>The robber locating game on a simple connected graph is a variant of pursuit evasion game where a cop wants to locate a mobile, invisible and omniscient robber hiding in the vertices of the graph using distance queries. This game has two versions depending on whether the robber is allowed to backtrack or not. The minimum numbers of rounds required to locate the robber, against any strategy he adopts, in the two versions of the game are referred to as backtrack and non-backtrack location numbers of the graph respectively. However, if the robber is immobile, then the minimum numbers of rounds required to locate the invisible robber is called the sequential metric dimension of the graph. This article gives a lower bound of the backtrack and non-backtrack location numbers in terms of the metric dimension and diameter. This has been done by establishing a lower bound of the sequential metric dimension. A hideout is a subgraph <span><math><mi>G</mi></math></span> of a graph <span><math><mi>H</mi></math></span> such that the mobile, invisible and omniscient robber can win by remaining on the vertices of <span><math><mi>G</mi></math></span>. We prove sufficient conditions for a graph to be a hideout for both backtracking and non-backtracking scenarios.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 281-286"},"PeriodicalIF":1.0,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980018","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Thick Forests","authors":"Martin Dyer ,&nbsp;Haiko Müller","doi":"10.1016/j.dam.2025.12.042","DOIUrl":"10.1016/j.dam.2025.12.042","url":null,"abstract":"<div><div>We consider classes of graphs, which we call <em>thick graphs</em>, that have the vertices of a corresponding <em>thin</em> graph replaced by cliques and the edges replaced by cobipartite graphs. In particular, we consider the case of <em>thick forests</em>, which we show to be the largest class of perfect thick graphs.</div><div>Recognising membership of a class of thick <span><math><mi>C</mi></math></span>-graphs is NP-complete unless the class <span><math><mi>C</mi></math></span> is triangle-free, so we focus on this case. Even then membership can be NP-complete. However, we show that the class of thick forests can be recognised in polynomial time.</div><div>We consider two well-studied combinatorial problems on thick graphs, independent sets and proper colourings. Since determining the independence or chromatic number of a perfect graph is known to be tractable, we examine the complexity of <em>counting</em> all independent sets and colourings in thick forests.</div><div>Finally, we consider two parametric extensions to larger classes of thick graphs: where the parameter is the size of the thin graph, and where the parameter is its treewidth.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 177-200"},"PeriodicalIF":1.0,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928978","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Neighbor connectivity of bubble-sort star graphs","authors":"Liying Zhao ,&nbsp;Shumin Zhang ,&nbsp;Bo Zhu ,&nbsp;Jou-Ming Chang","doi":"10.1016/j.dam.2026.01.027","DOIUrl":"10.1016/j.dam.2026.01.027","url":null,"abstract":"<div><div>In a spy network, when a particular spy is threatened, all associated partners in the surrounding vicinity may also be at risk. To better evaluate the reliability and security of the spy network, particularly in the context of the underground resistance movement, some early scholars introduced the concept of neighbor connectivity. This notion has recently been extended to assess the stability of interconnection networks, particularly in light of multi-processor or distributed computing systems. The topology of a network is typically modeled as a simple undirected graph. The neighbor connectivity of a graph <span><math><mi>G</mi></math></span>, denoted as <span><math><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mtext>NB</mtext></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> (resp. the edge neighbor connectivity, denoted as <span><math><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mtext>NB</mtext></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>), is defined as the minimum number of vertices (resp. edges) whose removal, along with their closed neighborhood, results in <span><math><mi>G</mi></math></span> becoming disconnected, empty, or complete (resp. trivial). Bubble-sort star graphs possess several favorable properties, including bipartiteness, fault tolerance, and vertex transitivity, making them attractive as network structures. In this paper, we establish that <span><math><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mtext>NB</mtext></mrow></msub><mrow><mo>(</mo><mi>B</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mrow><mo>⌊</mo><mfrac><mrow><mn>3</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>⌋</mo></mrow></mrow></math></span> for <span><math><mrow><mi>n</mi><mo>≥</mo><mn>4</mn></mrow></math></span> and <span><math><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mtext>NB</mtext></mrow></msub><mrow><mo>(</mo><mi>B</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>3</mn></mrow></math></span> for <span><math><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 386-396"},"PeriodicalIF":1.0,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023531","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}