{"title":"On the Nth 2-adic complexity of binary sequences identified with algebraic 2-adic integers","authors":"Zhixiong Chen , Arne Winterhof","doi":"10.1016/j.dam.2025.05.024","DOIUrl":"10.1016/j.dam.2025.05.024","url":null,"abstract":"<div><div>We identify a binary sequence <span><math><mrow><mi>S</mi><mo>=</mo><msubsup><mrow><mrow><mo>(</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>∞</mi></mrow></msubsup></mrow></math></span> with the 2-adic integer <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow><mo>=</mo><munderover><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>∞</mi></mrow></munderover><msub><mrow><mi>s</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span>. In the case that <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> is algebraic over <span><math><mi>Q</mi></math></span> of degree <span><math><mrow><mi>d</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, we prove that the <span><math><mi>N</mi></math></span>th 2-adic complexity of <span><math><mi>S</mi></math></span> is at least <span><math><mrow><mfrac><mrow><mi>N</mi></mrow><mrow><mi>d</mi></mrow></mfrac><mo>+</mo><mi>O</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>, where the implied constant depends only on the minimal polynomial of <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>. This result is an analog of the bound of Mérai and the second author on the linear complexity of automatic sequences, that is, sequences with algebraic <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span> over the rational function field <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span>.</div><div>We further discuss the most important case <span><math><mrow><mi>d</mi><mo>=</mo><mn>2</mn></mrow></math></span> in both settings and explain that the intersection of the set of 2-adic algebraic sequences and the set of automatic sequences is the set of (eventually) periodic sequences. Finally, we provide some experimental results supporting the conjecture that 2-adic algebraic sequences can have also a desirable <span><math><mi>N</mi></math></span>th linear complexity and automatic sequences a desirable <span><math><mi>N</mi></math></span>th 2-adic complexity, respectively.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"373 ","pages":"Pages 279-289"},"PeriodicalIF":1.0,"publicationDate":"2025-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143947561","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":"Unavoidable patterns in 2-colorings of the complete bipartite graph","authors":"Adriana Hansberg , Denae Ventura","doi":"10.1016/j.dam.2025.05.018","DOIUrl":"10.1016/j.dam.2025.05.018","url":null,"abstract":"<div><div>We determine the colored patterns that appear in any 2-edge coloring of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>, with <span><math><mi>n</mi></math></span> large enough and with sufficient edges in each color. We prove the existence of a positive integer <span><math><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> such that any 2-edge coloring of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> with at least <span><math><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> edges in each color contains at least one of these patterns. We give a general upper bound for <span><math><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and prove its tightness for some cases. We define the concepts of bipartite <span><math><mi>r</mi></math></span>-tonality and bipartite omnitonality using the complete bipartite graph as a base graph. We provide a characterization for bipartite <span><math><mi>r</mi></math></span>-tonal graphs and prove that every tree is bipartite omnitonal. Finally, we define the bipartite-balancing number and provide the exact bipartite-balancing number for paths and stars.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"374 ","pages":"Pages 50-60"},"PeriodicalIF":1.0,"publicationDate":"2025-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144069363","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":"Conditional matroidal edge connectivity of Cayley graphs","authors":"Li Wang , Mingxi Su , Liqing Lin , Xiaohui Hua","doi":"10.1016/j.dam.2025.05.011","DOIUrl":"10.1016/j.dam.2025.05.011","url":null,"abstract":"<div><div>Edge connectivity is an important indicator to evaluate edge fault tolerance. As for a host of networks, the edge connectivity is exactly equal to their minimum degree. Conditional matroidal edge connectivity is a new graph edge connectivity parameter that can be defined when a partition of the edge set is given. It is a good indicator to restrict the different kinds of faulty edges. However, the determination of conditional matroidal edge connectivity is often limited by the edge transitivity of the corresponding graph. In this paper, we use the algebraic technique to conquer the limitation of edge transitivity and give a new method to calculate the conditional matroidal edge connectivity of Cayley graphs. As an application, we not only show the conditional matroidal edge connectivity of varietal hypercube <span><math><mrow><mi>V</mi><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span>, but also verify the conditional matroidal edge connectivity of alternating group graph <span><math><mrow><mi>A</mi><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> that was studied by Zhang et al. (2022).