{"title":"Degree sequences for k-regulable ribbon realizations","authors":"Xia Guo , Jiyong Chen , Xian’an Jin","doi":"10.1016/j.dam.2025.05.044","DOIUrl":"10.1016/j.dam.2025.05.044","url":null,"abstract":"<div><div>The motivation question as to which ribbon graphs have a 4-regular checkerboard colorable twual is posed by Ellis-Monaghan and Moffatt. The ribbon graph <span><math><mi>G</mi></math></span> is a realization of the sequence <span><math><mi>D</mi></math></span> if its degree sequence is <span><math><mi>D</mi></math></span>. Furthermore, we refer to <span><math><mi>G</mi></math></span> as a <span><math><mi>k</mi></math></span>-regulable realization of <span><math><mi>D</mi></math></span> if the realization <span><math><mi>G</mi></math></span> of <span><math><mi>D</mi></math></span> has a <span><math><mi>k</mi></math></span>-regular partial dual. Since any Eulerian ribbon graph has a checkerboard colorable partial Petrial, we attempt to distinguish the ribbon graphs with <span><math><mi>k</mi></math></span>-regular partial duals directly from their degree sequences. Although the result is frustrating, some sequences lacking <span><math><mi>k</mi></math></span>-regulable realization are exposed. Moreover, we construct a family of <span><math><mi>k</mi></math></span>-regulable realizations for all sequences whose elements are greater than 1, except for the sequence <span><math><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>)</mo></mrow></math></span> when <span><math><mrow><mi>k</mi><mo>=</mo><mn>3</mn></mrow></math></span>, where the number of 2s and 3s are <span><math><mrow><mn>3</mn><msub><mrow><mi>j</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mn>2</mn><msub><mrow><mi>j</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mn>1</mn></mrow></math></span>, respectively, <span><math><mrow><msub><mrow><mi>j</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>j</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>></mo><mn>0</mn></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 50-65"},"PeriodicalIF":1.0,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144262442","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}
J. George Barnabas , Yegnanarayanan Venkataraman , Bryan Freyberg
{"title":"A note on prime distance graphs with chromatic number 3 or 4","authors":"J. George Barnabas , Yegnanarayanan Venkataraman , Bryan Freyberg","doi":"10.1016/j.dam.2025.05.030","DOIUrl":"10.1016/j.dam.2025.05.030","url":null,"abstract":"<div><div>Prime distance graphs (PDGs) are divided into four classes depending on whether their chromatic number is 1, 2, 3 or 4. It is still an open problem to completely characterize the family of PDGs with chromatic number 3 or 4. In this brief note we give necessary and sufficient conditions for fans and wheels to be PDGs. We also show (1) <span><math><mi>n</mi></math></span> edge-disjoint copies of a PDG may be chained together to form a PDG, and (2) the Cartesian product of a PDG with a path is a PDG.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 38-44"},"PeriodicalIF":1.0,"publicationDate":"2025-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144239379","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":"Performance of efficient variants of the 2-Opt heuristic for the traveling salesperson problem","authors":"Bodo Manthey, Jesse van Rhijn","doi":"10.1016/j.dam.2025.05.034","DOIUrl":"10.1016/j.dam.2025.05.034","url":null,"abstract":"<div><div>We analyze variants of the 2-opt local search heuristic for the Traveling Salesperson Problem (TSP) with guaranteed polynomial running-time. First we consider X-opt, a heuristic that removes intersecting pairs of edges from two-dimensional Euclidean instances. We show that the longest X-optimal tour may be approximately <span><math><mrow><mi>n</mi><mo>/</mo><mn>2</mn></mrow></math></span> times longer than the optimal tour in the worst case. Moreover, even when the instance consists of <span><math><mi>n</mi></math></span> points placed uniformly at random in the unit square, the longest tour is <span><math><mrow><mi>Ω</mi><mrow><mo>(</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>)</mo></mrow></mrow></math></span> times longer than the optimal tour. Next, we propose a new heuristic, which we call Y-opt, that is defined for all TSP instances, not just Euclidean ones. Y-opt has essentially the same approximation guarantees as the well-studied 2-opt. We furthermore evaluate the approximation performance of both X-opt and Y-opt numerically on random instances and compare them to 2-opt. While Y-opt behaves as predicted, we find that X-opt appears to have a constant approximation ratio on these instances in practice.