Shengjin Ji, Mengya He, Guang Li, Yingui Pan, Wenqian Zhang
{"title":"限制图形的强制总数","authors":"Shengjin Ji, Mengya He, Guang Li, Yingui Pan, Wenqian Zhang","doi":"10.1007/s10878-023-01089-4","DOIUrl":null,"url":null,"abstract":"<p>In recent years, a dynamic coloring, named as zero forcing, of the vertices in a graph have attracted many researchers. For a given <i>G</i> and a vertex subset <i>S</i>, assigning each vertex of <i>S</i> black and each vertex of <span>\\(V\\setminus S\\)</span> no color, if one vertex <span>\\(u\\in S\\)</span> has a unique neighbor <i>v</i> in <span>\\(V\\setminus S\\)</span>, then <i>u</i> forces <i>v</i> to color black. <i>S</i> is called a zero forcing set if <i>S</i> can be expanded to the entire vertex set <i>V</i> by repeating the above forcing process. <i>S</i> is regarded as a total forcing set if the subgraph <i>G</i>[<i>S</i>] satisfies <span>\\(\\delta (G[S])\\ge 1\\)</span>. The minimum cardinality of a total forcing set in <i>G</i>, denoted by <span>\\(F_t(G)\\)</span>, is named the total forcing number of <i>G</i>. For a graph <i>G</i>, <i>p</i>(<i>G</i>), <i>q</i>(<i>G</i>) and <span>\\(\\phi (G)\\)</span> denote the number of pendant vertices, the number of vertices with degree at least 3 meanwhile having one pendant path and the cyclomatic number of <i>G</i>, respectively. In the paper, by means of the total forcing set of a spanning tree regarding a graph <i>G</i>, we verify that <span>\\(F_t(G)\\le p(G)+q(G)+2\\phi (G)\\)</span>. Furthermore, all graphs achieving the equality are determined.\n</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Bounding the total forcing number of graphs\",\"authors\":\"Shengjin Ji, Mengya He, Guang Li, Yingui Pan, Wenqian Zhang\",\"doi\":\"10.1007/s10878-023-01089-4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In recent years, a dynamic coloring, named as zero forcing, of the vertices in a graph have attracted many researchers. For a given <i>G</i> and a vertex subset <i>S</i>, assigning each vertex of <i>S</i> black and each vertex of <span>\\\\(V\\\\setminus S\\\\)</span> no color, if one vertex <span>\\\\(u\\\\in S\\\\)</span> has a unique neighbor <i>v</i> in <span>\\\\(V\\\\setminus S\\\\)</span>, then <i>u</i> forces <i>v</i> to color black. <i>S</i> is called a zero forcing set if <i>S</i> can be expanded to the entire vertex set <i>V</i> by repeating the above forcing process. <i>S</i> is regarded as a total forcing set if the subgraph <i>G</i>[<i>S</i>] satisfies <span>\\\\(\\\\delta (G[S])\\\\ge 1\\\\)</span>. The minimum cardinality of a total forcing set in <i>G</i>, denoted by <span>\\\\(F_t(G)\\\\)</span>, is named the total forcing number of <i>G</i>. For a graph <i>G</i>, <i>p</i>(<i>G</i>), <i>q</i>(<i>G</i>) and <span>\\\\(\\\\phi (G)\\\\)</span> denote the number of pendant vertices, the number of vertices with degree at least 3 meanwhile having one pendant path and the cyclomatic number of <i>G</i>, respectively. In the paper, by means of the total forcing set of a spanning tree regarding a graph <i>G</i>, we verify that <span>\\\\(F_t(G)\\\\le p(G)+q(G)+2\\\\phi (G)\\\\)</span>. Furthermore, all graphs achieving the equality are determined.\\n</p>\",\"PeriodicalId\":50231,\"journal\":{\"name\":\"Journal of Combinatorial Optimization\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2023-11-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Optimization\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10878-023-01089-4\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Optimization","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10878-023-01089-4","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
In recent years, a dynamic coloring, named as zero forcing, of the vertices in a graph have attracted many researchers. For a given G and a vertex subset S, assigning each vertex of S black and each vertex of \(V\setminus S\) no color, if one vertex \(u\in S\) has a unique neighbor v in \(V\setminus S\), then u forces v to color black. S is called a zero forcing set if S can be expanded to the entire vertex set V by repeating the above forcing process. S is regarded as a total forcing set if the subgraph G[S] satisfies \(\delta (G[S])\ge 1\). The minimum cardinality of a total forcing set in G, denoted by \(F_t(G)\), is named the total forcing number of G. For a graph G, p(G), q(G) and \(\phi (G)\) denote the number of pendant vertices, the number of vertices with degree at least 3 meanwhile having one pendant path and the cyclomatic number of G, respectively. In the paper, by means of the total forcing set of a spanning tree regarding a graph G, we verify that \(F_t(G)\le p(G)+q(G)+2\phi (G)\). Furthermore, all graphs achieving the equality are determined.
期刊介绍:
The objective of Journal of Combinatorial Optimization is to advance and promote the theory and applications of combinatorial optimization, which is an area of research at the intersection of applied mathematics, computer science, and operations research and which overlaps with many other areas such as computation complexity, computational biology, VLSI design, communication networks, and management science. It includes complexity analysis and algorithm design for combinatorial optimization problems, numerical experiments and problem discovery with applications in science and engineering.
The Journal of Combinatorial Optimization publishes refereed papers dealing with all theoretical, computational and applied aspects of combinatorial optimization. It also publishes reviews of appropriate books and special issues of journals.