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"373 ","pages":"Pages 271-278"},"PeriodicalIF":1.0,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143947560","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":"Use of simple arithmetic operations to construct efficiently implementable Boolean functions possessing high nonlinearity and good resistance to algebraic attacks","authors":"Claude Carlet , Palash Sarkar","doi":"10.1016/j.dam.2025.05.004","DOIUrl":"10.1016/j.dam.2025.05.004","url":null,"abstract":"<div><div>We describe a new class of Boolean functions which provide the presently best known trade-off between low computational complexity, nonlinearity and (fast) algebraic immunity. In particular, for <span><math><mrow><mi>n</mi><mo>≤</mo><mn>20</mn></mrow></math></span>, we show that there are functions in the family achieving a combination of nonlinearity and (fast) algebraic immunity which is superior to what is achieved by any other efficiently implementable function. The main novelty of our approach is to apply a judicious combination of simple integer and binary field arithmetic to Boolean function construction.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"373 ","pages":"Pages 256-270"},"PeriodicalIF":1.0,"publicationDate":"2025-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143943126","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":"r-dynamic colorings and the spectral radius in graphs","authors":"Jiangdong Ai , Suil O. , Liwen Zhang","doi":"10.1016/j.dam.2025.05.003","DOIUrl":"10.1016/j.dam.2025.05.003","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>ρ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the chromatic number and spectral radius of <span><math><mi>G</mi></math></span>, respectively. In 1967, Wilf proved that for a graph <span><math><mi>G</mi></math></span>, we have <span><math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>1</mn><mo>+</mo><mi>ρ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. An <span><math><mi>r</mi></math></span>-dynamic <span><math><mi>k</mi></math></span>-coloring of a graph <span><math><mi>G</mi></math></span> is a proper <span><math><mi>k</mi></math></span>-coloring of <span><math><mi>G</mi></math></span> such that every vertex <span><math><mi>v</mi></math></span> in <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> has neighbors in at least <span><math><mrow><mo>min</mo><mrow><mo>{</mo><mi>d</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>,</mo><mi>r</mi><mo>}</mo></mrow></mrow></math></span> different color classes. The <span><math><mi>r</mi></math></span>-dynamic chromatic number of a graph <span><math><mi>G</mi></math></span>, written <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is the least <span><math><mi>k</mi></math></span> such that <span><math><mi>G</mi></math></span> has such a <span><math><mi>k</mi></math></span>-coloring. Note that <span><math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>χ</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>1</mn><mo>+</mo><mi>r</mi><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> (*) (Jahanbekama et al., 2016). By the inequality (*), we observe that for a positive integer <span><math><mrow><mi>r</mi><mo>≥</mo><mn>2</mn></mrow></math></span> and a connected graph <span><math><mi>G</mi></math></span>, we have <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>1</mn><mo>+</mo><mi>r</mi><msup><mrow><mi>ρ</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>.</mo></mrow></math></span></div><div>In this paper, for a positive integer <span><math><mrow><mi>k</mi><mo>></mo><msup><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span>, we provide graphs <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>r</mi></mrow></msub></math></span> with <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><m","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"373 ","pages":"Pages 249-255"},"PeriodicalIF":1.0,"publicationDate":"2025-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143935843","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":"Relation between the H-rank of a mixed graph and the girth of its underlying graph","authors":"Suliman Khan","doi":"10.1016/j.dam.2025.05.006","DOIUrl":"10.1016/j.dam.2025.05.006","url":null,"abstract":"<div><div>Let <span><math><mrow><msup><mrow><mi>Σ</mi></mrow><mrow><mi>π</mi></mrow></msup><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mrow><mo>(</mo><msup><mrow><mi>Σ</mi></mrow><mrow><mi>π</mi></mrow></msup><mo>)</mo></mrow><mo>,</mo><mi>E</mi><mrow><mo>(</mo><msup><mrow><mi>Σ</mi></mrow><mrow><mi>π</mi></mrow></msup><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> be a mixed graph obtained from a simple graph <span><math><mi>Γ</mi></math></span> with the same vertex set <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></mrow></mrow></math></span> and an edge set <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></mrow></mrow></math></span> containing undirected edges and arcs. Let <span><math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>Σ</mi></mrow><mrow><mi>π</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> be the (first kind of) Hermitian adjacency matrix of <span><math><msup><mrow><mi>Σ</mi></mrow><mrow><mi>π</mi></mrow></msup></math></span>. The <span><math><mi>H</mi></math></span>-rank of <span><math><msup><mrow><mi>Σ</mi></mrow><mrow><mi>π</mi></mrow></msup></math></span> is the rank of <span><math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>Σ</mi></mrow><mrow><mi>π</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>, denoted by <span><math><mrow><msup><mrow><mi>r</mi></mrow><mrow><mi>H</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>Σ</mi></mrow><mrow><mi>π</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>. The girth of <span><math><mi>Γ</mi></math></span> is the length of the shortest cycle in <span><math><mi>Γ</mi></math></span>, dented by <span><math><mrow><mi>g</mi><mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></mrow></mrow></math></span> (or simply by <span><math><mi>g</mi></math></span>). In this paper, we show that under some conditions the <span><math><mi>H</mi></math></span>-rank of a mixed graph is equal to the girth of its underlying graph. Moreover, we characterize mixed graphs with <span><math><mi>H</mi></math></span>-rank <span><math><mrow><mi>g</mi><mo>−</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>g</mi><mo>+</mo><mn>2</mn></mrow></math></span>, distinct from the characterization of <span><math><mi>T</mi></math></span>-gain graphs provided by Khan (2024).