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 7-16"},"PeriodicalIF":1.0,"publicationDate":"2025-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144229564","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":"Closures and heavy pairs for hamiltonicity","authors":"Wangyi Shang , Hajo Broersma , Shenggui Zhang , Binlong Li","doi":"10.1016/j.dam.2025.05.028","DOIUrl":"10.1016/j.dam.2025.05.028","url":null,"abstract":"<div><div>We say that a graph <span><math><mi>G</mi></math></span> on <span><math><mi>n</mi></math></span> vertices is <span><math><mrow><mo>{</mo><mi>H</mi><mo>,</mo><mi>F</mi><mo>}</mo></mrow></math></span>-<span><math><mi>o</mi></math></span>-heavy if every induced subgraph of <span><math><mi>G</mi></math></span> isomorphic to <span><math><mi>H</mi></math></span> or <span><math><mi>F</mi></math></span> contains two nonadjacent vertices with degree sum at least <span><math><mi>n</mi></math></span>. Generalizing earlier sufficient forbidden subgraph conditions for hamiltonicity, in 2012, Li, Ryjáček, Wang and Zhang determined all connected graphs <span><math><mi>R</mi></math></span> and <span><math><mi>S</mi></math></span> of order at least 3 other than <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> such that every 2-connected <span><math><mrow><mo>{</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>}</mo></mrow></math></span>-<span><math><mi>o</mi></math></span>-heavy graph is hamiltonian. In particular, they showed that, up to symmetry, <span><math><mi>R</mi></math></span> must be a claw and <span><math><mrow><mi>S</mi><mo>∈</mo><mrow><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi>B</mi><mo>,</mo><mi>N</mi><mo>,</mo><mi>W</mi><mo>}</mo></mrow></mrow></math></span>. In 2008, Čada extended Ryjáček’s closure concept for claw-free graphs by introducing what we call the <span><math><mi>c</mi></math></span>-closure for claw-<span><math><mi>o</mi></math></span>-heavy graphs. We apply it here to characterize the structure of the <span><math><mi>c</mi></math></span>-closure of 2-connected <span><math><mrow><mo>{</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>}</mo></mrow></math></span>-<span><math><mi>o</mi></math></span>-heavy graphs, where <span><math><mi>R</mi></math></span> and <span><math><mi>S</mi></math></span> are as above. Our main results extend or generalize several earlier results on hamiltonicity involving forbidden or <span><math><mi>o</mi></math></span>-heavy subgraphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 25-37"},"PeriodicalIF":1.0,"publicationDate":"2025-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144229539","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":"Determining some graph joins by the signless Laplacian spectrum","authors":"Jiachang Ye , Jianguo Qian , Zoran Stanić","doi":"10.1016/j.dam.2025.05.035","DOIUrl":"10.1016/j.dam.2025.05.035","url":null,"abstract":"<div><div>A graph is determined by its signless Laplacian spectrum if there is no other non-isomorphic graph sharing the same signless Laplacian spectrum. Let <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span>, <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span>, <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>l</mi><mo>−</mo><mi>s</mi></mrow></msub></math></span> be the cycle, the path, the complete graph and the complete bipartite graph with <span><math><mi>l</mi></math></span> vertices, respectively. We prove that <span><math><mrow><mi>G</mi><mo>≅</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∨</mo><mrow><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>∪</mo><msub><mrow><mi>C</mi></mrow><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>∪</mo><mo>⋯</mo><mo>∪</mo><msub><mrow><mi>C</mi></mrow><mrow><msub><mrow><mi>l</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></msub><mo>∪</mo><mi>s</mi><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mo>,</mo></mrow></math></span> with <span><math><mrow><mi>s</mi><mo>≥</mo><mn>0</mn><mo>,</mo><mi>t</mi><mo>≥</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>≥</mo><mn>22</mn></mrow></math></span>, is determined by the signless Laplacian spectrum if and only if either <span><math><mrow><mi>s</mi><mo>=</mo><mn>0</mn></mrow></math></span> or <span><math><mrow><mi>s</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><msub><mrow><mi>l</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≠</mo><mn>3</mn></mrow></math></span> holds for <span><math><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>t</mi></mrow></math></span>, where <span><math><mi>n</mi></math></span> is the order of <span><math><mi>G</mi></math></span>, and <span><math><mo>∪</mo></math></span> and <span><math><mo>∨</mo></math></span> stand for the disjoint union and the join of two graphs, respectively. Moreover, for <span><math><mrow><mi>s</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><msub><mrow><mi>l</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mn>3</mn></mrow></math></span>, <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∨</mo><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>3</mn></mrow></msub><mo>∪</mo><msub><mrow><mi>C</mi></mrow><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>∪</mo><msub><mrow><mi>C</mi></mrow><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>∪</mo><mo>⋯</mo><mo>∪</mo><msub><mrow><mi>C</mi></mrow><mrow><msub><mrow><mi>l</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub><mo>∪</mo><mrow><mo>(</mo><mi>s</mi><mo>−</mo><mn>1","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 17-24"},"PeriodicalIF":1.0,"publicationDate":"2025-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144229538","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 maximum induced forests of the balanced bipartite graphs","authors":"Ali Ghalavand, Xueliang Li","doi":"10.1016/j.dam.2025.05.029","DOIUrl":"10.1016/j.dam.2025.05.029","url":null,"abstract":"<div><div>Let <span><math><mi>B</mi></math></span> be a balanced bipartite graph with two parts, <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>, each containing <span><math><mi>n</mi></math></span> vertices, resulting in a total of <span><math><mrow><mn>2</mn><mi>n</mi></mrow></math></span> vertices. Recently, Wang and Wu conjectured that if the minimum degree of <span><math><mi>B</mi></math></span>, denoted as <span><math><mrow><mi>δ</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow></math></span>, is greater than or equal to <span><math><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mn>1</mn></mrow></math></span>, then the largest order of an induced forest in <span><math><mi>B</mi></math></span> is equal to <span><math><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></math></span>. In this paper, we prove this conjecture and show that the condition on the minimum degree cannot be relaxed in general terms. Furthermore, we determine that if <span><math><mrow><mi>δ</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow><mo>≥</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mn>1</mn></mrow></math></span>, then any subset <span><math><mi>S</mi></math></span> of vertices in <span><math><mi>B</mi></math></span> that induces a forest of size <span><math><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></math></span> will satisfy the conditions <span><math><mrow><mo>min</mo><mrow><mo>{</mo><mrow><mo>|</mo><mi>S</mi><mo>∩</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>|</mo></mrow><mo>,</mo><mrow><mo>|</mo><mi>S</mi><mo>∩</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>|</mo></mrow><mo>}</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span> when <span><math><mi>n</mi></math></span> is odd, and <span><math><mrow><mo>min</mo><mrow><mo>{</mo><mrow><mo>|</mo><mi>S</mi><mo>∩</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>|</mo></mrow><mo>,</mo><mrow><mo>|</mo><mi>S</mi><mo>∩</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>|</mo></mrow><mo>}</mo></mrow><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>}</mo></mrow></mrow></math></span> when <span><math><mi>n</mi></math></span> is even. Additionally, we identify infinitely many balanced bipartite graphs that meet these conditions.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 1-6"},"PeriodicalIF":1.0,"publicationDate":"2025-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144222226","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":"2-distance coloring of planar graphs without 4, 6-cycles","authors":"Yuehua Bu , Zhimin Bao , Hongguo Zhu","doi":"10.1016/j.dam.2025.05.023","DOIUrl":"10.1016/j.dam.2025.05.023","url":null,"abstract":"<div><div>For a graph <span><math><mi>G</mi></math></span> and a positive integer <span><math><mi>k</mi></math></span>, a 2-distance <span><math><mi>k</mi></math></span>-coloring of <span><math><mi>G</mi></math></span> is a proper <span><math><mi>k</mi></math></span>-coloring such that any two vertices at distance 2 cannot share the same color. In this paper, we prove that every planar graph without 4, 6-cycles and <span><math><mrow><mi>Δ</mi><mo>≥</mo><mn>20</mn></mrow></math></span> admits a 2-distance <span><math><mrow><mo>(</mo><mi>Δ</mi><mo>+</mo><mn>3</mn><mo>)</mo></mrow></math></span>-coloring.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"374 ","pages":"Pages 135-143"},"PeriodicalIF":1.0,"publicationDate":"2025-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144116418","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":"Efficient methods of constructing universal cycles for k-permutations","authors":"Zuling Chang , Lingyu Diao , Shujie Wang","doi":"10.1016/j.dam.2025.05.020","DOIUrl":"10.1016/j.dam.2025.05.020","url":null,"abstract":"<div><div>We present two efficient methods of constructing universal cycles for the set of all <span><math><mi>k</mi></math></span>-permutations of the <span><math><mi>n</mi></math></span>-set <span><math><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></mrow></math></span> for <span><math><mrow><mi>n</mi><mo>></mo><mi>k</mi><mo>≥</mo><mn>3</mn></mrow></math></span>. These two methods generate universal cycles for <span><math><mi>k</mi></math></span>-permutations from pure cycling registers with feedback function <span><math><mrow><mi>f</mi><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></math></span>, using the cycle joining method. Here we design two classes of successor rules that build upon a framework proposed by Gabric et al. (2020), each of which produces <span><math><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></math></span> shift inequivalent universal cycles for <span><math><mi>k</mi></math></span>-permutations. Each universal cycle for <span><math><mi>k</mi></math></span>-permutations can be generated in <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> time per symbol using <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> space.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"374 ","pages":"Pages 120-134"},"PeriodicalIF":1.0,"publicationDate":"2025-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144107445","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 Clow , Melissa A. Huggan , M.E. Messinger
