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}
Pub Date : 2025-12-22DOI: 10.1016/j.dam.2025.12.037
Nino Bašić , Patrick W. Fowler , Maxine M. McCarthy , Primož Potočnik
A nut graph is a simple graph whose kernel is spanned by a single full vector (i.e., the adjacency matrix has a single zero eigenvalue and all non-zero kernel eigenvectors have no zero entry). We classify generalisations of nut graphs to nut digraphs: a digraph whose kernel (resp. co-kernel) is spanned by a full vector is dextro-nut (resp. laevo-nut); a bi-nut digraph is both laevo- and dextro-nut; an ambi-nut digraph is a bi-nut digraph where kernel and co-kernel are spanned by the same vector; a digraph is inter-nut if the intersection of the kernel and co-kernel is spanned by a full vector. It is known that a nut graph is connected, leafless and non-bipartite. It is shown here that an ambi-nut digraph is strongly connected, non-bipartite (i.e., has a non-bipartite underlying graph) and has minimum in-degree and minimum out-degree of at least 2. Refined notions of core and core-forbidden vertices apply to singular digraphs. Infinite families of nut digraphs and systematic coalescence, crossover and multiplier constructions are introduced. Relevance of nut digraphs to topological physics is discussed.
{"title":"Nut digraphs","authors":"Nino Bašić , Patrick W. Fowler , Maxine M. McCarthy , Primož Potočnik","doi":"10.1016/j.dam.2025.12.037","DOIUrl":"10.1016/j.dam.2025.12.037","url":null,"abstract":"<div><div>A <em>nut graph</em> is a simple graph whose kernel is spanned by a single full vector (i.e., the adjacency matrix has a single zero eigenvalue and all non-zero kernel eigenvectors have no zero entry). We classify generalisations of nut graphs to nut digraphs: a digraph whose kernel (resp. co-kernel) is spanned by a full vector is <em>dextro-nut</em> (resp. <em>laevo-nut</em>); a <em>bi-nut</em> digraph is both laevo- and dextro-nut; an <em>ambi-nut</em> digraph is a bi-nut digraph where kernel and co-kernel are spanned by the same vector; a digraph is <em>inter-nut</em> if the intersection of the kernel and co-kernel is spanned by a full vector. It is known that a nut graph is connected, leafless and non-bipartite. It is shown here that an ambi-nut digraph is strongly connected, non-bipartite (i.e., has a non-bipartite underlying graph) and has minimum in-degree and minimum out-degree of at least 2. Refined notions of core and core-forbidden vertices apply to singular digraphs. Infinite families of nut digraphs and systematic coalescence, crossover and multiplier constructions are introduced. Relevance of nut digraphs to topological physics is discussed.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 203-226"},"PeriodicalIF":1.0,"publicationDate":"2025-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842074","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 : 2025-12-22DOI: 10.1016/j.dam.2025.12.027
Michitaka Furuya , Mikio Kano
Let be a bipartite graph with bipartition , and let and be functions. In this paper, we give a sufficient condition for to have a factor satisfying for all and for all . Our theorem modifies a result in Addario-Berry et al. (2008).
