Discrete Applied Mathematics最新文献

{"title":"The completely independent spanning trees in P4-free graphs","authors":"Jun Yuan , Jiya Hao , Aixia Liu , Shan Liu , Shuchang Chai , Shangwei Lin","doi":"10.1016/j.dam.2025.08.048","DOIUrl":"10.1016/j.dam.2025.08.048","url":null,"abstract":"<div><div>Hasunuma has conjectured that every <span><math><mrow><mn>2</mn><mi>k</mi></mrow></math></span>-connected graph admits <span><math><mi>k</mi></math></span> completely independent spanning trees. Pai et al. showed the conjecture is true for complete graphs, complete bipartite graphs and complete tripartite graphs, all of which are <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-free. Chen et al. showed there exist two completely independent spanning trees in every 4-connected <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-free graph. In this paper, we further discuss the completely independent spanning trees in both complete multipartite graphs and <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-free graphs. Let <span><math><msub><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></msub></math></span> be a complete <span><math><mi>m</mi></math></span>-partite graph, where <span><math><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≤</mo><mo>⋯</mo><mo>≤</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>.</mo></mrow></math></span> We first prove that <span><math><msub><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></msub></math></span> contains <span><math><mrow><mo>min</mo><mrow><mo>{</mo><mrow><mo>⌊</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><munderover><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></munderover><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>⌋</mo></mrow><mo>,</mo><mrow><mo>⌊</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac><munderover><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>⌋</mo></mrow><mo>}</mo></mrow></mrow></math></span> completely independent spanning trees. This implies that every <span><math><mrow><mn>2</mn><mi>k</mi></mrow></math></span>-connected complete <span><math><mi>m</mi></math></span>-partite graph admits <span><math><mi>k</mi></math></span> completely independent spanning trees if its order is at least <span><math><mrow><mn>3</mn><mi>k</mi><mo>.</mo></mrow></math></span> We further provide the upper and lower bounds of the maximum number of completely independent spanning trees in complete <span><math><mi>m</mi></math></span>-partite graphs, and dem","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 573-585"},"PeriodicalIF":1.0,"publicationDate":"2025-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144907765","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 rich-neighbor edge-coloring of sparse graphs","authors":"Lily Chen ,&nbsp;Chenghao Nan ,&nbsp;Xiangqian Zhou","doi":"10.1016/j.dam.2025.08.043","DOIUrl":"10.1016/j.dam.2025.08.043","url":null,"abstract":"<div><div>Let <span><math><mi>ϕ</mi></math></span> be a proper edge-coloring of a graph <span><math><mi>G</mi></math></span>. An edge <span><math><mi>e</mi></math></span> is rich if the edges adjacent to <span><math><mi>e</mi></math></span> receive distinct colors. In particular, a pendant edge and an isolated edge are both rich. Petruševski and Škrekovski (2024) introduced the concept of rich-neighbor edge-coloring as a weakening of strong edge-coloring. A proper <span><math><mi>k</mi></math></span>-edge-coloring <span><math><mi>ϕ</mi></math></span> is a rich-neighbor <span><math><mi>k</mi></math></span>-coloring if each non-isolated edge is adjacent to at least one rich edge. Petruševski and Škrekovski (2024) conjectured that every connected subcubic graph admits a rich-neighbor 5-coloring except for <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>. We show that if <span><math><mi>G</mi></math></span> is a subcubic graph, then <ul><li><span>•</span><span><div>[(1)] if <span><math><mrow><mi>m</mi><mi>a</mi><mi>d</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>&lt;</mo><mfrac><mrow><mn>36</mn></mrow><mrow><mn>13</mn></mrow></mfrac></mrow></math></span>, then <span><math><mi>G</mi></math></span> has a rich-neighbor 6-coloring; and</div></span></li><li><span>•</span><span><div>[(2)] if <span><math><mi>G</mi></math></span> is claw-free, then <span><math><mi>G</mi></math></span> has a rich-neighbor 6-coloring.</div></span></li></ul></div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"378 ","pages":"Pages 538-546"},"PeriodicalIF":1.0,"publicationDate":"2025-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144908617","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 path sequence of a graph","authors":"Yirong Cai,&nbsp;Hanyuan Deng","doi":"10.1016/j.dam.2025.08.046","DOIUrl":"10.1016/j.dam.2025.08.046","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> be the path sequence of a graph <span><math><mi>G</mi></math></span>, where <span><math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is the number of paths with length <span><math><mi>i</mi></math></span> and <span><math><mi>ρ</mi></math></span> is the length of a longest path in <span><math><mi>G</mi></math></span>. In this paper, we first give the path sequences of some graphs and show that the number of paths with length <span><math><mi>h</mi></math></span> in a starlike tree is completely determined by its branches of length not more than <span><math><mrow><mi>h</mi><mo>−</mo><mn>2</mn></mrow></math></span>. And then we consider whether the path sequence characterizes a graph from a different point of view and find that any two graphs in some graph families are isomorphic if and only if they have the same path sequence.