Discrete Applied Mathematics最新文献

{"title":"A graph discretization of vector Laplacian","authors":"Shu Li ,&nbsp;Lu Lu ,&nbsp;Jianfeng Wang","doi":"10.1016/j.dam.2025.09.019","DOIUrl":"10.1016/j.dam.2025.09.019","url":null,"abstract":"<div><div>As known, the scalar Laplacian gives the celebrated Laplacian matrix of a graph. In this paper, we determine the graph matrix presentation of vector Laplacian (or Helmholtz operator), named as Helmholtzian matrix. To compare the difference and similarity with previous graph matrices, we study the limit points of spectral radius and characterize the connected graphs with spectral radius at most 4.38+ via Helmholtzian matrix. Finally, we discuss the potential applications of Helmholtzian spectra of graphs in the simplicial networks and small-world networks.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"379 ","pages":"Pages 446-460"},"PeriodicalIF":1.0,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145117834","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":"Signless Laplacian spectral conditions for extremal quadrilateral and star embeddings","authors":"Zhe Wei,&nbsp;Zhenzhen Lou,&nbsp;Changxiang He","doi":"10.1016/j.dam.2025.09.016","DOIUrl":"10.1016/j.dam.2025.09.016","url":null,"abstract":"<div><div>The signless Laplacian spectral radius has emerged as a crucial spectral parameter in network science. This paper establishes new extremal results in spectral graph theory by investigating the signless Laplacian spectral radius (<span><math><mi>Q</mi></math></span>-index) of graphs with forbidden subgraphs. We present a <span><math><mi>Q</mi></math></span>-spectral analog of classical Nosal-type theorems, providing sharp conditions that guarantee the existence of either a 4-cycle or a large star <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>m</mi><mo>−</mo><mi>k</mi></mrow></msub></math></span> in such graphs. The main theorem states that for integers <span><math><mrow><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></math></span> and graphs <span><math><mi>G</mi></math></span> with <span><math><mi>m</mi></math></span> edges where <span><math><mrow><mi>m</mi><mo>≥</mo><mo>max</mo><mrow><mo>{</mo><mn>7</mn><mi>k</mi><mo>+</mo><mn>31</mn><mo>,</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>8</mn><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>}</mo></mrow></mrow></math></span>, if <span><math><mrow><mi>q</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mi>q</mi><mrow><mo>(</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msubsup><mo>)</mo></mrow></mrow></math></span>, then <span><math><mi>G</mi></math></span> must contain a 4-cycle or <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>m</mi><mo>−</mo><mi>k</mi></mrow></msub></math></span>, unless <span><math><mi>G</mi></math></span> is isomorphic to the extremal graph <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> formed by adding <span><math><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></math></span> independent edges to the star <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>m</mi><mo>−</mo><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span>. This result refines related previous work on star embeddings by Wang and Guo (2024), and completes the <span><math><mi>Q</mi></math></span>-spectral counterpart to Wang’s adjacency spectral theorem for 4-cycle containment (Wang, 2022). Our analysis reveals new insights into how signless Laplacian eigenvalues encode graph structure, with tight bounds demonstrated through explicit extremal graph constructions and asymptotic analysis.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"379 ","pages":"Pages 461-468"},"PeriodicalIF":1.0,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145117833","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":"Distribution of statistics on separable permutations restricted by a flat POP","authors":"Alice L.L. Gao ,&nbsp;Sergey Kitaev ,&nbsp;Ya-Xing Li ,&nbsp;Xuan Ruan","doi":"10.1016/j.dam.2025.09.017","DOIUrl":"10.1016/j.dam.2025.09.017","url":null,"abstract":"<div><div>Finding distributions of statistics in pattern-avoiding permutations has attracted significant attention in the literature. In particular, Chen, Kitaev, and Zhang derived functional equations for the joint distributions of any subset of classical minima and maxima statistics, as well as for the joint distributions of ascents and descents in separable permutations. Meanwhile, partially ordered patterns (POPs) have also been extensively studied. Notably, so-called flat POPs played a key role, via the notion of shape-Wilf-equivalence, in proving a conjecture on pattern-avoiding permutations.