设G为二分图（X，Y），设a，b:X→Z≥0，c:Y→Z≥0为函数。本文给出了G有一个因子F满足对所有x∈x degF(x)∈{a(x),b(x)}，对所有y∈y degF(y)∈{c(y),c(y)+1}的充分条件。我们的定理修正了adario - berry et al.（2008）的一个结果。
{"title":"Factors of bipartite graphs with degree conditions imposed on each partite set","authors":"Michitaka Furuya , Mikio Kano","doi":"10.1016/j.dam.2025.12.027","DOIUrl":"10.1016/j.dam.2025.12.027","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a bipartite graph with bipartition <span><math><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></mrow></math></span>, and let <span><math><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>X</mi><mo>→</mo><msub><mrow><mi>Z</mi></mrow><mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow></math></span> and <span><math><mrow><mi>c</mi><mo>:</mo><mi>Y</mi><mo>→</mo><msub><mrow><mi>Z</mi></mrow><mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow></math></span> be functions. In this paper, we give a sufficient condition for <span><math><mi>G</mi></math></span> to have a factor <span><math><mi>F</mi></math></span> satisfying <span><math><mrow><msub><mrow><mo>deg</mo></mrow><mrow><mi>F</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>∈</mo><mrow><mo>{</mo><mi>a</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo><mi>b</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>}</mo></mrow></mrow></math></span> for all <span><math><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow></math></span> and <span><math><mrow><msub><mrow><mo>deg</mo></mrow><mrow><mi>F</mi></mrow></msub><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>∈</mo><mrow><mo>{</mo><mi>c</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>,</mo><mi>c</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>+</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span> for all <span><math><mrow><mi>y</mi><mo>∈</mo><mi>Y</mi></mrow></math></span>. Our theorem modifies a result in Addario-Berry et al. (2008).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 165-168"},"PeriodicalIF":1.0,"publicationDate":"2025-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842055","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 : 2025-12-22DOI: 10.1016/j.dam.2025.12.028
Jaroslav Garvardt , Christian Komusiewicz, Nils Morawietz
In the NP-hard Weighted Cluster Deletion problem, the input is an undirected graph and an edge-weight function , and the task is to partition the vertex set into cliques so that the total weight of edges in the cliques is maximized. Recently, it has been shown that Weighted Cluster Deletion is NP-hard on some graph classes where Cluster Deletion, the special case where every edge has unit weight, can be solved in polynomial time. We study the influence of the value of the largest edge weight assigned by on the problem complexity for such graph classes. Our main results are that Weighted Cluster Deletion is fixed-parameter tractable with respect to on graph classes whose graphs consist of well-separated clusters that are connected by a sparse periphery. Concrete examples for such classes are split graphs and graphs that are close to cluster graphs. We complement our results by strengthening previous hardness results for Weighted Cluster Deletion. For example, we show that Weighted Cluster Deletion is NP-hard on restricted subclasses of cographs even when every edge has weight 1 or 2.
{"title":"When can Cluster Deletion with bounded weights be solved efficiently?","authors":"Jaroslav Garvardt , Christian Komusiewicz, Nils Morawietz","doi":"10.1016/j.dam.2025.12.028","DOIUrl":"10.1016/j.dam.2025.12.028","url":null,"abstract":"<div><div>In the NP-hard <span>Weighted Cluster Deletion</span> problem, the input is 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> and an edge-weight function <span><math><mrow><mi>ω</mi><mo>:</mo><mi>E</mi><mo>→</mo><mi>N</mi></mrow></math></span>, and the task is to partition the vertex set <span><math><mi>V</mi></math></span> into cliques so that the total weight of edges in the cliques is maximized. Recently, it has been shown that <span>Weighted Cluster Deletion</span> is NP-hard on some graph classes where <span>Cluster Deletion</span>, the special case where every edge has unit weight, can be solved in polynomial time. We study the influence of the value <span><math><mi>t</mi></math></span> of the largest edge weight assigned by <span><math><mi>ω</mi></math></span> on the problem complexity for such graph classes. Our main results are that <span>Weighted Cluster Deletion</span> is fixed-parameter tractable with respect to <span><math><mi>t</mi></math></span> on graph classes whose graphs consist of well-separated clusters that are connected by a sparse periphery. Concrete examples for such classes are split graphs and graphs that are close to cluster graphs. We complement our results by strengthening previous hardness results for <span>Weighted Cluster Deletion</span>. For example, we show that <span>Weighted Cluster Deletion</span> is NP-hard on restricted subclasses of cographs even when every edge has weight 1 or 2.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 169-183"},"PeriodicalIF":1.0,"publicationDate":"2025-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842056","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 : 2025-12-22DOI: 10.1016/j.dam.2025.12.026
Stefan Kratsch , Van Bang Le
A stable cutset in a graph is a set such that vertices of are pairwise non-adjacent and such that is disconnected, i.e., it is both stable (or independent) set and a cutset (or separator). Unlike general cutsets, it is -complete to determine whether a given graph has any stable cutset. Recently, Rauch et al. [FCT 2023 & JCSS 2025] gave a number of fixed-parameter tractable (FPT) algorithms, running in time , for Stable Cutset under a variety of parameters such as the size of a (given) dominating set, the size of an odd cycle transversal, or the deletion distance to -free graphs. Earlier works imply FPT algorithms relative to clique-width and relative to solution size.
