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Title: | Bounds on the connected local dimension of graphs in terms of the marked dimension and the clique number |
Authors: | Isariyapalakul S. Pho-on W. Khemmani V. |
Keywords: | 05C12 connected local dimension Connected local resolving set marked dimension true twin graph |
Issue Date: | 2022 |
Publisher: | Taylor and Francis Ltd. |
Abstract: | Let G be a connected graph and let v be a vertex of G. The representation of v with respect to an ordered set (Formula presented.) is the k-vector (Formula presented.) where (Formula presented.) is a distance between v and wi for (Formula presented.) If the representations of any two adjacent vertices of G with respect to W are distinct and the induced subgraph (Formula presented.) is connected, then W is called a connected local resolving set of G. The minimum cardinality of connected local resolving sets of G is referred to as the connected local dimension of G, denoted by (Formula presented.) A connected local resolving set of cardinality (Formula presented.) is called a minimum connected local resolving set or a connected local basis of G. The true twin graph tG of G is obtained by true twin equivalence classes of G such that the vertex set of tG consists of every true twin equivalence class of G and any two distinct vertices of tG are adjacent if the distance of them in G is 1. A connected local resolving set of tG containing all marked vertices is called a marked set of tG. A marked set of tG having minimum cardinality is called a minimum marked set or a marked basis of tG and this cardinality is called the marked dimension of tG, which is denoted by (Formula presented.) In this work, we investigate the connected local dimension of G by using the marked dimension of its true twin graph tG. The bounds for the connected local dimension of G are presented in terms of the marked dimension of tG and the clique number of a set of all marked vertices of tG. © 2022 The Author(s). Published with license by Taylor & Francis Group, LLC. |
URI: | https://www.scopus.com/inward/record.uri?eid=2-s2.0-85132664449&doi=10.1080%2f09728600.2022.2066490&partnerID=40&md5=4bd2fbd6472461e0bffc63ee38a3e0e0 https://ir.swu.ac.th/jspui/handle/123456789/27604 |
ISSN: | 9728600 |
Appears in Collections: | Scopus 2022 |
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