</div><div>In this paper, we study flat POP-avoiding separable permutations, where the maximum element in a flat POP receives the largest label. Avoiding such a POP imposes restrictions on the position of the maximum element in a separable permutation, forcing it to be positioned to the left. We establish a system of functional equations describing the joint distribution of six classical statistics in the most general case, extending the work of Chen, Kitaev, and Zhang.</div><div>As a specialization, when the POP has length 3, we recover a joint distribution result of Han and Kitaev on permutations avoiding classical patterns of length 3. As another specialization, for the flat POP of length 4, we derive an explicit rational generating function that captures the distribution of six statistics, with a numerator containing 100 monomials and a denominator containing 19 monomials.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"380 ","pages":"Pages 155-174"},"PeriodicalIF":1.0,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145119921","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":"Reconstruction of caterpillar tanglegrams","authors":"Ann Clifton ,&nbsp;Éva Czabarka ,&nbsp;Kevin Liu ,&nbsp;Sarah Loeb ,&nbsp;Utku Okur ,&nbsp;László Székely ,&nbsp;Kristina Wicke","doi":"10.1016/j.dam.2025.09.022","DOIUrl":"10.1016/j.dam.2025.09.022","url":null,"abstract":"<div><div>A tanglegram consists of two rooted binary trees with the same number of leaves and a perfect matching between the leaves of the trees. Given a size-<span><math><mi>n</mi></math></span> tanglegram, i.e., a tanglegram for two trees with <span><math><mi>n</mi></math></span> leaves, a multiset of induced size-<span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span> tanglegrams is obtained by deleting a pair of matched leaves in every possible way. Here, we analyze whether a size-<span><math><mi>n</mi></math></span> tanglegram is uniquely encoded by this multiset of size-<span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span> tanglegrams. We answer this question affirmatively in the case that at least one of the two trees of the tanglegram is a caterpillar tree.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"378 ","pages":"Pages 647-661"},"PeriodicalIF":1.0,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145117980","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":"Incorporating predictions in online graph coloring algorithms","authors":"Antonios Antoniadis,&nbsp;Hajo Broersma,&nbsp;Yang Meng","doi":"10.1016/j.dam.2025.08.063","DOIUrl":"10.1016/j.dam.2025.08.063","url":null,"abstract":"<div><div>We focus on learning augmented algorithms for the online graph coloring problem. We consider incorporating predictions in such algorithms to improve their performance. We apply this strategy in particular to the well-known greedy online graph coloring algorithm <span>FirstFit</span>. Although <span>FirstFit</span> is known to perform poorly in the worst case, we are able to establish a relationship between the structure of the input graph <span><math><mi>G</mi></math></span> that is revealed online and the number of colors that <span>FirstFit</span> uses for <span><math><mi>G</mi></math></span>. Based on this relationship, we propose an online coloring algorithm <span>FirstFitPredictions</span> that extends <span>FirstFit</span> while making use of machine learned predictions. We show that <span>FirstFitPredictions</span> is both <em>consistent</em> and <em>smooth</em>. Moreover, we develop a novel framework for combining online algorithms at runtime specifically for the online graph coloring problem. Finally, we show how this framework can be used to robustify <span>FirstFitPredictions</span> by combining it with any classical online coloring algorithm (that disregards the predictions).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"379 ","pages":"Pages 434-445"},"PeriodicalIF":1.0,"publicationDate":"2025-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145118241","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":"Joint equidistributions of mesh patterns 123 and 132 with minus antipodal shadings","authors":"Shuzhen Lv,&nbsp;Philip B. Zhang","doi":"10.1016/j.dam.2025.09.002","DOIUrl":"10.1016/j.dam.2025.09.002","url":null,"abstract":"<div><div>The study of joint equidistributions of mesh patterns 123 and 132 with the same symmetric shadings was recently initiated by Kitaev and Lv, where 75 of 80 potential joint equidistributions were proven. In this paper, we prove 112 out of 126 potential joint equidistributions of mesh patterns 123 and 132 with the same minus antipodal shadings. As a byproduct, we present 562 joint equidistribution results for non-symmetric and non-minus-antipodal shadings. To achieve this, we construct bijections, find recurrence relations, and obtain generating functions. Moreover, we demonstrate that the joint distributions of several pairs of mesh patterns are related to the unsigned Stirling numbers of the first kind.