We complement these findings by giving the first results on the existence of polynomial kernelizations for Stable Cutset, i.e., efficient preprocessing algorithms that return an equivalent instance of size polynomial in the parameter value. Under the standard assumption that , we show that no polynomial kernelization is possible relative to the deletion distance to a single path, generalizing deletion distance to various graph classes, nor by the size of a (given) dominating set. We also show that under the same assumption no polynomial kernelization is possible relative to solution size, i.e., given answering whether there is a stable cutset of size at most . On the positive side, we show polynomial kernelizations for parameterization by modulators to a single clique, to a cluster or a co-cluster graph, and by twin cover.
{"title":"On polynomial kernelization for Stable Cutset","authors":"Stefan Kratsch , Van Bang Le","doi":"10.1016/j.dam.2025.12.026","DOIUrl":"10.1016/j.dam.2025.12.026","url":null,"abstract":"<div><div>A <em>stable cutset</em> in a graph <span><math><mi>G</mi></math></span> is a set <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> such that vertices of <span><math><mi>S</mi></math></span> are pairwise non-adjacent and such that <span><math><mrow><mi>G</mi><mo>−</mo><mi>S</mi></mrow></math></span> is disconnected, i.e., it is both stable (or independent) set and a cutset (or separator). Unlike general cutsets, it is <span><math><mi>NP</mi></math></span>-complete to determine whether a given graph <span><math><mi>G</mi></math></span> has any stable cutset. Recently, Rauch et al. [FCT 2023 & JCSS 2025] gave a number of fixed-parameter tractable (FPT) algorithms, running in time <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow><mi>⋅</mi><msup><mrow><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow><mrow><mi>c</mi></mrow></msup></mrow></math></span>, for <span>Stable Cutset</span> under a variety of parameters <span><math><mi>k</mi></math></span> such as the size of a (given) dominating set, the size of an odd cycle transversal, or the deletion distance to <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>-free graphs. Earlier works imply FPT algorithms relative to clique-width and relative to solution size.</div><div>We complement these findings by giving the first results on the existence of polynomial kernelizations for <span>Stable Cutset</span>, i.e., efficient preprocessing algorithms that return an equivalent instance of size polynomial in the parameter value. Under the standard assumption that <span><math><mi>NP ⊈ coNP/poly</mi></math></span>, we show that no polynomial kernelization is possible relative to the deletion distance to a single path, generalizing deletion distance to various graph classes, nor by the size of a (given) dominating set. We also show that under the same assumption no polynomial kernelization is possible relative to solution size, i.e., given <span><math><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span> answering whether there is a stable cutset of size at most <span><math><mi>k</mi></math></span>. On the positive side, we show polynomial kernelizations for parameterization by modulators to a single clique, to a cluster or a co-cluster graph, and by twin cover.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 184-202"},"PeriodicalIF":1.0,"publicationDate":"2025-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842073","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 : 2025-12-19DOI: 10.1016/j.dam.2025.12.038
Yongjiang Wu , Yongtao Li , Zhiyi Liu , Lihua Feng , Tingzeng Wu
Two families of sets and are said to be cross -union if for any and , . In 2021, Frankl and Wong proved that if are non-empty cross -union, then Moreover, for , equality holds if and only if . In this paper, we give a new method to prove this result. Our method also allows us to establish a vector space version and a hereditary family extension. As a byproduct, we revisit the vector space version of the Katona -union theorem due to Frankl and Tokushige, and characterize the extremal families for the case .
