Discrete Applied Mathematics最新文献

{"title":"A survey of the monotonicity and non-contradiction of consensus methods and supertree methods","authors":"Mareike Fischer ,&nbsp;Michael Hendriksen","doi":"10.1016/j.dam.2025.02.007","DOIUrl":"10.1016/j.dam.2025.02.007","url":null,"abstract":"<div><div>In a recent study, Bryant, Francis and Steel investigated the concept of “future-proofing” consensus methods in phylogenetics. That is, they investigated if such methods can be robust against the introduction of additional data like added trees or new species. In the present manuscript, we analyze consensus methods under a different aspect of introducing new data, namely concerning the discovery of new clades. In evolutionary biology, often formerly unresolved clades get resolved by refined reconstruction methods or new genetic data analyses. In our manuscript we investigate which properties of consensus methods can guarantee that such new insights do not disagree with previously found consensus trees, but merely refine them, a property termed <em>monotonicity</em>. Along the lines of analyzing monotonicity, we also study two established supertree methods, namely Matrix Representation with Parsimony (MRP) and Matrix Representation with Compatibility (MRC), which have also been suggested as consensus methods in the literature. While we (just like Bryant, Francis and Steel in their recent study) unfortunately have to conclude some negative answers concerning general consensus methods, we also state some relevant and positive results concerning the majority rule (<span><math><mstyle><mi>M</mi><mi>R</mi></mstyle></math></span>) and strict consensus methods, which are amongst the most frequently used consensus methods. Moreover, we show that there exist infinitely many consensus methods which are monotonic and have some other desirable properties.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"368 ","pages":"Pages 1-17"},"PeriodicalIF":1.0,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143444322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The matching-connectivity of a graph","authors":"Hengzhe Li ,&nbsp;Menghan Ma ,&nbsp;Shuli Zhao ,&nbsp;Xiao Zhao ,&nbsp;Xiaohui Hua ,&nbsp;Yingbin Ma ,&nbsp;Hong-Jian Lai","doi":"10.1016/j.dam.2025.02.013","DOIUrl":"10.1016/j.dam.2025.02.013","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; be a connected subgraph of a connected graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. The &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-structure connectivity of the graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, denoted by &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;κ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, is the minimum cardinality of a set of disjoint subgraphs &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; in &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, such that every &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is isomorphic to &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∪&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is disconnected or trivial. By definition, the vertex connectivity of a graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; equals its &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-structure connectivity, that is, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;κ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;κ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. Define &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;κ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;κ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, known as the &lt;em&gt;matching-connectivity&lt;/em&gt; of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;In this paper, we prove that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;κ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is well-defined if and only if &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;∉&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. For a connected graph &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;∉&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, we prove &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;⌈&lt;/mo&gt;&lt;mi&gt;κ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;⌉&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"367 ","pages":"Pages 210-217"},"PeriodicalIF":1.0,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143427960","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 results for two kinds of conflict-free edge-coloring of graphs","authors":"Ping Li","doi":"10.1016/j.dam.2025.02.016","DOIUrl":"10.1016/j.dam.2025.02.016","url":null,"abstract":"<div><div>An edge-coloring of graph <span><math><mi>G</mi></math></span> is called <em>closed-neighborhood</em> (resp. <em>open-neighborhood</em>) <em>conflict-free edge-coloring</em> if for every edge <span><math><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, there is a color assigned to exactly one edge among <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>∪</mo><mi>E</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span> (resp. <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>∪</mo><mi>E</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>−</mo><mrow><mo>{</mo><mi>u</mi><mi>v</mi><mo>}</mo></mrow></mrow></math></span>). The