{"title":"Domination number, independent domination number and k-independence number in trees","authors":"Qing Cui, Xu Zou","doi":"10.1016/j.dam.2025.01.036","DOIUrl":"10.1016/j.dam.2025.01.036","url":null,"abstract":"<div><div>For any graph <span><math><mi>G</mi></math></span>, let <span><math><mrow><mi>γ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>i</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> denote the domination number and the independent domination number of <span><math><mi>G</mi></math></span>, respectively. For any positive integer <span><math><mi>k</mi></math></span>, a subset <span><math><mi>S</mi></math></span> of vertices in a graph <span><math><mi>G</mi></math></span> is said to be a <span><math><mi>k</mi></math></span>-independent set of <span><math><mi>G</mi></math></span> if <span><math><mrow><mi>G</mi><mrow><mo>[</mo><mi>S</mi><mo>]</mo></mrow></mrow></math></span> has maximum degree less than <span><math><mi>k</mi></math></span>. The <span><math><mi>k</mi></math></span>-independence number of <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is the maximum cardinality of a <span><math><mi>k</mi></math></span>-independent set of <span><math><mi>G</mi></math></span>. Let <span><math><mi>T</mi></math></span> be any tree with <span><math><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></math></span> vertices. Dehgardi et al. proved that <span><math><mrow><mi>i</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>γ</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>+</mo><mi>i</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn></mrow></mfrac><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow></mrow></math></span>. Later, Zhang and Wu extended the former result of Dehgardi et al. by showing that <span><math><mrow><mi>i</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><mi>k</mi></mrow></mfrac><msub><mrow><mi>α</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow></mrow></math></span>, and conjectured that the latter one can also be generalized to <span><math><mrow><mi>γ</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>+</mo><mi>i</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mn>2</mn><mi>k</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>−</mo><mn>1</mn></mrow></mfrac><msub><mrow><mi>α</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow></mrow></math></span>. In this paper, we prove this conjecture, and moreover, we characterize all extremal trees for which the equality holds.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"366 ","pages":"Pages 176-184"},"PeriodicalIF":1.0,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143161753","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":"Perfect out-forest problem and directed Steiner cycle packing problem","authors":"Yuefang Sun , Zemin Jin","doi":"10.1016/j.dam.2025.01.027","DOIUrl":"10.1016/j.dam.2025.01.027","url":null,"abstract":"<div><div>The perfect out-forest problem generalizes the perfect matching problem, and the directed Steiner cycle packing problem generalizes the Hamiltonian cycle decomposition problem and is a variant of the directed Steiner tree packing problem. In this paper, we study the complexity of these two types of digraph packing problems.</div><div>For the perfect out-forest problem, Gutin and Yeo proved that it is NP-complete to decide whether a given strong digraph contains a 0-perfect out-forest. We show that this result also holds for the 1-perfect out-forest problem. However, when restricted to a semicomplete digraph <span><math><mi>D</mi></math></span>, the problem of deciding whether <span><math><mi>D</mi></math></span> contains an <span><math><mi>i</mi></math></span>-perfect out-forest becomes polynomial-time solvable, where <span><math><mrow><mi>i</mi><mo>∈</mo><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span>. In addition, we prove that it is NP-hard to find a 0-perfect out-forest of maximum size in a connected acyclic digraph, and it is NP-hard to find a 1-perfect out-forest of maximum size in a connected digraph.