{"title":"Qd的(d−2)泄漏强迫数和GP的(n,1)泄漏强迫数","authors":"Rebekah Herrman","doi":"10.1016/j.disopt.2022.100744","DOIUrl":null,"url":null,"abstract":"<div><p>Leaky-forcing is a recently introduced variant of zero forcing that has been studied for families of graphs including paths, cycles, wheels, grids, and trees. In this paper, we extend previous results on the leaky forcing number of the <span><math><mi>d</mi></math></span>-dimensional hypercube, <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span>, to show that the <span><math><mrow><mo>(</mo><mi>d</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow></math></span>-leaky forcing number of <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> is <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>. We also examine a question about the relationship between the size of a minimum <span><math><mi>ℓ</mi></math></span>-leaky-forcing set and a minimum zero forcing set for a graph <span><math><mi>G</mi></math></span>.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The (d−2)-leaky forcing number of Qd and ℓ-leaky forcing number of GP(n,1)\",\"authors\":\"Rebekah Herrman\",\"doi\":\"10.1016/j.disopt.2022.100744\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Leaky-forcing is a recently introduced variant of zero forcing that has been studied for families of graphs including paths, cycles, wheels, grids, and trees. In this paper, we extend previous results on the leaky forcing number of the <span><math><mi>d</mi></math></span>-dimensional hypercube, <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span>, to show that the <span><math><mrow><mo>(</mo><mi>d</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow></math></span>-leaky forcing number of <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> is <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>. We also examine a question about the relationship between the size of a minimum <span><math><mi>ℓ</mi></math></span>-leaky-forcing set and a minimum zero forcing set for a graph <span><math><mi>G</mi></math></span>.</p></div>\",\"PeriodicalId\":50571,\"journal\":{\"name\":\"Discrete Optimization\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2022-11-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete Optimization\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1572528622000494\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Optimization","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1572528622000494","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
The (d−2)-leaky forcing number of Qd and ℓ-leaky forcing number of GP(n,1)
Leaky-forcing is a recently introduced variant of zero forcing that has been studied for families of graphs including paths, cycles, wheels, grids, and trees. In this paper, we extend previous results on the leaky forcing number of the -dimensional hypercube, , to show that the -leaky forcing number of is . We also examine a question about the relationship between the size of a minimum -leaky-forcing set and a minimum zero forcing set for a graph .
期刊介绍:
Discrete Optimization publishes research papers on the mathematical, computational and applied aspects of all areas of integer programming and combinatorial optimization. In addition to reports on mathematical results pertinent to discrete optimization, the journal welcomes submissions on algorithmic developments, computational experiments, and novel applications (in particular, large-scale and real-time applications). The journal also publishes clearly labelled surveys, reviews, short notes, and open problems. Manuscripts submitted for possible publication to Discrete Optimization should report on original research, should not have been previously published, and should not be under consideration for publication by any other journal.