{"title":"Cops and attacking robbers with cycle constraints","authors":"Alexander Clow , Melissa A. Huggan , M.E. Messinger","doi":"10.1016/j.dam.2025.05.019","DOIUrl":"10.1016/j.dam.2025.05.019","url":null,"abstract":"<div><div>This paper considers the Cops and Attacking Robbers game, a variant of Cops and Robbers, where the robber is empowered to attack a cop in the same way a cop can capture the robber. In a graph <span><math><mi>G</mi></math></span>, the number of cops required to capture a robber in the Cops and Attacking Robbers game is denoted by <span><math><mrow><mo>cc</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. We give a sufficient condition for a triangle-free graph to have attacking cop number at most 2 and we characterise when outerplanar graphs have attacking cop number 2. We also prove that all bipartite planar graphs <span><math><mi>G</mi></math></span> have <span><math><mrow><mo>cc</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>4</mn></mrow></math></span> and show this is tight by constructing a bipartite planar graph <span><math><mi>G</mi></math></span> with <span><math><mrow><mo>cc</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mn>4</mn></mrow></math></span>. Finally we construct 17 non-isomorphic graphs <span><math><mi>H</mi></math></span> of order 58 with <span><math><mrow><mo>cc</mo><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>=</mo><mn>6</mn></mrow></math></span> and <span><math><mrow><mo>c</mo><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn></mrow></math></span>. This provides the first example of a graph <span><math><mi>H</mi></math></span> with <span><math><mrow><mo>cc</mo><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>−</mo><mo>c</mo><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>≥</mo><mn>3</mn></mrow></math></span>, extending work by Bonato et al. (2014). We conclude with a list of conjectures and open problems.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"373 ","pages":"Pages 327-342"},"PeriodicalIF":1.0,"publicationDate":"2025-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144098556","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":"Fixed-parameter algorithms for cardinality-constrained graph partitioning problems on sparse graphs","authors":"Suguru Yamada , Tesshu Hanaka","doi":"10.1016/j.dam.2025.05.012","DOIUrl":"10.1016/j.dam.2025.05.012","url":null,"abstract":"<div><div>For an undirected and edge-weighted graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> and a vertex subset <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi></mrow></math></span>, we define a function <span><math><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow><mo>≔</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow><mi>⋅</mi><mi>w</mi><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow><mo>+</mo><mi>α</mi><mi>⋅</mi><mi>w</mi><mrow><mo>(</mo><mi>S</mi><mo>,</mo><mi>V</mi><mo>∖</mo><mi>S</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></math></span> is a real number, <span><math><mrow><mi>w</mi><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow></mrow></math></span> is the sum of the weights of edges having two endpoints in <span><math><mi>S</mi></math></span>, and <span><math><mrow><mi>w</mi><mrow><mo>(</mo><mi>S</mi><mo>,</mo><mi>V</mi><mo>∖</mo><mi>S</mi><mo>)</mo></mrow></mrow></math></span> is the sum of the weights of edges having one endpoint in <span><math><mi>S</mi></math></span> and the other in <span><math><mrow><mi>V</mi><mo>∖</mo><mi>S</mi></mrow></math></span>. Then, given an undirected and edge-weighted graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> and a positive integer <span><math><mi>k</mi></math></span>, <span>Max (Min)</span>\u0000 <span><math><mi>α</mi></math></span>-<span>Fixed Cardinality Graph Partitioning (Max (Min)</span>\u0000 <span><math><mi>α</mi></math></span>-<span>FCGP)</span> is the problem to find a vertex subset <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi></mrow></math></span> of size <span><math><mi>k</mi></math></span> that maximizes (minimizes) <span><math><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow></mrow></math></span>. In this paper, we first show that <span>Max</span>\u0000 <span><math><mi>α</mi></math></span>-<span>FCGP</span> with <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>[</mo><mn>1</mn><mo>/</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></math></span> and <span>Min</span>\u0000 <span><math><mi>α</mi></math></span>-<span>FCGP</span> with <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>/</mo><mn>3</mn><mo>]</mo></mrow></mrow></math></span> can be solved in time <span><math><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>o</mi><mrow><mo>(</mo><mi>k</mi><mi>d</mi><mo>+</mo><mi>k</mi><mo>)</mo></mrow></mrow></msup><msup><mrow><mrow><mo>(</mo><mi>e</mi><mo>+</mo><mi>e</mi><mi>d</mi><mo>)</mo></mrow></mrow><mrow><mi>k</mi></mrow></msup><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup></mrow></math></span> where <span><math><","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"373 ","pages":"Pages 343-354"},"PeriodicalIF":1.0,"publicationDate":"2025-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144098557","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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