smallest number of colors needed in any possible closed-neighborhood (resp. open-neighborhood) conflict-free edge-coloring of <span><math><mi>G</mi></math></span>, denoted <span><math><mrow><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>C</mi><mi>F</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mrow><mo>[</mo><mi>G</mi><mo>]</mo></mrow></mrow></math></span> (resp. <span><math><mrow><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>C</mi><mi>F</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>), is called the <em>closed-neighborhood</em> (resp. <em>open-neighborhood</em>) <em>conflict-free index</em> of <span><math><mi>G</mi></math></span>. In this paper, we prove that decide whether <span><math><mrow><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>C</mi><mi>F</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mrow><mo>[</mo><mi>G</mi><mo>]</mo></mrow><mo>=</mo><mn>2</mn></mrow></math></span> or <span><math><mrow><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>C</mi><mi>F</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mn>2</mn></mrow></math></span> is NP-complete, even if <span><math><mi>G</mi></math></span> is a bipartite graph.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"367 ","pages":"Pages 218-225"},"PeriodicalIF":1.0,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143427964","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":"Minimum degree and size conditions for the graphs of proper connection number 2","authors":"Zhenzhen Li,&nbsp;Baoyindureng Wu","doi":"10.1016/j.dam.2025.02.011","DOIUrl":"10.1016/j.dam.2025.02.011","url":null,"abstract":"<div><div>An edge-colored graph <span><math><mi>G</mi></math></span> is called properly connected if every pair of distinct vertices of <span><math><mi>G</mi></math></span> is connected by a proper path. For a connected graph <span><math><mi>G</mi></math></span>, the proper connection number of <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><mi>p</mi><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is defined as the smallest number of colors that are needed in order to make <span><math><mi>G</mi></math></span> properly connected. Guan, Xue, Cheng and Yang (Guan et al., 2019) conjectured that if <span><math><mi>G</mi></math></span> is a connected graph of order <span><math><mi>n</mi></math></span>, <span><math><mrow><mi>δ</mi><mo>≥</mo><mn>3</mn></mrow></math></span> and <span><math><mrow><mrow><mo>|</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>≥</mo><mfenced><mfrac><mrow><mi>n</mi><mo>−</mo><mi>p</mi><mo>−</mo><mrow><mo>(</mo><mn>3</mn><mo>−</mo><mi>p</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>δ</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></mfrac></mfenced><mo>+</mo><mrow><mo>(</mo><mn>3</mn><mo>−</mo><mi>p</mi><mo>)</mo></mrow><mfenced><mfrac><mrow><mi>δ</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mfenced><mo>+</mo><mn>4</mn></mrow></math></span>, then <span><math><mrow><mi>p</mi><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>2</mn></mrow></math></span>, where <span><math><mi>p</mi></math></span> takes the value 1 if <span><math><mrow><mi>δ</mi><mo>=</mo><mn>3</mn></mrow></math></span> and 0 if <span><math><mrow><mi>δ</mi><mo>≥</mo><mn>4</mn></mrow></math></span>. We confirm the validity of the conjecture.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"367 ","pages":"Pages 179-194"},"PeriodicalIF":1.0,"publicationDate":"2025-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143422692","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":"Scheduling with a discounted profit criterion on identical machines","authors":"Weidong Li ,&nbsp;Yaru Yang ,&nbsp;Man Xiao ,&nbsp;Xin Chen ,&nbsp;Małgorzata Sterna ,&nbsp;Jacek Błażewicz","doi":"10.1016/j.dam.2025.02.015","DOIUrl":"10.1016/j.dam.2025.02.015","url":null,"abstract":"<div><div>In this paper, we introduce a novel scheduling criterion named as a discounted profit (to be maximized), which could be considered as a generalization of early work (also to be maximized). The goals of such scheduling models are to maximize <span><math><mrow><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mrow><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>+</mo><mi>δ</mi><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, where <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>\u0000 (<span><math><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>) is the early (late) work of job <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>, and <span><math><mrow><mn>0</mn><mo>≤</mo><mi>δ</mi><mo>&lt;</mo><mn>1</mn></mrow></math></span> is a discount factor. When <span><math><mrow><mi>δ</mi><mo>=</mo><mn>0</mn></mrow></math></span>, these models are reduced to the ones with early work maximization. We focus on the models of scheduling on identical machines when jobs share a common due date. For the online case, we prove that the competitive ratio of the classical List Scheduling (LS) algorithm is exactly <span><math><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn><mo>+</mo><mi>δ</mi></mrow></mfrac></math></span>, improving the seminal result (<span><math><msqrt><mrow><mn>2</mn></mrow></msqrt></math></span>) and covering the very recent result (<span><math><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math></span>) when <span><math><mrow><mi>δ</mi><mo>=</mo><mn>0</mn></mrow></math></span>. Moreover, when the number of machines <span><math><mrow><mi>m</mi><mo>=</mo><mn>2</mn></mrow></math></span>, we propose a new optimal online algorithm with a competitive ratio <span><math><mfrac><mrow><msqrt><mrow><mn>2</mn><mi>δ</mi><mo>+</mo><mn>5</mn></mrow></msqrt><mo>+</mo><mn>2</mn><mi>δ</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>δ</mi><msqrt><mrow><mn>2</mn><mi>δ</mi><mo>+</mo><mn>5</mn></mrow></msqrt><mo>+</mo><mn>1</mn></mrow></mfrac></math></span>, matching the previous best known result (<span><math><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt><mo>−</mo><mn>1</mn></mrow></math></span>) when <span><math><mrow><mi>δ</mi><mo>=</mo><mn>0</mn></mrow></math></span>. For the offline case, we prove that the Longest Processing Time first (LPT) algorithm has an approximation ratio <span><math><mfrac><mrow><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><mo>+</mo><mrow><mo>(</mo><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mi>δ</mi></mrow></mfrac></math></span>, extending the existed results when <span><math><mrow><mi>δ</mi><mo>=</mo><mn>0</mn></mrow></math></span> and <span><math><mrow><mi>m</mi><mo>=</mo><mn>2</mn></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"367 ","pages":"Pages 195-209"},"PeriodicalIF":1.0,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143422691","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":"Spectral versions on Lovász’s (a,b)-parity factor theorem in graphs","authors":"Huicai Jia ,&nbsp;Jing Lou ,&nbsp;Ruifang Liu","doi":"10.1016/j.dam.2025.02.014","DOIUrl":"10.1016/j.dam.2025.02.014","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; be a graph, and let &lt;span&gt;&lt;math&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; be two integer-valued functions defined on &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; such that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; for every &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; A &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;em&gt;-parity factor&lt;/em&gt; of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is a spanning subgraph &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; such that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;≡&lt;/mo&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;≡&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;mod&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; for every &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; In particular, a &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;-parity factor is called an &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;em&gt;-parity factor&lt;/em&gt; if &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;≡&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;≡&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; are two positive integers satisfying &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;≡&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;≡&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;mod&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; In recent years, many interesting researches focus on establishing sufficient conditions to ensure that a graph contains an &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;-parity factor. Based on Lovász’s &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;-parity factor theorem and technical distance spectral methods, whi","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"367 ","pages":"Pages 165-178"},"PeriodicalIF":1.0,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143403418","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 hardness of short and sign-compatible circuit walks","authors":"Steffen Borgwardt ,&nbsp;Weston Grewe ,&nbsp;Sean Kafer ,&nbsp;Jon Lee ,&nbsp;Laura Sanità","doi":"10.1016/j.dam.2025.02.009","DOIUrl":"10.1016/j.dam.2025.02.009","url":null,"abstract":"<div><div>The circuits of a polyhedron are a superset of its edge directions. Circuit walks, a sequence of steps along circuits, generalize edge walks and are “short” if they have few steps or small total length. Both interpretations of short are relevant to the theory and application of linear programming.</div><div>We study the hardness of several problems relating to the construction of short circuit walks. We establish that for a pair of vertices of a <span><math><mrow><mn>0</mn><mo>/</mo><mn>1</mn></mrow></math></span>-network-flow polytope, it is NP-complete to determine the length of a shortest circuit walk, even if we add the requirement that the walk must be sign-compatible. Our results also imply that determining the minimal number of circuits needed for a sign-compatible decomposition is NP-complete. Further, we show that it is NP-complete to determine the smallest total length (for <span><math><mi>p</mi></math></span>-norms <span><math><mrow><mo>‖</mo><mi>⋅</mi><msub><mrow><mo>‖</mo></mrow><mrow><mi>p</mi></mrow></msub></mrow></math></span>, <span><math><mrow><mn>1</mn><mo>&lt;</mo><mi>p</mi><mo>≤</mo><mi>∞</mi></mrow></math></span>) of a circuit walk between a pair of vertices. One method to construct a short circuit walk is to pick up a correct facet at each step, which generalizes a non-revisiting walk. We prove that it is NP-complete to determine if there is a circuit direction that picks up a correct facet; in contrast, this problem can be solved in polynomial time for TU polyhedra.