Discrete Applied Mathematics最新文献

{"title":"Minimized compact automaton for clumps over degenerate patterns","authors":"E. Furletova ,&nbsp;J. Holub ,&nbsp;M. Régnier","doi":"10.1016/j.dam.2025.08.049","DOIUrl":"10.1016/j.dam.2025.08.049","url":null,"abstract":"<div><div>Clumps are sequences of overlapping occurrences of a given pattern that play a vital role in the study of distribution of pattern occurrences. These distributions are used for finding functional fragments in biological sequences. In this paper we present a minimized compacted automaton (Overlap walking automaton, <em>OWA</em>) recognizing all the possible clumps for degenerate patterns and its usage for computation of probabilities of sets of clumps. We also present <span>Aho–Corasick</span> like automaton, <em>RMinPatAut</em>, recognizing all the sequences ending with pattern occurrences. The states of <em>RMinPatAut</em> are equivalence classes on the prefixes of the pattern words. We use <em>RMinPatAut</em> as an auxiliary structure for <em>OWA</em> construction. For degenerate patterns, <em>RMinPatAut</em> is Nerode-minimal, i.e., minimal in classical sense. In this case <em>RMinPatAut</em> can be constructed in linear time on the number of its states (it is bounded by <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></math></span>, where <span><math><mi>m</mi></math></span> the length of pattern words). <em>OWA</em> can be constructed in linear time on the sum of its size and <em>RMinPatAut</em> size.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"380 ","pages":"Pages 51-67"},"PeriodicalIF":1.0,"publicationDate":"2025-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145061295","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":"An approximate dynamic programming approach for multi-stage stochastic lot-sizing under a Decision–Hazard–Decision information structure","authors":"Victor Spitzer ,&nbsp;Céline Gicquel ,&nbsp;Evgeny Gurevsky ,&nbsp;François Sanson","doi":"10.1016/j.dam.2025.08.051","DOIUrl":"10.1016/j.dam.2025.08.051","url":null,"abstract":"<div><div>This work studies a combinatorial optimization problem encountered in industrial production planning: the single-item multi-resource lot-sizing problem with inventory bounds and lost sales. The demand to be satisfied by the production plan is subject to uncertainty and only probabilistically known. We consider a multi-stage decision process with a Decision–Hazard–Decision information structure in which decisions are made at each stage both before and after the uncertainty is revealed. Such a setting has not yet been studied for stochastic lot-sizing problems, and the resulting problem is modeled as a multi-stage stochastic integer program. We propose a solution approach based on an approximate stochastic dynamic programming algorithm. It relies on a decomposition of the problem into single-stage sub-problems and on the estimation at each stage of the expected future costs. Due to the Decision–Hazard–Decision information structure, each nested single-stage sub-problem is itself a two-stage stochastic integer program. We therefore introduce a Benders decomposition scheme to reduce the computational effort required to solve each nested sub-problem, and present a special-purpose polynomial-time algorithm to efficiently solve the single-scenario second-stage sub-problems involved in the Benders decomposition. The results of extensive simulation experiments carried out on large-size randomly generated instances are reported. They demonstrate the practical benefit, in terms of the actual production cost, of using the proposed approach as compared to a naive deterministic optimization approach based on the expected demand.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"379 ","pages":"Pages 355-378"},"PeriodicalIF":1.0,"publicationDate":"2025-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145049050","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":"Hardness transitions of star colouring and restricted star colouring","authors":"Shalu M.A. ,&nbsp;Cyriac Antony","doi":"10.1016/j.dam.2025.08.056","DOIUrl":"10.1016/j.dam.2025.08.056","url":null,"abstract":"<div><div>We study how the complexity of the graph colouring problems star colouring and restricted star colouring vary with the maximum degree of the graph. Restricted star colouring (in short, rs colouring) is a variant of star colouring, as the name implies. For <span><math><mrow><mi>k</mi><mo>∈</mo><mi>N</mi></mrow></math></span>, a <span><math><mi>k</mi></math></span>-colouring of a graph <span><math><mi>G</mi></math></span> is a function <span><math><mrow><mi>f</mi><mo>:</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>→</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span> such that <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>≠</mo><mi>f</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span> for every edge <span><math><mrow><mi>u</mi><mi>v</mi></mrow></math></span> of <span><math><mi>G</mi></math></span>. A <span><math><mi>k</mi></math></span>-colouring of <span><math><mi>G</mi></math></span> is called a <span><math><mi>k</mi></math></span>-star colouring of <span><math><mi>G</mi></math></span> if there is no path <span><math><mrow><mi>u</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo>,</mo><mi>x</mi></mrow></math></span> in <span><math><mi>G</mi></math></span> with <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span>. A <span><math><mi>k</mi></math></span>-colouring of <span><math><mi>G</mi></math></span> is called a <span><math><mi>k</mi></math></span>-rs colouring of <span><math><mi>G</mi></math></span> if there is no path <span><math><mrow><mi>u</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi>w</mi></mrow></math></span> in <span><math><mi>G</mi></math></span> with <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>&gt;</mo><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow></mrow></math></span>. For <span><math><mrow><mi>k</mi><mo>∈</mo><mi>N</mi></mrow></math></span>, the problem <span><math><mi>k</mi></math></span>-<span>Star Colourability</span> takes a graph <span><math><mi>G</mi></math></span> as input and asks whether <span><math><mi>G</mi></math></span> admits a <span><math><mi>k</mi></math></span>-star colouring. The problem <span><math><mi>k</mi></math></span>-RS <span>Colourability</span> is defined similarly. Recently, Brause et al. (Electron. J. Comb., 2022) investigated the complexity of 3-star colouring with respect to the graph diameter. We study the complexity of <span><math><mi>k</mi></math></span>-star colouring and <span><math><mi>k</mi></math></span>-rs colouring with respect to the maximum degree for all <span><math><mrow><mi>k</mi><mo>≥</mo><mn>3</mn></mrow></math></span>. For <span><math><mr","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"380 ","pages":"Pages 1-33"},"PeriodicalIF":1.0,"publicationDate":"2025-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145050667","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":"Maximal polyomino chains with respect to the Kirchhoff index","authors":"Wensheng Sun ,&nbsp;Yujun Yang ,&nbsp;Shou-Jun Xu","doi":"10.1016/j.dam.2025.08.060","DOIUrl":"10.1016/j.dam.2025.08.060","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a connected graph. The resistance distance between two vertices <span><math><mi>u</mi></math></span> and <span><math><mi>v</mi></math></span> of <span><math><mi>G</mi></math></span> is defined as the potential difference generated between <span><math><mi>u</mi></math></span> and <span><math><mi>v</mi></math></span> induced by the unique <span><math><mrow><mi>u</mi><mo>→</mo><mi>v</mi></mrow></math></span> flow when a unit current flows in from node <span><math><mi>u</mi></math></span> and flows out from node <span><math><mi>v</mi></math></span>. The Kirchhoff index of <span><math><mi>G</mi></math></span> is defined as the sum of all the resistance distances pairs of <span><math><mi>G</mi></math></span>. Polyomino chains, as an important geometric structure, have been widely studied in statistical physics and mathematical chemistry. In this paper, by employing standard techniques from electrical networks and using comparison results on the Kirchhoff index of <span><math><mrow><mi>S</mi><mo>,</mo><mi>T</mi></mrow></math></span>-isomers, we first show that among all polyomino chains with <span><math><mi>n</mi></math></span> squares, the maximum Kirchhoff index is attained only when the polyomino chain is a “bend-free” chain. Furthermore, according to the recursion formula for the resistance distances, “bend-free” chains with maximum and minimum Kirchhoff index are characterized. As a result, the polyomino chains with maximum Kirchhoff index are obtained.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"380 ","pages":"Pages 34-50"},"PeriodicalIF":1.0,"publicationDate":"2025-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145050666","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":"k-edge-Hamilton-laceable bipartite graphs","authors":"Huimei Guo ,&nbsp;Rong-Xia Hao ,&nbsp;Jou-Ming Chang","doi":"10.1016/j.dam.2025.08.061","DOIUrl":"10.1016/j.dam.2025.08.061","url":null,"abstract":"<div><div>Kužel et al. expanded the concept of Hamilton-connectivity of a non-bipartite graph <span><math><mi>G</mi></math></span> to the so-called <span><math><mi>k</mi></math></span>-edge-Hamilton-connectivity (<span><math><mi>k</mi></math></span>-EHC for short) (Kužel et al., 2012). For a graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span>, let <span><math><mrow><msup><mrow><mi>E</mi></mrow><mrow><mo>+</mo></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><mi>u</mi><mi>v</mi><mo>:</mo><mspace></mspace><mi>u</mi><mo>,</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><mi>u</mi><mo>≠</mo><mi>v</mi><mo>}</mo></mrow></mrow></math></span> and define <span><math><mrow><mi>G</mi><mo>+</mo><mi>X</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>∪</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>X</mi><mo>⊂</mo><msup><mrow><mi>E</mi></mrow><mrow><mo>+</mo></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is called a