</div><div>For the directed Steiner cycle packing problem, when both <span><math><mrow><mi>k</mi><mo>≥</mo><mn>2</mn><mo>,</mo><mi>ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math></span> are fixed integers, we show that the problem of deciding whether there are at least <span><math><mi>ℓ</mi></math></span> internally disjoint directed <span><math><mi>S</mi></math></span>-Steiner cycles in a digraph <span><math><mi>D</mi></math></span> is NP-complete, where <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow><mo>=</mo><mi>k</mi></mrow></math></span>. However, when we consider the class of symmetric digraphs, the problem becomes polynomial-time solvable. We also show that the problem of deciding whether there are at least <span><math><mi>ℓ</mi></math></span> arc-disjoint directed <span><math><mi>S</mi></math></span>-Steiner cycles in a given digraph <span><math><mi>D</mi></math></span> is NP-complete, where <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow><mo>=</mo><mi>k</mi></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"366 ","pages":"Pages 201-209"},"PeriodicalIF":1.0,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143161754","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 biased edge coloring game","authors":"Runze Wang","doi":"10.1016/j.dam.2025.01.028","DOIUrl":"10.1016/j.dam.2025.01.028","url":null,"abstract":"<div><div>We combine the ideas of edge coloring games and asymmetric graph coloring games and define the <span><math><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span>-<em>edge coloring game</em>, which is alternatively played by two players Maker and Breaker on a finite simple graph <span><math><mi>G</mi></math></span> with a set of colors <span><math><mi>X</mi></math></span>. Maker plays first and colors <span><math><mi>m</mi></math></span> uncolored edges on each turn. Breaker colors only one uncolored edge on each turn. They make sure that adjacent edges get distinct colors. Maker wins if eventually every edge is colored; Breaker wins if at some point, the player who is playing cannot color any edge. We define the <span><math><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span>-<em>game chromatic index</em> of <span><math><mi>G</mi></math></span> to be the smallest nonnegative integer <span><math><mi>k</mi></math></span> such that Maker has a winning strategy with <span><math><mrow><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow><mo>=</mo><mi>k</mi></mrow></math></span>. We give some general upper bounds on the <span><math><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span>-game chromatic indices of trees, determine the exact <span><math><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span>-game chromatic indices of some caterpillars and all wheels, and show that larger <span><math><mi>m</mi></math></span> does not necessarily give us smaller <span><math><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span>-game chromatic index.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"366 ","pages":"Pages 193-200"},"PeriodicalIF":1.0,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143160547","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":"The k-general d-position problem for graphs","authors":"Brent Cody, Garrett Moore","doi":"10.1016/j.dam.2025.01.025","DOIUrl":"10.1016/j.dam.2025.01.025","url":null,"abstract":"<div><div>A set of vertices of a graph is said to be in general position if no three vertices from the set lie on a common shortest path. Recently Klavžar, Rall and Yero generalized this notion by defining a set of vertices to be in general <span><math><mi>d</mi></math></span>-position if no three vertices from the set lie on a common shortest path of length at most <span><math><mi>d</mi></math></span>. We generalize this notion further by defining a set of vertices to be in <span><math><mi>k</mi></math></span>-general <span><math><mi>d</mi></math></span>-position if no <span><math><mi>k</mi></math></span> vertices of the set lie on a common shortest path of length at most <span><math><mi>d</mi></math></span>. The <span><math><mi>k</mi></math></span>-general <span><math><mi>d</mi></math></span>-position number of a graph is the largest cardinality of a <span><math><mi>k</mi></math></span>-general <span><math><mi>d</mi></math></span>-position set. We provide upper and lower bounds on the <span><math><mi>k</mi></math></span>-general <span><math><mi>d</mi></math></span>-position number of graphs in terms of the <span><math><mi>k</mi></math></span>-general <span><math><mi>d</mi></math></span>-position number of certain kinds of subgraphs. We compute the <span><math><mi>k</mi></math></span>-general <span><math><mi>d</mi></math></span>-position number of finite paths and cycles. Along the way we establish that the maximally even subsets of cycles, which were introduced in Clough and Douthett’s work on music theory, provide the largest possible <span><math><mi>k</mi></math></span>-general <span><math><mi>d</mi></math></span>-position sets in <span><math><mi>n</mi></math></span>-cycles. We generalize Klavžar and Manuel’s notion of monotone-geodesic labeling to that of <span><math><mi>k</mi></math></span>-monotone-geodesic labeling in order to calculate the <span><math><mi>k</mi></math></span>-general <span><math><mi>d</mi></math></span>-position number of the infinite two-dimensional grid. We also prove a formula for the <span><math><mi>k</mi></math></span>-general <span><math><mi>d</mi></math></span>-position number of certain thin finite grids, providing a partial answer to a question asked by Klavžar, Rall and Yero.