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"367 ","pages":"Pages 129-149"},"PeriodicalIF":1.0,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143403365","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":"Hereditary Nordhaus–Gaddum graphs","authors":"Vaidy Sivaraman ,&nbsp;Rebecca Whitman","doi":"10.1016/j.dam.2025.02.010","DOIUrl":"10.1016/j.dam.2025.02.010","url":null,"abstract":"<div><div>Nordhaus and Gaddum proved in 1956 that the sum of the chromatic number <span><math><mi>χ</mi></math></span> of a graph <span><math><mi>G</mi></math></span> and its complement is at most <span><math><mrow><mrow><mo>|</mo><mi>G</mi><mo>|</mo></mrow><mo>+</mo><mn>1</mn></mrow></math></span>. The Nordhaus–Gaddum graphs are the class of graphs satisfying this inequality with equality, and are well-understood. In this paper we consider a hereditary generalization: graphs <span><math><mi>G</mi></math></span> for which all induced subgraphs <span><math><mi>H</mi></math></span> of <span><math><mi>G</mi></math></span> satisfy <span><math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>+</mo><mi>χ</mi><mrow><mo>(</mo><mover><mrow><mi>H</mi></mrow><mo>¯</mo></mover><mo>)</mo></mrow><mo>≥</mo><mrow><mo>|</mo><mi>H</mi><mo>|</mo></mrow></mrow></math></span>. We characterize the forbidden induced subgraphs of this class and find its intersection with a number of common classes, including line graphs. We also discuss <span><math><mi>χ</mi></math></span>-boundedness and algorithmic results.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"367 ","pages":"Pages 150-164"},"PeriodicalIF":1.0,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143395488","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Further results on the mixed metric dimension of graphs","authors":"Hongbo Hua ,&nbsp;Yaojun Chen ,&nbsp;Xinying Hua","doi":"10.1016/j.dam.2025.02.012","DOIUrl":"10.1016/j.dam.2025.02.012","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a graph with vertex set <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and edge set <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. The <em>mixed metric dimension</em> of a connected graph <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><msub><mrow><mi>dim</mi></mrow><mrow><mi>m</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is the minimum cardinality of a subset <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> such that for any two <span><math><mrow><mi>u</mi><mo>,</mo><mspace></mspace><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>∪</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, there exists <span><math><mrow><mi>w</mi><mo>∈</mo><mi>S</mi></mrow></math></span> so that the distance between <span><math><mi>w</mi></math></span> and <span><math><mi>u</mi></math></span> is not equal to the distance between <span><math><mi>w</mi></math></span> and <span><math><mi>v</mi></math></span>. In this paper, we present further results on the mixed metric dimension. First, we give a sharp upper bound on the mixed metric dimension for a graph in terms of the number of cut vertices of this graph. Second, we compare the mixed metric dimension with geodesic transversal number for trees, unicyclic graphs and block graphs. Finally, we provide some new results about a conjecture, due to Sedlar and Škrekovski (Sedlar and Škrekovski, 2021), on the mixed metric dimension.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"367 ","pages":"Pages 99-106"},"PeriodicalIF":1.0,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143387168","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":"Orientable burning number of graphs","authors":"Julien Courtiel ,&nbsp;Paul Dorbec ,&nbsp;Tatsuya Gima ,&nbsp;Romain Lecoq ,&nbsp;Yota Otachi","doi":"10.1016/j.dam.2025.02.004","DOIUrl":"10.1016/j.dam.2025.02.004","url":null,"abstract":"<div><div>In this paper, we introduce the problem of finding an orientation of a given undirected graph that maximizes the burning number of the resulting directed graph. We show that the problem is polynomial-time solvable on Kőnig–Egerváry graphs (and thus on bipartite graphs) and that an almost optimal solution can be computed in polynomial time for perfect graphs. On the other hand, we show that the problem is NP-hard in general and W[1]-hard parameterized by the target burning number. The hardness results are complemented by several fixed-parameter tractable results parameterized by structural parameters. Our main result in this direction shows that the problem is fixed-parameter tractable parameterized by cluster vertex deletion number plus clique number (and thus also by vertex cover number).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"367 ","pages":"Pages 116-128"},"PeriodicalIF":1.0,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143387169","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}