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"378 ","pages":"Pages 553-567"},"PeriodicalIF":1.0,"publicationDate":"2025-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144913551","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 generalized Turán number of graphs with bounded matching number","authors":"Yisai Xue ,&nbsp;Liying Kang","doi":"10.1016/j.dam.2025.08.029","DOIUrl":"10.1016/j.dam.2025.08.029","url":null,"abstract":"<div><div>The generalized Turán number <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> is defined as the maximum number of copies of a graph <span><math><mi>H</mi></math></span> in an <span><math><mi>n</mi></math></span>-vertex graph that does not contain any graph <span><math><mrow><mi>F</mi><mo>∈</mo><mi>F</mi></mrow></math></span>. Alon and Frankl initiated the study of Turán problems with a bounded matching number. In this paper, we establish stability results for generalized Turán problems with bounded matching number. Using the stability results, we provide exact values of <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mrow><mo>{</mo><mi>F</mi><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span> for <span><math><mi>F</mi></math></span> being any non-bipartite graph or a path.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 586-597"},"PeriodicalIF":1.0,"publicationDate":"2025-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144911853","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":"Recursive characterization of maximal bipartite planar graphs","authors":"Metrose Metsidik ,&nbsp;Helin Gong","doi":"10.1016/j.dam.2025.08.032","DOIUrl":"10.1016/j.dam.2025.08.032","url":null,"abstract":"<div><div>In this paper, we initiate our study by demonstrating that bipartite-minors, derived through vertex/edge deletions and edge contractions within a bond, inherently preserve the bipartite property. Leveraging this, we establish a well-quasi-ordering on the class of bipartite graphs. Subsequently, we provide elegant and succinct characterizations of bipartite graphs, planar bipartite graphs, and outer-planar bipartite graphs, all formulated in terms of excluded bipartite-minors. Finally, we introduce a novel planar operation that enables the recursive construction of all maximal bipartite planar graphs, starting from a single vertex.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"378 ","pages":"Pages 547-552"},"PeriodicalIF":1.0,"publicationDate":"2025-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144913553","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":"Chords of longest cycles in 3-connected graphs with some special circumferences","authors":"Danning Wang,&nbsp;Jun Yue","doi":"10.1016/j.dam.2025.08.040","DOIUrl":"10.1016/j.dam.2025.08.040","url":null,"abstract":"<div><div>Thomassen’s chord conjecture from 1976 states that every longest cycle in a 3-connected graph has a chord. In this paper, we prove that it holds for any 3-connected graph with a small induced circumference or a large circumference.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 541-551"},"PeriodicalIF":1.0,"publicationDate":"2025-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144904508","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":"New results on the 1-isolation number of graphs without short cycles","authors":"Yirui Huang,&nbsp;Gang Zhang,&nbsp;Xian’an Jin","doi":"10.1016/j.dam.2025.08.033","DOIUrl":"10.1016/j.dam.2025.08.033","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a graph. A subset <span><math><mrow><mi>D</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is called a 1-isolating set of <span><math><mi>G</mi></math></span> if <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>−</mo><mi>N</mi><mrow><mo>[</mo><mi>D</mi><mo>]</mo></mrow><mo>)</mo></mrow><mo>≤</mo><mn>1</mn></mrow></math></span>, that is, <span><math><mrow><mi>G</mi><mo>−</mo><mi>N</mi><mrow><mo>[</mo><mi>D</mi><mo>]</mo></mrow></mrow></math></span> consists of isolated edges and isolated vertices only. The 1-isolation number of <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><msub><mrow><mi>ι</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is the cardinality of a smallest 1-isolating set of <span><math><mi>G</mi></math></span>. In this paper, we prove that if <span><math><mrow><mi>G</mi><mo>∉</mo><mrow><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>11</mn></mrow></msub><mo>}</mo></mrow></mrow></math></span> is a connected graph of order <span><math><mi>n</mi></math></span> without 6-cycles, or without induced 5- and 6-cycles, then <span><math><mrow><msub><mrow><mi>ι</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></math></span>. Both bounds are sharp.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"379 ","pages":"Pages 222-235"},"PeriodicalIF":1.0,"publicationDate":"2025-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144904658","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":"Egan conjecture holds","authors":"Sergei Drozdov","doi":"10.1016/j.dam.2025.08.034","DOIUrl":"10.1016/j.dam.2025.08.034","url":null,"abstract":"<div><div>Given a Euclidean simplex of dimension <span><math><mrow><mi>n</mi><mo>⩾</mo><mn>2</mn></mrow></math></span> let its radii of inscribed and circumscribed spheres be <span><math><mi>r</mi></math></span> and <span><math><mi>R</mi></math></span>, and the distance between the centers of the inscribed and circumscribed spheres be <span><math><mrow><mi>d</mi><mo>.</mo></mrow></math></span> Then,</div><div><span><math><mrow><mrow><mo>(</mo><mi>R</mi><mo>−</mo><mi>n</mi><mi>r</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>R</mi><mo>+</mo><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow><mi>r</mi><mo>)</mo></mrow><mo>⩾</mo><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>.