Discrete Applied Mathematics最新文献

{"title":"Some new bounds for the energy of graphs","authors":"Jiuying Dong,&nbsp;Yingying Yao","doi":"10.1016/j.dam.2025.03.022","DOIUrl":"10.1016/j.dam.2025.03.022","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a graph with <span><math><mi>n</mi></math></span> vertices and <span><math><mi>m</mi></math></span> edges. The energy of a graph <span><math><mi>G</mi></math></span> is defined as the sum of absolute values of the eigenvalues about its adjacency matrix, i.e. <span><math><mrow><mi>ɛ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mrow><mo>|</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow></mrow></math></span>. In this paper, we derive some new upper bounds on the graph energy based on a new formula and some inequalities for calculating the graph energy, and characterize the extremal graphs. In addition, we propose some new lower bounds for the graph energy involving order <span><math><mi>n</mi></math></span>, the size <span><math><mi>m</mi></math></span>, the eigenvalue with maximum absolute value <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and the eigenvalue with minimum absolute value <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of the graph <span><math><mi>G</mi></math></span>, and characterize the extremal graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 73-79"},"PeriodicalIF":1.0,"publicationDate":"2025-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143715514","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":"Preprocessing complexity for some graph problems parameterized by structural parameters","authors":"Manuel Lafond,&nbsp;Weidong Luo","doi":"10.1016/j.dam.2025.03.023","DOIUrl":"10.1016/j.dam.2025.03.023","url":null,"abstract":"<div><div>Structural graph parameters play an important role in parameterized complexity, including in kernelization. Notably, vertex cover, neighborhood diversity, twin-cover, and modular-width have been studied extensively in the last few years. However, there are many fundamental problems whose preprocessing complexity is not fully understood under these parameters. Indeed, the existence of polynomial kernels or polynomial Turing kernels for famous problems such as <span>Clique</span>, <span>Chromatic Number</span>, and <span>Steiner Tree</span> has only been established for a subset of structural parameters. In this work, we use several techniques to obtain a complete preprocessing complexity landscape for over a dozen of fundamental algorithmic problems.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 46-59"},"PeriodicalIF":1.0,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143696945","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":"Bounds on the Aα-spectral radius of uniform hypergraphs with some vertices deleted","authors":"Peng-Li Zhang ,&nbsp;Xiao-Dong Zhang","doi":"10.1016/j.dam.2025.03.020","DOIUrl":"10.1016/j.dam.2025.03.020","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the diagonal and adjacency tensors of a <span><math><mi>k</mi></math></span>-uniform hypergraph <span><math><mrow><mi>G</mi><mo>,</mo></mrow></math></span> respectively. The <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-spectral radius of <span><math><mi>G</mi></math></span> is defined as the spectral radius of the tensor <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>α</mi><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span> where <span><math><mrow><mn>0</mn><mo>≤</mo><mi>α</mi><mo>&lt;</mo><mn>1</mn><mo>.</mo></mrow></math></span> In this paper, we obtain an interlacing inequality on the spectral radius of a principal subtensor for a nonnegative weakly irreducible symmetric tensor, which is used to present several sharp lower bounds for the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-spectral radius of any subhypergraph <span><math><mrow><mi>G</mi><mo>−</mo><mi>S</mi></mrow></math></span> of a connected <span><math><mi>k</mi></math></span>-uniform hypergraph <span><math><mi>G</mi></math></span> in terms of the principal eigenvector associated with the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-spectral radius of <span><math><mi>G</mi></math></span>, degrees and co-degrees, where <span><math><mi>S</mi></math></span> is a subset of <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. They extend and strengthen some known results.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 1-16"},"PeriodicalIF":1.0,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143696939","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 disjunction convex hulls by big-M lifting","authors":"Yushan Qu,&nbsp;Jon Lee","doi":"10.1016/j.dam.2025.03.013","DOIUrl":"10.1016/j.dam.2025.03.013","url":null,"abstract":"<div><div>We study the natural extended-variable formulation for the disjunction of <span><math><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></math></span> polytopes in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>. We demonstrate that the convex hull <span><math><mi>D</mi></math></span> in the natural extended-variable space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi><mo>+</mo><mi>n</mi></mrow></msup></math></span> is given by full optimal big-M lifting (i) when <span><math><mrow><mi>d</mi><mo>≤</mo><mn>2</mn></mrow></math></span> (and that it is not generally true for <span><math><mrow><mi>d</mi><mo>≥</mo><mn>3</mn></mrow></math></span>), and also (ii) under some technical conditions, when the polytopes have a common facet-describing constraint matrix, for arbitrary <span><math><mrow><mi>d</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math></span>. We give a broad family of examples with <span><math><mrow><mi>d</mi><mo>≥</mo><mn>3</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow></math></span>, where the convex hull is not described after employing all full optimal big-M lifting inequalities, but it is described after one round of MIR inequalities. Additionally, we give some general results on the polyhedral structure of <span><math><mi>D</mi></math></span>, and we demonstrate that all facets of <span><math><mi>D</mi></math></span> can be enumerated in polynomial time when <span><math><mi>d</mi></math></span> is fixed.