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"373 ","pages":"Pages 239-248"},"PeriodicalIF":1.0,"publicationDate":"2025-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143943125","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":"Some results on (1,2)-rainbow connection number","authors":"Yingbin Ma, Yuyu Zhao","doi":"10.1016/j.dam.2025.05.008","DOIUrl":"10.1016/j.dam.2025.05.008","url":null,"abstract":"<div><div>In Li et al., (2018), proved the sharp upper bound of <span><math><mrow><mi>r</mi><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mi>r</mi><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mrow><mo>(</mo><mover><mrow><mi>G</mi></mrow><mo>¯</mo></mover><mo>)</mo></mrow></mrow></math></span> and left a problem of the sharp lower bound of <span><math><mrow><mi>r</mi><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mi>r</mi><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mrow><mo>(</mo><mover><mrow><mi>G</mi></mrow><mo>¯</mo></mover><mo>)</mo></mrow></mrow></math></span>. In this article, we solve this problem and show some sharp examples. In Doan and Do (2023), proved that <span><math><mrow><mi>r</mi><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>3</mn></mrow></math></span> for a graph <span><math><mi>G</mi></math></span> with large clique number <span><math><mrow><mi>ω</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mi>n</mi><mo>−</mo><mn>3</mn></mrow></math></span>. Then we completely determine the <span><math><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></math></span>-rainbow connection number of <span><math><mi>G</mi></math></span> with <span><math><mrow><mi>ω</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mi>n</mi><mo>−</mo><mn>3</mn></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"373 ","pages":"Pages 231-238"},"PeriodicalIF":1.0,"publicationDate":"2025-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143943127","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 unimodality of Zhang-Zhang polynomials of parallelogram chains","authors":"Guanru Li , Yi Wang , Qiqi Xiao","doi":"10.1016/j.dam.2025.05.007","DOIUrl":"10.1016/j.dam.2025.05.007","url":null,"abstract":"<div><div>The Zhang-Zhang polynomial counts Clar covers of a hexagonal system and is closely related to many important topological invariants. Heping Zhang and Fuji Zhang conjectured that the Zhang-Zhang polynomial of any hexagonal system has unimodal coefficients. In this paper we show that Zhang-Zhang polynomials of the parallelogram chains have some stronger unimodality properties.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"373 ","pages":"Pages 224-230"},"PeriodicalIF":1.0,"publicationDate":"2025-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143935842","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":"Parameterized complexity of locally minimal defensive alliances","authors":"Ajinkya Gaikwad, Soumen Maity, Shuvam Kant Tripathi","doi":"10.1016/j.dam.2025.05.001","DOIUrl":"10.1016/j.dam.2025.05.001","url":null,"abstract":"<div><div>A set <span><math><mi>S</mi></math></span> of vertices of a graph is a defensive alliance if, for each element of <span><math><mi>S</mi></math></span>, the majority of its neighbours is in <span><math><mi>S</mi></math></span>. We consider the notion of local minimality in this paper. We are interested in locally minimal defensive alliance of maximum size. This problem is known to be NP-hard but its parameterized complexity remains open until now. We enhance our understanding of the problem from the viewpoint of parameterized complexity. The main results of the paper are the following: (1) <span>Locally Minimal Defensive Alliance</span> is NP-complete, even when restricted to planar graphs, (2) a randomized FPT algorithm for <span>Exact Connected Locally Minimal Defensive Alliance</span> parameterized by solution size, (3) <span>Locally Minimal Defensive Alliance</span> is fixed-parameter tractable (FPT) when parameterized by neighbourhood diversity, (4) <span>Locally Minimal Defensive Alliance</span> parameterized by treewidth is W[1]-hard and thus not FPT (unless <span><math><mrow><mtext>FPT</mtext><mo>=</mo><mtext>W[1]</mtext></mrow></math></span>), (5) <span>Locally Minimal Defensive Alliance</span> can be solved in polynomial time for graphs of bounded treewidth.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"372 ","pages":"Pages 324-340"},"PeriodicalIF":1.0,"publicationDate":"2025-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143932108","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":"Approximating Graphic Min-Max and Minimum Cycle/Path/Tree Cover Problems","authors":"Wei Yu, Zhaohui Liu","doi":"10.1016/j.dam.2025.05.002","DOIUrl":"10.1016/j.dam.2025.05.002","url":null,"abstract":"<div><div>In this work we consider the Graphic Min-Max Cycle/Path/Tree Cover Problem and the Graphic Minimum Cycle/Path/Tree Cover Problem, some of which generalize the famous Graphic TSP. For all six problems, we obtain approximation algorithms with better ratios than the corresponding problems defined on general metrics. For the Graphic Minimum Path Cover Problem, we even show a best possible approximation ratio of 2, assuming <span><math><mrow><mi>P</mi><mo>≠</mo><mi>N</mi><mi>P</mi></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"372 ","pages":"Pages 314-323"},"PeriodicalIF":1.0,"publicationDate":"2025-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143929494","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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