</mo></mrow></math></span></div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 562-572"},"PeriodicalIF":1.0,"publicationDate":"2025-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144904507","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":"Complexity and structural results for the hull and convexity numbers in cycle convexity for graph products","authors":"Bijo S. Anand ,&nbsp;Ullas Chandran S.V. ,&nbsp;Julliano R. Nascimento ,&nbsp;Revathy S. Nair","doi":"10.1016/j.dam.2025.08.039","DOIUrl":"10.1016/j.dam.2025.08.039","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a graph and <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. In the cycle convexity, we say that <span><math><mi>S</mi></math></span> is <em>cycle convex</em> if for any <span><math><mrow><mi>u</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>∖</mo><mi>S</mi></mrow></math></span>, the induced subgraph of <span><math><mrow><mi>S</mi><mo>∪</mo><mrow><mo>{</mo><mi>u</mi><mo>}</mo></mrow></mrow></math></span> contains no cycle that includes <span><math><mi>u</mi></math></span>. The <em>cycle convex hull</em> of <span><math><mi>S</mi></math></span> is the smallest convex set containing <span><math><mi>S</mi></math></span>. The <em>cycle hull number</em> of <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><mo>hncc</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is the cardinality of the smallest set <span><math><mi>S</mi></math></span> such that the convex hull of <span><math><mi>S</mi></math></span> is <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. The <em>cycle convexity number</em> of <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><mo>concc</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is the maximum cardinality of a proper cycle convex set of <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. This paper studies cycle convexity in graph products. We show that the cycle hull number is always two for strong and lexicographic products. For the Cartesian, we establish tight bounds for this product and provide a closed formula when the factors are trees, generalizing an existing result for grid graphs. In addition, given a graph <span><math><mi>G</mi></math></span> and an integer <span><math><mi>k</mi></math></span>, we prove that <span><math><mrow><mo>hncc</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>k</mi></mrow></math></span> is NP-complete even if <span><math><mi>G</mi></math></span> is a bipartite Cartesian product graph, addressing an open question in the literature. Furthermore, we present exact formulas for the cycle convexity number in those three graph products. That leads to the NP-completeness of, given a graph <span><math><mi>G</mi></math></span> and an integer <span><math><mi>k</mi></math></span>, deciding whether <span><math><mrow><mo>concc</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mi>k</mi></mrow></math></span>, when <span><math><mi>G</mi></math></span> is a Cartesian, strong or lexicographic product graph.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 552-561"},"PeriodicalIF":1.0,"publicationDate":"2025-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144904509","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":"Reducing regular graphs to partition their vertices into a total dominating set and an independent dominating set","authors":"Teresa W. Haynes ,&nbsp;Michael A. Henning","doi":"10.1016/j.dam.2025.08.035","DOIUrl":"10.1016/j.dam.2025.08.035","url":null,"abstract":"<div><div>A graph <span><math><mi>G</mi></math></span> whose vertex set can be partitioned into a total dominating set and an independent dominating set is called a TI-graph. There exist infinite families of graphs that are not TI-graphs. We define the TI-reduction number <span><math><mrow><msup><mrow><mi>ti</mi></mrow><mrow><mo>−</mo></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of a graph <span><math><mi>G</mi></math></span> to be the minimum number of edges that must be removed from <span><math><mi>G</mi></math></span> to ensure that the resulting graph is a TI-graph. We observe that the TI-reduction number exists for all connected graphs with minimum degree at least 2 with one exception, namely the 5-cycle. We show that if <span><math><mi>G</mi></math></span> is a cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with <span><math><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≠</mo><mn>5</mn></mrow></math></span>, then <span><math><mrow><msup><mrow><mi>ti</mi></mrow><mrow><mo>−</mo></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>2</mn></mrow></math></span>, with equality if and only if <span><math><mrow><mi>n</mi><mo>≡</mo><mn>2</mn><mspace></mspace><mrow><mo>(</mo><mi>mod</mi><mspace></mspace><mn>3</mn><mo>)</mo></mrow></mrow></math></span>. If <span><math><mi>G</mi></math></span> is a cubic graph of order <span><math><mi>n</mi></math></span>, then we show that <span><math><mrow><msup><mrow><mi>ti</mi></mrow><mrow><mo>−</mo></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mi>n</mi></mrow></math></span>. For <span><math><mrow><mi>r</mi><mo>≥</mo><mn>4</mn></mrow></math></span> we show that if <span><math><mi>G</mi></math></span> is an <span><math><mi>r</mi></math></span>-regular graph of order <span><math><mi>n</mi></math></span>, then <span><math><mrow><msup><mrow><mi>ti</mi></mrow><mrow><mo>−</mo></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mrow><mo>(</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mfenced><mrow><mfrac><mrow><mi>r</mi></mrow><mrow><mn>3</mn><mi>r</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></mfenced><mi>n</mi></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"379 ","pages":"Pages 209-221"},"PeriodicalIF":1.0,"publicationDate":"2025-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144904660","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}