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 31-45"},"PeriodicalIF":1.0,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683253","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":"Grundy packing coloring of graphs","authors":"Didem Gözüpek ,&nbsp;Iztok Peterin","doi":"10.1016/j.dam.2025.03.024","DOIUrl":"10.1016/j.dam.2025.03.024","url":null,"abstract":"<div><div>A map <span><math><mrow><mi>c</mi><mo>:</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>→</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></mrow></math></span> of a graph <span><math><mi>G</mi></math></span> is a packing <span><math><mi>k</mi></math></span>-coloring if every two different vertices of the same color <span><math><mrow><mi>i</mi><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></mrow></math></span> are at distance more than <span><math><mi>i</mi></math></span>. The packing chromatic number <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mi>G</mi></math></span> is the smallest integer <span><math><mi>k</mi></math></span> such that there exists a packing <span><math><mi>k</mi></math></span>-coloring. In this paper we introduce the notion of <em>Grundy packing chromatic number</em>, analogous to the Grundy chromatic number of a graph. We first present a polynomial-time algorithm that is based on a greedy approach and gives a packing coloring of any graph <span><math><mi>G</mi></math></span>. We then define the Grundy packing chromatic number <span><math><mrow><msub><mrow><mi>Γ</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of a graph <span><math><mi>G</mi></math></span> as the maximum value that this algorithm yields in <span><math><mi>G</mi></math></span>. We present several properties of <span><math><mrow><msub><mrow><mi>Γ</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, provide results on the complexity of the problem as well as bounds and some exact results for <span><math><mrow><msub><mrow><mi>Γ</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 17-30"},"PeriodicalIF":1.0,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683268","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":"Pareto-scheduling of two competing agents with total weighted tardiness being one criterion","authors":"Jinwen Sun ,&nbsp;Rubing Chen ,&nbsp;Qiulan Zhao","doi":"10.1016/j.dam.2025.03.026","DOIUrl":"10.1016/j.dam.2025.03.026","url":null,"abstract":"<div><div>We study the Pareto-scheduling of two competing agents on a single machine, in which the jobs of at least one agent have their own equal processing times. When the criterion of one agent is the total weighted tardiness and the criterion of the other agent is the total completion time, the total tardiness or the total weighted completion time, the exact complexities of these problems remain open as posed by Chen et al. (2022). In this paper, we design a unified algorithm for solving these problems. As consequences, we show that these problems are solvable either in polynomial time or in pseudo-polynomial time. Combining the known results in the literature, we determine the complexity classification of nine problems.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"369 ","pages":"Pages 137-148"},"PeriodicalIF":1.0,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143681869","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 planar Turán number of double stars S2,l","authors":"Xin Xu ,&nbsp;Jiawei Shao ,&nbsp;Qiang Zhou","doi":"10.1016/j.dam.2025.03.016","DOIUrl":"10.1016/j.dam.2025.03.016","url":null,"abstract":"<div><div>The planar Turán number <span><math><mrow><msub><mrow><mi>ex</mi></mrow><mrow><mi>P</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mi>H</mi></math></span> is the maximum number of edges in an <span><math><mi>H</mi></math></span>-free planar graph on <span><math><mi>n</mi></math></span> vertices. The double star <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>l</mi></mrow></msub></math></span> is obtained from joining the centers of two stars each having <span><math><mi>k</mi></math></span> leaves and <span><math><mi>l</mi></math></span> leaves, respectively. In this paper, we give the exact value of <span><math><mrow><msub><mrow><mi>ex</mi></mrow><mrow><mi>P</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn><mo>,</mo><mn>5</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span>, which determines the planar Turán number for all double stars <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>l</mi></mrow></msub></math></span> when <span><math><mrow><mi>l</mi><mo>≥</mo><mn>2</mn></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"369 ","pages":"Pages 131-136"},"PeriodicalIF":1.0,"publicationDate":"2025-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143681914","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":"Fault-tolerability analysis of hypercubes based on 3-component path-structure connectivity","authors":"Bo Zhu ,&nbsp;Shumin Zhang ,&nbsp;Jou-Ming Chang ,&nbsp;Jinyu Zou","doi":"10.1016/j.dam.2025.03.021","DOIUrl":"10.1016/j.dam.2025.03.021","url":null,"abstract":"<div><div>Interconnection networks are essential in parallel computing and network science nowadays. Network failures are inevitable during operation and result in inestimable losses. Hence, designing an interconnection network with excellent performance is necessary. Reliability is a key indicator of the performance of interconnection networks, and its research originated from the first telecommunication switching network system. The failure of elements in a network system reduces overall communication capacity, leading to network congestion and system failure, typically measured by connectivity. In this paper, we