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"379 ","pages":"Pages 419-433"},"PeriodicalIF":1.0,"publicationDate":"2025-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145105086","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 partitioning a bipartite graph into cycles and degenerated cycles","authors":"Shuya Chiba ,&nbsp;Koshin Yoshida","doi":"10.1016/j.dam.2025.09.003","DOIUrl":"10.1016/j.dam.2025.09.003","url":null,"abstract":"<div><div>For a bipartite graph <span><math><mi>G</mi></math></span>, let <span><math><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the minimum degree sum of two non-adjacent vertices in different partite sets of <span><math><mi>G</mi></math></span>. We prove the following result: If <span><math><mi>G</mi></math></span> is a balanced bipartite graph of order <span><math><mrow><mn>2</mn><mi>n</mi><mo>≥</mo><mi>k</mi></mrow></math></span>, and if <span><math><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mi>n</mi><mo>−</mo><mfenced><mrow><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>/</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mn>1</mn></mrow></math></span>, then one of the following (i)–(iv) holds: (i) <span><math><mi>G</mi></math></span> contains <span><math><mi>k</mi></math></span> vertex-disjoint subgraphs <span><math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span> such that <span><math><mrow><msub><mrow><mo>⋃</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></mrow></msub><mi>V</mi><mrow><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and for each <span><math><mi>i</mi></math></span>, <span><math><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></mrow></math></span>, <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is a cycle or <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> or <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>; (ii) <span><math><mrow><mi>G</mi><mo>≅</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>6</mn></mrow></msub></mrow></math></span> and <span><math><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow></math></span>; (iii) <span><math><mrow><mi>G</mi><mo>≅</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>8</mn></mrow></msub></mrow></math></span> and <span><math><mrow><mi>k</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></math></span>; (iv) <span><math><mrow><mi>G</mi><mo>≅</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>10</mn></mrow></msub></mrow></math></span> and <span><math><mrow><mi>k</mi><mo>=</mo><mn>4</mn></mrow></math></span>. This result is a bipartite graph version of the result of Enomoto and Li (2004). We actually prove a stronger result which gives us control on the number of cycles in the <span><math><mi>k</mi></math></span> vertex-disjoint subgraphs of (i).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"378 ","pages":"Pages 635-646"},"PeriodicalIF":1.0,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145105020","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":"Cover numbers by certain graph families","authors":"Márton Marits","doi":"10.1016/j.dam.2025.09.009","DOIUrl":"10.1016/j.dam.2025.09.009","url":null,"abstract":"<div><div>We define the <em>cover number</em> of a graph <span><math><mi>G</mi></math></span> by a graph class <span><math><mi>P</mi></math></span> as the minimum number of graphs of class <span><math><mi>P</mi></math></span> required to cover the edge set of <span><math><mi>G</mi></math></span>. Taking inspiration from a paper by Harary et al. (1977), we find an exact formula for the cover number by the graph classes <span><math><mrow><mo>{</mo><mi>G</mi><mo>∣</mo><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>f</mi><mrow><mo>(</mo><mi>ω</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>}</mo></mrow></math></span> for all non-decreasing functions <span><math><mi>f</mi></math></span>.</div><div>After this, we establish a chain of inequalities between five cover numbers, the one by the class <span><math><mrow><mo>{</mo><mi>G</mi><mo>∣</mo><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>ω</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>}</mo></mrow></math></span>, by the class of perfect graphs, generalized split graphs, co-unipolar graphs and finally the cover number by bipartite graphs. We prove that at each inequality, the difference between the two sides can grow arbitrarily large. We also prove that the cover number by unipolar graphs cannot be expressed in terms of the chromatic or the clique number.