{"title":"Maximal non-empty cross s-union families","authors":"Yongjiang Wu , Yongtao Li , Zhiyi Liu , Lihua Feng , Tingzeng Wu","doi":"10.1016/j.dam.2025.12.038","DOIUrl":"10.1016/j.dam.2025.12.038","url":null,"abstract":"<div><div>Two families of sets <span><math><mi>F</mi></math></span> and <span><math><mi>G</mi></math></span> are said to be cross <span><math><mi>s</mi></math></span>-union if for any <span><math><mrow><mi>F</mi><mo>∈</mo><mi>F</mi></mrow></math></span> and <span><math><mrow><mi>G</mi><mo>∈</mo><mi>G</mi></mrow></math></span>, <span><math><mrow><mrow><mo>|</mo><mi>F</mi><mo>∪</mo><mi>G</mi><mo>|</mo></mrow><mo>≤</mo><mi>s</mi></mrow></math></span>. In 2021, Frankl and Wong proved that if <span><math><mrow><mi>F</mi><mo>,</mo><mi>G</mi><mo>⊆</mo><msup><mrow><mn>2</mn></mrow><mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow></msup></mrow></math></span> are non-empty cross <span><math><mi>s</mi></math></span>-union, then <span><math><mrow><mrow><mo>|</mo><mi>F</mi><mo>|</mo></mrow><mo>+</mo><mrow><mo>|</mo><mi>G</mi><mo>|</mo></mrow><mo>≤</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mfenced><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></mfrac></mrow></mfenced><mo>+</mo><mn>1</mn><mo>.</mo></mrow></math></span> Moreover, for <span><math><mrow><mi>s</mi><mo><</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span>, equality holds if and only if <span><math><mrow><mfenced><mrow><mi>F</mi><mo>,</mo><mi>G</mi></mrow></mfenced><mo>=</mo><mfenced><mrow><mrow><mo>{</mo><mo>0̸</mo><mo>}</mo></mrow><mo>,</mo><mrow><mo>{</mo><mi>G</mi><mo>⊆</mo><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mo>:</mo><mrow><mo>|</mo><mi>G</mi><mo>|</mo></mrow><mo>≤</mo><mi>s</mi><mo>}</mo></mrow></mrow></mfenced></mrow></math></span>. In this paper, we give a new method to prove this result. Our method also allows us to establish a vector space version and a hereditary family extension. As a byproduct, we revisit the vector space version of the Katona <span><math><mi>s</mi></math></span>-union theorem due to Frankl and Tokushige, and characterize the extremal families for the case <span><math><mrow><mi>s</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 130-138"},"PeriodicalIF":1.0,"publicationDate":"2025-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792158","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 : 2025-12-19DOI: 10.1016/j.dam.2025.12.039
Daniel A. Jaume , Vadim E. Levit , Eugen Mandrescu , Gonzalo Molina , Kevin Pereyra
A graph is said to be Kőnig–Egerváry if its matching number equals its vertex cover number. The difference between these two graph parameters, the vertex cover number minus the matching number, measures, in some sense, how far a graph is from being a Kőnig–Egerváry graph. Several properties of this difference, called the Kőnig–Egerváry index or Kőnig deficiency, are presented, including some nontrivial structural characterizations. Furthermore, it is shown that various statements involving Kőnig–Egerváry graphs are, in fact, general statements about graphs that can be expressed in terms of their Kőnig–Egerváry indices.
{"title":"On the Kőnig–Egerváry index of a graph","authors":"Daniel A. Jaume , Vadim E. Levit , Eugen Mandrescu , Gonzalo Molina , Kevin Pereyra","doi":"10.1016/j.dam.2025.12.039","DOIUrl":"10.1016/j.dam.2025.12.039","url":null,"abstract":"<div><div>A graph is said to be Kőnig–Egerváry if its matching number equals its vertex cover number. The difference between these two graph parameters, the vertex cover number minus the matching number, measures, in some sense, how far a graph is from being a Kőnig–Egerváry graph. Several properties of this difference, called the Kőnig–Egerváry index or Kőnig deficiency, are presented, including some nontrivial structural characterizations. Furthermore, it is shown that various statements involving Kőnig–Egerváry graphs are, in fact, general statements about graphs that can be expressed in terms of their Kőnig–Egerváry indices.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 139-151"},"PeriodicalIF":1.0,"publicationDate":"2025-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792159","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