introduce a new type of conditional connectivity of a graph <span><math><mi>G</mi></math></span>, termed <span><math><mi>r</mi></math></span>-component <span><math><mi>H</mi></math></span>-structure connectivity and denoted as <span><math><mrow><mi>c</mi><msub><mrow><mi>κ</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>;</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span>. Then, we investigate 3-component <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-structure connectivity for hypercube networks <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and acquire the result <span><span><span><math><mrow><mi>c</mi><msub><mrow><mi>κ</mi></mrow><mrow><mn>3</mn></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>;</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mfenced><mrow><mtable><mtr><mtd><mn>2</mn><mi>n</mi><mo>−</mo><mn>4</mn><mspace></mspace></mtd><mtd><mtext>if </mtext><mi>k</mi><mo>=</mo><mn>2</mn><mo>;</mo></mtd></mtr><mtr><mtd><mrow><mo>⌈</mo><mrow><mfrac><mrow><mn>4</mn><mi>n</mi><mo>−</mo><mn>5</mn></mrow><mrow><mi>k</mi></mrow></mfrac></mrow><mo>⌉</mo></mrow><mspace></mspace></mtd><mtd><mtext>for </mtext><mi>k</mi><mo>≥</mo><mn>4</mn><mtext> even</mtext><mo>;</mo></mtd></mtr><mtr><mtd><mrow><mo>⌈</mo><mrow><mfrac><mrow><mn>4</mn><mi>n</mi><mo>−</mo><mn>5</mn></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><mo>⌉</mo></mrow><mspace></mspace></mtd><mtd><mtext>for </mtext><mi>k</mi><mo>≥</mo><mn>3</mn><mtext> odd</mtext><mo>.</mo></mtd></mtr></mtable></mrow></mfenced></mrow></math></span></span></span></div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"370 ","pages":"Pages 111-123"},"PeriodicalIF":1.0,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143680942","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":"Clique Cover on L-EPG representations of graphs","authors":"Kedson Alves Silva ,&nbsp;Tanilson Dias dos Santos ,&nbsp;Uéverton dos Santos Souza","doi":"10.1016/j.dam.2025.03.027","DOIUrl":"10.1016/j.dam.2025.03.027","url":null,"abstract":"<div><div><span>Clique Cover</span> is a classical graph theory problem where we are 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>, an integer <span><math><mi>k</mi></math></span>, and asked if <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> can be partitioned into at most <span><math><mi>k</mi></math></span> cliques. Edge intersection graphs of paths in grids (EPG graphs) are graphs whose vertices can be represented as nontrivial paths in a grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. When the paths have at most one change of direction (bend), these graphs are called <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-EPG graphs; in addition, they are called <span><math><mi>⌞</mi></math></span>-EPG graphs (or <span><math><mi>⌞</mi></math></span>-shaped graphs) if all the paths are represented with one of the following shapes: “<span><math><mi>⌞</mi></math></span>”, “–”, or “<span><math><mo>∣</mo></math></span>”. Such a representation is called <span><math><mi>⌞</mi></math></span>-EPG representation. The class of <span><math><mi>⌞</mi></math></span>-EPG graphs is a natural superclass of interval graphs. Since all maximal cliques of a <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-EPG graph <span><math><mi>G</mi></math></span> can be computed in polynomial time, and <span>Clique Cover</span> on interval graphs is polynomial-time solvable; in this paper, we study the complexity of <span>Clique Cover</span> on <span><math><mi>⌞</mi></math></span>-EPG graphs. We show that given an <span><math><mi>⌞</mi></math></span>-EPG representation of a graph <span><math><mi>G</mi></math></span>, it is NP-complete to determine whether <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> can be partitioned into at most <span><math><mi>k</mi></math></span> cliques, but it can be solved in FPT time when parameterized by <span><math><mi>k</mi></math></span>. Furthermore, we adapt our algorithm to the case where the input is a graph without its <span><math><mi>⌞</mi></math></span>-EPG representation. Finally, we also show that <span>Clique Cover</span> on <span><math><mi>∂</mi></math></span>EPG<span><math><msub><mrow></mrow><mrow><mn>2</mn></mrow></msub></math></span> representations (a particular class of <span><math><mi>⌞</mi></math></span>-EPG representations) is polynomial-time solvable.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"370 ","pages":"Pages 145-156"},"PeriodicalIF":1.0,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143681990","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":"PSPACE-completeness of k-Atropos","authors":"Chao Yang ,&nbsp;Zhujun Zhang","doi":"10.1016/j.dam.2025.03.010","DOIUrl":"10.1016/j.dam.2025.03.010","url":null,"abstract":"<div><div>Burke and Teng introduced a two-player combinatorial game Atropos based on Sperner’s lemma, and showed that deciding whether one has a winning strategy for Atropos is <span>PSPACE</span>-complete. In the original Atropos game, the players must color a node adjacent to the last colored node. Burke and Teng also mentioned a variant <span><math><mi>k</mi></math></span>-Atropos in which each move is at most of distance <span><math><mi>k</mi></math></span> of the previous move, and asked a question on determining the computational complexity of this variant. In this paper, we answer this question by showing that for any fixed integer <span><math><mrow><mi>k</mi><mspace></mspace><mrow><mo>(</mo><mi>k</mi><mo>≥</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>, the associated decision problem <span><math><mi>k</mi></math></span>-<span>Atropos</span> is <span>PSPACE</span>-complete by reduction from True Quantified Boolean Formula (<span>TQBF</span>).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"368 ","pages":"Pages 190-198"},"PeriodicalIF":1.0,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143686565","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}