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"366 ","pages":"Pages 135-151"},"PeriodicalIF":1.0,"publicationDate":"2025-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143160890","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}
Alexander Firbas , Alexander Dobler , Fabian Holzer, Jakob Schafellner, Manuel Sorge , Anaïs Villedieu, Monika Wißmann
{"title":"The complexity of cluster vertex splitting and company","authors":"Alexander Firbas , Alexander Dobler , Fabian Holzer, Jakob Schafellner, Manuel Sorge , Anaïs Villedieu, Monika Wißmann","doi":"10.1016/j.dam.2025.01.012","DOIUrl":"10.1016/j.dam.2025.01.012","url":null,"abstract":"<div><div>Clustering a graph when the clusters can overlap can be seen from three different angles: We may look for cliques that cover the edges of the graph with bounded overlap, we may look to add or delete few edges to uncover the cluster structure, or we may split vertices to separate the clusters from each other. Splitting a vertex <span><math><mi>v</mi></math></span> means to remove it and to add two new copies of <span><math><mi>v</mi></math></span> and to make each previous neighbor of <span><math><mi>v</mi></math></span> adjacent with at least one of the copies. In this work, we study underlying computational problems regarding the three angles to overlapping clusterings, in particular when the overlap is small. We show that the above-mentioned covering problem is <span>NP</span>-complete. We then make structural observations that show that the covering viewpoint and the vertex-splitting viewpoint are equivalent, yielding NP-hardness for the vertex-splitting problem. On the positive side, we show that splitting at most <span><math><mi>k</mi></math></span> vertices to obtain a cluster graph has a problem kernel with <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span> vertices. Finally, we observe that combining our hardness results with structural observations and a so-called critical-clique lemma yields a simple alternative NP-hardness proof for the <span>Cluster Editing With Vertex Splitting</span> problem, where we add or delete edges and split vertices to obtain a cluster graph.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"365 ","pages":"Pages 190-207"},"PeriodicalIF":1.0,"publicationDate":"2025-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150697","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Testing popularity in linear time via maximum matching","authors":"Erika Bérczi-Kovács , Kata Kosztolányi","doi":"10.1016/j.dam.2025.01.014","DOIUrl":"10.1016/j.dam.2025.01.014","url":null,"abstract":"<div><div>Popularity is an approach in mechanism design to find fair structures in a graph, based on the votes of the nodes. Popular matchings are the relaxation of stable matchings: 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> with strict preferences on the neighbors of the nodes, a matching <span><math><mi>M</mi></math></span> is popular if there is no other matching <span><math><msup><mrow><mi>M</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> such that the number of nodes preferring <span><math><msup><mrow><mi>M</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> is more than those preferring <span><math><mi>M</mi></math></span>. This paper considers the popularity testing problem, when the task is to decide whether a given matching is popular or not. Previous algorithms applied reductions to maximum weight matchings. We give a new algorithm for testing popularity by reducing the problem to maximum matching testing, thus attaining a linear running time <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mo>|</mo><mi>E</mi><mo>|</mo><mo>)</mo></mrow></mrow></math></span>.</div><div>Linear programming-based characterization of popularity is often applied for proving the popularity of a certain matching. As a consequence of our algorithm we derive a more structured dual witness than previous ones. Based on this result we give a combinatorial characterization of fractional popular matchings, which is a special class of popular matchings.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"366 ","pages":"Pages 152-160"},"PeriodicalIF":1.0,"publicationDate":"2025-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143161750","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Cristina Bazgan , Pinar Heggernes , André Nichterlein , Thomas Pontoizeau
