{"title":"图形中方块的隔离","authors":"","doi":"10.1016/j.disc.2024.114161","DOIUrl":null,"url":null,"abstract":"<div><p>Given a set <span><math><mi>F</mi></math></span> of graphs, we call a copy of a graph in <span><math><mi>F</mi></math></span> an <span><math><mi>F</mi></math></span>-graph. The <span><math><mi>F</mi></math></span>-isolation number of a graph <em>G</em>, denoted by <span><math><mi>ι</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span>, is the size of a smallest subset <em>D</em> of the vertex set <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> such that the closed neighbourhood of <em>D</em> intersects the vertex sets of the <span><math><mi>F</mi></math></span>-graphs contained by <em>G</em> (equivalently, <span><math><mi>G</mi><mo>−</mo><mi>N</mi><mo>[</mo><mi>D</mi><mo>]</mo></math></span> contains no <span><math><mi>F</mi></math></span>-graph). Thus, <span><math><mi>ι</mi><mo>(</mo><mi>G</mi><mo>,</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>}</mo><mo>)</mo></math></span> is the domination number of <em>G</em>. The second author showed that if <span><math><mi>F</mi></math></span> is the set of cycles and <em>G</em> is a connected <em>n</em>-vertex graph that is not a triangle, then <span><math><mi>ι</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>≤</mo><mrow><mo>⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mo>⌋</mo></mrow></math></span>. This bound is attainable for every <em>n</em> and solved a problem of Caro and Hansberg. A question that arises immediately is how much smaller an upper bound can be if <span><math><mi>F</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></math></span> for some <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>, where <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is a cycle of length <em>k</em>. The problem is to determine the smallest real number <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> (if it exists) such that for some finite set <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> of graphs, <span><math><mi>ι</mi><mo>(</mo><mi>G</mi><mo>,</mo><mo>{</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo><mo>)</mo><mo>≤</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo></math></span> for every connected graph <em>G</em> that is not an <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-graph. The above-mentioned result yields <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></math></span> and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mo>{</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>}</mo></math></span>. The second author also showed that if <span><math><mi>k</mi><mo>≥</mo><mn>5</mn></math></span> and <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> exists, then <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>≥</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></span>. We prove that <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>5</mn></mrow></mfrac></math></span> and determine <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>, which consists of three 4-vertex graphs and six 9-vertex graphs. The 9-vertex graphs in <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> were fully determined by means of a computer program. A method that has the potential of yielding similar results is introduced.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Isolation of squares in graphs\",\"authors\":\"\",\"doi\":\"10.1016/j.disc.2024.114161\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Given a set <span><math><mi>F</mi></math></span> of graphs, we call a copy of a graph in <span><math><mi>F</mi></math></span> an <span><math><mi>F</mi></math></span>-graph. The <span><math><mi>F</mi></math></span>-isolation number of a graph <em>G</em>, denoted by <span><math><mi>ι</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span>, is the size of a smallest subset <em>D</em> of the vertex set <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> such that the closed neighbourhood of <em>D</em> intersects the vertex sets of the <span><math><mi>F</mi></math></span>-graphs contained by <em>G</em> (equivalently, <span><math><mi>G</mi><mo>−</mo><mi>N</mi><mo>[</mo><mi>D</mi><mo>]</mo></math></span> contains no <span><math><mi>F</mi></math></span>-graph). Thus, <span><math><mi>ι</mi><mo>(</mo><mi>G</mi><mo>,</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>}</mo><mo>)</mo></math></span> is the domination number of <em>G</em>. The second author showed that if <span><math><mi>F</mi></math></span> is the set of cycles and <em>G</em> is a connected <em>n</em>-vertex graph that is not a triangle, then <span><math><mi>ι</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>≤</mo><mrow><mo>⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mo>⌋</mo></mrow></math></span>. This bound is attainable for every <em>n</em> and solved a problem of Caro and Hansberg. A question that arises immediately is how much smaller an upper bound can be if <span><math><mi>F</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></math></span> for some <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>, where <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is a cycle of length <em>k</em>. The problem is to determine the smallest real number <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> (if it exists) such that for some finite set <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> of graphs, <span><math><mi>ι</mi><mo>(</mo><mi>G</mi><mo>,</mo><mo>{</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo><mo>)</mo><mo>≤</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo></math></span> for every connected graph <em>G</em> that is not an <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-graph. The above-mentioned result yields <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></math></span> and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mo>{</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>}</mo></math></span>. The second author also showed that if <span><math><mi>k</mi><mo>≥</mo><mn>5</mn></math></span> and <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> exists, then <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>≥</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></span>. We prove that <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>5</mn></mrow></mfrac></math></span> and determine <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>, which consists of three 4-vertex graphs and six 9-vertex graphs. The 9-vertex graphs in <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> were fully determined by means of a computer program. A method that has the potential of yielding similar results is introduced.</p></div>\",\"PeriodicalId\":50572,\"journal\":{\"name\":\"Discrete Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-07-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0012365X24002929\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24002929","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
给定一个图集 F,我们称 F 中一个图的副本为 F 图。图 G 的 F 隔离数用 ι(G,F)表示,是顶点集 V(G)的最小子集 D 的大小,D 的封闭邻域与 G 所包含的 F 图的顶点集相交(等价地,G-N[D] 不包含任何 F 图)。因此,ι(G,{K1}) 是 G 的支配数。第二位作者证明,如果 F 是循环集,而 G 是一个非三角形的 n 顶点连通图,那么 ι(G,F)≤⌊n4⌋。对于每一个 n,这个界限都是可以达到的,并且解决了卡罗和汉斯伯格的一个问题。随即产生的一个问题是,如果 F={Ck} 为某个 k≥3,其中 Ck 是长度为 k 的循环,那么上限还能小多少?问题是确定最小实数 ck(如果存在),使得对于某个有限图集 Ek,ι(G,{Ck})≤ck|V(G)| 对于每个非 Ek 图的连通图 G,ι(G,{Ck})≤ck|V(G)|。根据上述结果可以得出 c3=14 和 E3={C3}。第二位作者还证明,如果 k≥5 且 ck 存在,则 ck≥22k+1。我们证明 c4=15 并确定 E4,它由三个 4 顶点图和六个 9 顶点图组成。E4 中的 9 顶点图是通过计算机程序完全确定的。本文介绍了一种有可能得出类似结果的方法。
Given a set of graphs, we call a copy of a graph in an -graph. The -isolation number of a graph G, denoted by , is the size of a smallest subset D of the vertex set such that the closed neighbourhood of D intersects the vertex sets of the -graphs contained by G (equivalently, contains no -graph). Thus, is the domination number of G. The second author showed that if is the set of cycles and G is a connected n-vertex graph that is not a triangle, then . This bound is attainable for every n and solved a problem of Caro and Hansberg. A question that arises immediately is how much smaller an upper bound can be if for some , where is a cycle of length k. The problem is to determine the smallest real number (if it exists) such that for some finite set of graphs, for every connected graph G that is not an -graph. The above-mentioned result yields and . The second author also showed that if and exists, then . We prove that and determine , which consists of three 4-vertex graphs and six 9-vertex graphs. The 9-vertex graphs in were fully determined by means of a computer program. A method that has the potential of yielding similar results is introduced.
期刊介绍:
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.