potential edge set. A graph <span><math><mi>G</mi></math></span> is <span><math><mi>k</mi></math></span>-EHC if, for any potential edge set <span><math><mi>X</mi></math></span> with <span><math><mrow><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow><mo>≤</mo><mi>k</mi></mrow></math></span> such that every component of the graph induced by <span><math><mi>X</mi></math></span> is a path, the graph <span><math><mrow><mi>G</mi><mo>+</mo><mi>X</mi></mrow></math></span> has a Hamiltonian cycle passing all edges of <span><math><mi>X</mi></math></span>. This paper extends the analogous notion of <span><math><mi>k</mi></math></span>-EHC to a bipartite graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> to acquire a property called <span><math><mi>k</mi></math></span>-edge-Hamilton-laceability (<span><math><mi>k</mi></math></span>-EHL for short), in which the potential edge set <span><math><mi>X</mi></math></span> must fulfill the balanced condition (i.e., the number of potential edges with both ends appearing at <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> differs by at most one). We then characterize <span><math><mi>k</mi></math></span>-EHL bipartite graphs through three conditions related to fault-tolerant Hamilton-laceability, balanced paired <span><math><mi>k</mi></math></span>-disjoint coverage, and the inheritance property of <span><math><mi>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"379 ","pages":"Pages 379-389"},"PeriodicalIF":1.0,"publicationDate":"2025-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145049049","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":"Claw-free cubic graphs are (1, 1, 1, 3)-packing edge-colorable","authors":"Jingxi Hou ,&nbsp;Tao Wang ,&nbsp;Xiaojing Yang","doi":"10.1016/j.dam.2025.08.062","DOIUrl":"10.1016/j.dam.2025.08.062","url":null,"abstract":"<div><div>For a non-decreasing positive integer sequence <span><math><mrow><mi>S</mi><mo>=</mo><mrow><mo>(</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, an <span><math><mi>S</mi></math></span>-packing edge-coloring of a graph <span><math><mi>G</mi></math></span> is a partition of the edge set of <span><math><mi>G</mi></math></span> into subsets <span><math><mrow><msub><mrow><mi>E</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span> such that for each <span><math><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></mrow></math></span>, the distance between any two distinct edges <span><math><mrow><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></math></span> is at least <span><math><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><mn>1</mn></mrow></math></span>. Gastineau and Togni conjectured that cubic graphs, except the Petersen and Tietze graphs, admit <span><math><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>)</mo></mrow></math></span>-packing edge-colorings. In this paper, we prove that every claw-free cubic graph admits such a coloring.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"379 ","pages":"Pages 332-338"},"PeriodicalIF":1.0,"publicationDate":"2025-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145049047","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 vertices belonging to all edge metric bases","authors":"Anni Hakanen ,&nbsp;Ville Junnila ,&nbsp;Tero Laihonen ,&nbsp;Ismael G. Yero","doi":"10.1016/j.dam.2025.08.054","DOIUrl":"10.1016/j.dam.2025.08.054","url":null,"abstract":"<div><div>An edge metric basis of a connected graph <span><math><mi>G</mi></math></span> is a smallest possible set of vertices <span><math><mi>S</mi></math></span> of <span><math><mi>G</mi></math></span> satisfying the following: for any two edges <span><math><mrow><mi>e</mi><mo>,</mo><mi>f</mi></mrow></math></span> of <span><math><mi>G</mi></math></span> there is a vertex <span><math><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></math></span> such that the distances from <span><math><mi>s</mi></math></span> to <span><math><mi>e</mi></math></span> and <span><math><mi>f</mi></math></span> differ. The cardinality of an edge metric basis is the edge metric dimension of <span><math><mi>G</mi></math></span>. In this article we consider the existence of vertices in a graph <span><math><mi>G</mi></math></span> such that they must belong to each edge metric basis of <span><math><mi>G</mi></math></span>, and we call them <em>edge basis forced vertices</em>. On the other hand, we name <em>edge void vertices</em> those vertices which do not belong to any edge metric basis. Among other results, we first deal with the computational complexity of deciding whether a given vertex is an edge basis forced vertex or an edge void vertex. We also establish some tight bounds on the number of edge basis forced vertices of a graph, as well as, on the number of edges in a graph having at least one edge basis forced vertex. Moreover, we show some realization results concerning which values for the integers <span><math><mi>n</mi></math></span>, <span><math><mi>k</mi></math></span> and <span><math><mi>f</mi></math></span> allow to confirm the existence of a graph <span><math><mi>G</mi></math></span> with <span><math><mi>n</mi></math></span> vertices, <span><math><mi>f</mi></math></span> edge basis forced vertices and edge metric dimension <span><math><mi>k</mi></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"379 ","pages":"Pages 339-354"},"PeriodicalIF":1.0,"publicationDate":"2025-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145049048","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":"Locating eigenvalues of matrogenic graphs in linear time","authors":"Nelson Assis Junior,&nbsp;Vilmar Trevisan","doi":"10.1016/j.dam.2025.08.050","DOIUrl":"10.1016/j.dam.2025.08.050","url":null,"abstract":"<div><div>Using a decomposition due to Tyshkevich, we proved that matrogenic graphs have clique-width at most 4 and this allowed us to compute, in linear time, some of spectral parameters, such as the spectral radius and the algebraic connectivity, for subfamilies of these graphs. This was achieved using the Diagonalize Algorithm developed by M. Fürer et al. in 2019. The techniques applied here may provide valuable tools to further explore spectral properties of matrogenic graphs and related graph classes as well.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"379 ","pages":"Pages 316-331"},"PeriodicalIF":1.0,"publicationDate":"2025-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145027003","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 obtaining long m-sequences from low-degree primitive polynomials","authors":"Dimitri Kagaris","doi":"10.1016/j.dam.2025.08.058","DOIUrl":"10.1016/j.dam.2025.08.058","url":null,"abstract":"<div><div>Maximum-length sequences of length <span><math><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn></mrow></math></span> (m-sequences) are typically obtained by starting from a primitive polynomial of degree <span><math><mi>n</mi></math></span> over <span><math><mrow><mi>G</mi><mi>F</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> and configuring a Linear Feedback Shift Register (LFSR) based on that polynomial. In this study, we investigate the generation of <em>long</em> m-sequences based on a primitive polynomial of <em>low</em> degree. Specifically, we investigate a very simple form of an LFSR structure, referred to as <em>Two-Multiplier Split LFSR (2M-SLFSR)</em>, that consists of <span><math><mi>m</mi></math></span>\u0000 <span><math><mi>δ</mi></math></span>-bit cells and is based on a single low-degree primitive polynomial of degree <span><math><mrow><mi>δ</mi><mo>≥</mo><mn>2</mn></mrow></math></span> over <span><math><mrow><mi>G</mi><mi>F</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> and which can generate, with proper configuration, an m-sequence of length <span><math><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mi>δ</mi></mrow></msup><mo>−</mo><mn>1</mn></mrow></math></span>. For example, we show that starting from the primitive polynomial <span><math><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span> over <span><math><mrow><mi>G</mi><mi>F</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>, a 2M-SLFSR with <span><math><mrow><mi>m</mi><mo>=</mo><mn>599</mn></mrow></math></span> 2-bit cells can be constructed that yields an m-sequence of length <span><math><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>1198</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow></math></span>. M-sequences of large length such as <span><math><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>512</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow></math></span> obtained from low degree primitive polynomials via LFSR structures akin to 2M-SLFSR find current applications in stream ciphers like those used in SNOW-V and SNOW-Vi.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"379 ","pages":"Pages 272-287"},"PeriodicalIF":1.0,"publicationDate":"2025-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145019021","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 minimum number of maximal dissociation sets in unicyclic graphs","authors":"Junxia Zhang ,&nbsp;Xiangyu Ren ,&nbsp;Maoqun Wang","doi":"10.1016/j.dam.2025.08.053","DOIUrl":"10.1016/j.dam.2025.08.053","url":null,"abstract":"<div><div>A subset of vertices in a graph <span><math><mi>G</mi></math></span> is considered a maximal dissociation set if it induces a subgraph with vertex degree at most 1 and it is not contained within any other dissociation sets. In this paper, we show that every unicyclic graph of order <span><math><mi>n</mi></math></span> with <span><math><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></math></span> contains at least <span><math><mrow><mrow><mo>⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo></mrow><mo>+</mo><mn>2</mn></mrow></math></span> maximal dissociation sets. We also characterize the graphs attaining this lower bound.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"378 ","pages":"Pages 627-634"},"PeriodicalIF":1.0,"publicationDate":"2025-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145019115","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}