{"title":"On the hardness of problems around s-clubs on split graphs","authors":"Cristina Bazgan , Pinar Heggernes , André Nichterlein , Thomas Pontoizeau","doi":"10.1016/j.dam.2025.01.023","DOIUrl":"10.1016/j.dam.2025.01.023","url":null,"abstract":"<div><div>We investigate the complexity of problems related to <span><math><mi>s</mi></math></span>-clubs. Given a graph, an <span><math><mi>s</mi></math></span>-club is a subset of vertices such that the subgraph induced by it has diameter at most <span><math><mi>s</mi></math></span>. We show that partitioning a split graph into two 2-clubs is NP-hard. Moreover, we prove that finding the minimum number of edges to add to a split graph in order to obtain a diameter of at most 2 is W[2]-hard with respect to the number of edges to add. Finally we show that finding the minimum number of edges to keep within a split graph of diameter 2 or 3 in order to maintain its diameter is NP-complete.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"364 ","pages":"Pages 247-254"},"PeriodicalIF":1.0,"publicationDate":"2025-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143142513","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Gowers U3 norm of one family of cubic power permutations","authors":"Zhaole Li , Deng Tang","doi":"10.1016/j.dam.2025.01.024","DOIUrl":"10.1016/j.dam.2025.01.024","url":null,"abstract":"<div><div>The Gowers uniformity norm has emerged as a significant metric in the evaluation of Boolean functions employed in symmetric-key encryptions, particularly in assessing their resilience against low degree approximation attacks. Beyond cryptography, this norm plays a pivotal role in theoretical computer science, including pseudorandomness and property testing of Boolean functions. However, the determination of the Gowers <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> norm for <span><math><mrow><mi>k</mi><mo>≥</mo><mn>3</mn></mrow></math></span> for general Boolean functions presents substantial computational and theoretical challenges. Recently, the relationship between the Gowers uniformity norm and the higher-order differential spectrum of Boolean functions has been derived. Furthermore, the fact that the Gowers uniformity norm of a power permutation can be determined by the Gowers uniformity norm of its any component function. In this paper, we focus on determining the Gowers <span><math><msub><mrow><mi>U</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> norm of one family of power permutations, thereby assessing their resistance to quadratic approximation attacks. The family contains five classes of power permutations, which are <span><math><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>+</mo><mn>3</mn></mrow></msup></math></span> with even <span><math><mrow><mi>n</mi><mo>≥</mo><mn>4</mn></mrow></math></span>, <span><math><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>s</mi></mrow></msup><mo>+</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>s</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></msup></math></span> with <span><math><mrow><mn>1</mn><mo>≤</mo><mi>s</mi><mo>≤</mo><mrow><mo>⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo></mrow></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, <span><math><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></mrow></msup></math></span> with <span><math><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, <span><math><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mrow><mn>2</mn></mrow><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>+</mo><mn>1</mn></mrow></msup></math></span> with even <span><math><mrow><mi>n</mi><mo>≥</mo><mn>4</mn></mrow></math></span>, and <span><math><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mrow><mn>2</mn></mrow><mrow><mfrac><mrow><mi>n</mi><mo>−<","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"365 ","pages":"Pages 208-222"},"PeriodicalIF":1.0,"publicationDate":"2025-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150143","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 almost bipartite non-König–Egerváry graphs","authors":"Vadim E. Levit , Eugen