{"title":"On the Number of $$ k $$ -Dominating Independent Sets in Planar Graphs","authors":"D. S. Taletskii","doi":"10.1134/s1990478924010149","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> A set\n<span>\\( J_k \\)</span> of graph vertices is said to be\n<span>\\( k \\)</span>-dominating independent (\n<span>\\( k \\geq 1 \\)</span>) if its vertices are pairwise adjacent and every vertex not in\n<span>\\( J_k \\)</span> is adjacent to at least\n<span>\\( k \\)</span> vertices in\n<span>\\( J_k \\)</span>. In the present paper, we obtain new upper bounds for the number of\n<span>\\( k \\)</span>-dominating independent sets for\n<span>\\( k \\geq 2 \\)</span> in some planar graph classes.\n</p>","PeriodicalId":607,"journal":{"name":"Journal of Applied and Industrial Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5800,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Applied and Industrial Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s1990478924010149","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Engineering","Score":null,"Total":0}
引用次数: 0
Abstract
A set
\( J_k \) of graph vertices is said to be
\( k \)-dominating independent (
\( k \geq 1 \)) if its vertices are pairwise adjacent and every vertex not in
\( J_k \) is adjacent to at least
\( k \) vertices in
\( J_k \). In the present paper, we obtain new upper bounds for the number of
\( k \)-dominating independent sets for
\( k \geq 2 \) in some planar graph classes.
期刊介绍：
Journal of Applied and Industrial Mathematics is a journal that publishes original and review articles containing theoretical results and those of interest for applications in various branches of industry. The journal topics include the qualitative theory of differential equations in application to mechanics, physics, chemistry, biology, technical and natural processes; mathematical modeling in mechanics, physics, engineering, chemistry, biology, ecology, medicine, etc.; control theory; discrete optimization; discrete structures and extremum problems; combinatorics; control and reliability of discrete circuits; mathematical programming; mathematical models and methods for making optimal decisions; models of theory of scheduling, location and replacement of equipment; modeling the control processes; development and analysis of algorithms; synthesis and complexity of control systems; automata theory; graph theory; game theory and its applications; coding theory; scheduling theory; and theory of circuits.
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