{"title":"Fault-tolerant resolvability of some graphs of convex polytopes","authors":"S. Sharma, H. Raza, V. K. Bhat","doi":"10.1515/dma-2023-0016","DOIUrl":null,"url":null,"abstract":"Abstract The fault-tolerant resolvability is an extension of metric resolvability in graphs with several intelligent systems applications, for example, network optimization, robot navigation, and sensor networking. The graphs of convex polytopes, which are rotationally symmetric, are essential in intelligent networks due to the uniform rate of data transformation to all nodes. A resolving set is an ordered set 𝕎 of vertices of a connected graph G in which the vector of distances to the vertices in 𝕎 uniquely determines all the vertices of the graph G. The minimum cardinality of a resolving set of G is known as the metric dimension of G. If 𝕎 ∖ ρ is also a resolving set for each ρ in 𝕎. In that case, 𝕎 is said to be a fault-tolerant resolving set. The fault-tolerant metric dimension of G is the minimum cardinality of such a set 𝕎. The metric dimension and the fault-tolerant metric dimension for three families of convex polytope graphs are studied. Our main results affirm that three families, as mentioned above, have constant fault-tolerant resolvability structures.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"33 1","pages":"177 - 187"},"PeriodicalIF":0.3000,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics and Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/dma-2023-0016","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 1
Abstract
Abstract The fault-tolerant resolvability is an extension of metric resolvability in graphs with several intelligent systems applications, for example, network optimization, robot navigation, and sensor networking. The graphs of convex polytopes, which are rotationally symmetric, are essential in intelligent networks due to the uniform rate of data transformation to all nodes. A resolving set is an ordered set 𝕎 of vertices of a connected graph G in which the vector of distances to the vertices in 𝕎 uniquely determines all the vertices of the graph G. The minimum cardinality of a resolving set of G is known as the metric dimension of G. If 𝕎 ∖ ρ is also a resolving set for each ρ in 𝕎. In that case, 𝕎 is said to be a fault-tolerant resolving set. The fault-tolerant metric dimension of G is the minimum cardinality of such a set 𝕎. The metric dimension and the fault-tolerant metric dimension for three families of convex polytope graphs are studied. Our main results affirm that three families, as mentioned above, have constant fault-tolerant resolvability structures.
期刊介绍:
The aim of this journal is to provide the latest information on the development of discrete mathematics in the former USSR to a world-wide readership. The journal will contain papers from the Russian-language journal Diskretnaya Matematika, the only journal of the Russian Academy of Sciences devoted to this field of mathematics. Discrete Mathematics and Applications will cover various subjects in the fields such as combinatorial analysis, graph theory, functional systems theory, cryptology, coding, probabilistic problems of discrete mathematics, algorithms and their complexity, combinatorial and computational problems of number theory and of algebra.