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"379 ","pages":"Pages 400-404"},"PeriodicalIF":1.0,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145105085","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 tractability of defensive alliance problem","authors":"Sangam Balchandar Reddy,&nbsp;Anjeneya Swami Kare","doi":"10.1016/j.dam.2025.09.008","DOIUrl":"10.1016/j.dam.2025.09.008","url":null,"abstract":"<div><div>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>, a non-empty set <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi></mrow></math></span> is a defensive alliance if, for every vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>S</mi></mrow></math></span>, the majority of vertices in its closed neighbourhood belong to <span><math><mi>S</mi></math></span>; that is, <span><math><mrow><mrow><mo>|</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>[</mo><mi>v</mi><mo>]</mo></mrow><mo>∩</mo><mi>S</mi><mo>|</mo></mrow><mo>≥</mo><mrow><mo>|</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>[</mo><mi>v</mi><mo>]</mo></mrow><mo>∖</mo><mi>S</mi><mo>|</mo></mrow></mrow></math></span>. The Defensive Alliance problem (<span>Defensive Alliance</span>) asks for a defensive alliance of minimum cardinality. The decision version of the problem is known to be NP-complete even when restricted to split graphs and bipartite graphs. From a parameterized complexity perspective, the <span>Defensive Alliance</span> is known to be fixed-parameter tractable (FPT) when parameterized by the solution size, the vertex cover number, or the neighbourhood diversity of the input graph. In contrast, the problem is W[1]-hard when parameterized by the treewidth or the feedback vertex set number.</div><div>In this paper, we investigate the complexity of the <span>Defensive Alliance</span> on bounded degree graphs. We prove that the problem is <em>polynomial-time solvable</em> on graphs with maximum degree at most 5 but becomes NP-complete when the maximum degree is 6. This result rules out fixed-parameter tractability with respect to the maximum degree. Additionally, we analyse the problem from the perspective of parameterized complexity and present an FPT algorithm parameterized by twin cover number, thereby resolving an open question posed in Gaikwad and Maity (2022).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"380 ","pages":"Pages 116-127"},"PeriodicalIF":1.0,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145097619","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 toll walk transit function of a graph: Axiomatic characterizations and first-order non-definability","authors":"Manoj Changat ,&nbsp;Jeny Jacob ,&nbsp;Lekshmi Kamal K. Sheela ,&nbsp;Iztok Peterin","doi":"10.1016/j.dam.2025.09.006","DOIUrl":"10.1016/j.dam.2025.09.006","url":null,"abstract":"<div><div>A walk <span><math><mrow><mi>W</mi><mo>=</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>…</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span>, <span><math><mrow><mi>k</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, is called a toll walk if <span><math><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≠</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span> and <span><math><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span> are the only neighbors of <span><math><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> on <span><math><mi>W</mi></math></span> in a graph <span><math><mi>G</mi></math></span>. A toll walk interval <span><math><mrow><mi>T</mi><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>u</mi><mo>,</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, contains all the vertices that belong to a toll walk between <span><math><mi>u</mi></math></span> and <span><math><mi>v</mi></math></span>. The toll walk intervals yield a toll walk transit function <span><math><mrow><mi>T</mi><mo>:</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>×</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>→</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></msup></mrow></math></span>. We represent several axioms that characterize the toll walk transit function among chordal graphs, trees, asteroidal triple-free graphs, Ptolemaic graphs, and distance-hereditary graphs. We also show that the toll walk transit function cannot be described in the language of first-order logic for an arbitrary graph.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"380 ","pages":"Pages 128-145"},"PeriodicalIF":1.0,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145097642","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}