Mandrescu","doi":"10.1016/j.dam.2025.01.022","DOIUrl":"10.1016/j.dam.2025.01.022","url":null,"abstract":"<div><div>A set <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi></mrow></math></span> is <em>independent</em> in a graph <span><math><mrow><mi>G</mi><mo>=</mo><mfenced><mrow><mi>V</mi><mo>,</mo><mi>E</mi></mrow></mfenced></mrow></math></span> if no two vertices from <span><math><mi>S</mi></math></span> are adjacent. The <em>independence number</em> <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is the cardinality of a maximum independent set, while <span><math><mrow><mi>μ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is the size of a maximum matching in <span><math><mi>G</mi></math></span>. If <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mi>μ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> equals the order of <span><math><mi>G</mi></math></span>, then <span><math><mi>G</mi></math></span> is a <em>König–Egerváry graph</em> (Deming, 1979; Gavril, 1977; Sterboul, 1979). The number <span><math><mrow><mi>d</mi><mfenced><mrow><mi>G</mi></mrow></mfenced><mo>=</mo><mo>max</mo><mrow><mo>{</mo><mfenced><mrow><mi>A</mi></mrow></mfenced><mo>−</mo><mfenced><mrow><mi>N</mi><mfenced><mrow><mi>A</mi></mrow></mfenced></mrow></mfenced><mo>:</mo><mi>A</mi><mo>⊆</mo><mi>V</mi><mo>}</mo></mrow></mrow></math></span> is the <em>critical difference</em> of <span><math><mi>G</mi></math></span> (Zhang, 1990) (where <span><math><mrow><mi>N</mi><mfenced><mrow><mi>A</mi></mrow></mfenced><mo>=</mo><mfenced><mrow><mi>v</mi><mo>:</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mo>,</mo><mi>N</mi><mfenced><mrow><mi>v</mi></mrow></mfenced><mo>∩</mo><mi>A</mi><mo>≠</mo><mo>0̸</mo></mrow></mfenced></mrow></math></span>). It is known that the inequality <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>−</mo><mi>μ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>d</mi><mfenced><mrow><mi>G</mi></mrow></mfenced></mrow></math></span> holds for every graph (Levit and Mandrescu, 2012; Lorentzen, 1966; Schrijver, 2003).</div><div>A graph <span><math><mi>G</mi></math></span> is <em>(i) unicyclic</em> if it has a unique cycle, <em>(ii) almost bipartite</em> if it has only one odd cycle. Let <span><math><mrow><mi>ker</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mo>⋂</mo><mrow><mo>{</mo><mi>S</mi><mo>:</mo><mi>S</mi></mrow></mrow></math></span> <em>is a critical independent set of</em> <span><math><mrow><mi>G</mi><mo>}</mo></mrow></math></span>, core<span><math><mfenced><mrow><mi>G</mi></mrow></mfenced></math></span> be the intersection of all maximum independent sets, and corona<span><math><mfenced><mrow><mi>G</mi></mrow></mfenced></math></span> be the union of all maximum independent sets of <span><math><mi>G</mi></math></span>. It is known that <span><math><mrow><mi>ker</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>⊆</mo><mi>core</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> for every graph (Levit and","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"366 ","pages":"Pages 127-134"},"PeriodicalIF":1.0,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143160891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Zero forcing of generalized hierarchical products","authors":"Sarah E. Anderson, Brenda K. Kroschel","doi":"10.1016/j.dam.2025.01.010","DOIUrl":"10.1016/j.dam.2025.01.010","url":null,"abstract":"<div><div>Zero forcing is a process that passes information along a network. In particular, a vertex that has information may pass that information to one of its neighbors only if that neighbor is its only neighbor that does not have that information. If the information is given to an initial set of vertices that eventually passes the information to the entire network, then that initial set is called a zero forcing set. The zero forcing number of a graph is the size of a minimum zero forcing set of that graph. In this paper, bounds on the zero forcing number of generalized hierarchical products, which are a generalization of the Cartesian product, are provided. In particular, the zero forcing number is determined exactly for certain hierarchical products by constructing zero forcing sets in the product that utilize knowledge about forcing chains for each factor of the product.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"366 ","pages":"Pages 120-126"},"PeriodicalIF":1.0,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143160892","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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