TY - JOUR
T1 - A Central Local Metric Dimension of Generalized Fan Graph, Generalized Broken Fan Graph, and CM ☉ K̅M
AU - Listiana, Yuni
AU - Susilowati, Liliek
AU - Slamin,
N1 - Publisher Copyright:
© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (https://creativecommons.org/licenses/by/4.0/).
PY - 2024/3/11
Y1 - 2024/3/11
N2 - The central local metric dimension is a new variation of local metric dimension that introduced in 2023. The central local metric dimension is a new concept that enriches research studies in graph theory, especially in the field of metric dimension. This concept combines the concept of local metric dimension by involving central points in the local metric set so that the existence of central points can strengthen the position of the local metric set in distinguishing every two neighboring points in a graph. The methodology of this research is study literature and observation. We find the central vertex of each graph and also find the local metric set of its graph, then we applied it to the related theorem to find the lower bound of central local metric dimension. Let G be a connected graph with order n and vertex set is V(G). A subset W = {x1, x2, …, xk} ⊆ V(G) is a local metric set of graph G if the metric code of every two adjacent vertices u, v in G are r(u|W) ≠ r(v|W), where r(u|W) = (d(u, x1), d(u, x2), …, d(u, xk)) and r(v|W) = (d(v, x1), d(v, x2), …, d(v, xk)). A vertex x ∈ V(G) is a central vertex in G if x have the the shorthest distance to the another all vertices in G. If W consist of all central vertices in G, then W is called a central local metric set of G. The minimal cardinality of W is called a central basis local set of G and its cardinality is called central local metric dimension of G or denoted by lmds(G). In this paper we explored the central local metric dimension on generalized fan graph, generalized broken fan graph, and a graph resulting from corona operation Cm ☉ K̅n. The result show that the central local metric dimension of generalized fan graph is equal with its order because the diameter and radius are equal. Different with it, the central local metric dimension of generalized broken fan is equal with its local metric dimension plus cardinality of central set, and the central local metric dimension of Cm ☉ K̅n are equal with the central local metric dimension of Cm,.
AB - The central local metric dimension is a new variation of local metric dimension that introduced in 2023. The central local metric dimension is a new concept that enriches research studies in graph theory, especially in the field of metric dimension. This concept combines the concept of local metric dimension by involving central points in the local metric set so that the existence of central points can strengthen the position of the local metric set in distinguishing every two neighboring points in a graph. The methodology of this research is study literature and observation. We find the central vertex of each graph and also find the local metric set of its graph, then we applied it to the related theorem to find the lower bound of central local metric dimension. Let G be a connected graph with order n and vertex set is V(G). A subset W = {x1, x2, …, xk} ⊆ V(G) is a local metric set of graph G if the metric code of every two adjacent vertices u, v in G are r(u|W) ≠ r(v|W), where r(u|W) = (d(u, x1), d(u, x2), …, d(u, xk)) and r(v|W) = (d(v, x1), d(v, x2), …, d(v, xk)). A vertex x ∈ V(G) is a central vertex in G if x have the the shorthest distance to the another all vertices in G. If W consist of all central vertices in G, then W is called a central local metric set of G. The minimal cardinality of W is called a central basis local set of G and its cardinality is called central local metric dimension of G or denoted by lmds(G). In this paper we explored the central local metric dimension on generalized fan graph, generalized broken fan graph, and a graph resulting from corona operation Cm ☉ K̅n. The result show that the central local metric dimension of generalized fan graph is equal with its order because the diameter and radius are equal. Different with it, the central local metric dimension of generalized broken fan is equal with its local metric dimension plus cardinality of central set, and the central local metric dimension of Cm ☉ K̅n are equal with the central local metric dimension of Cm,.
UR - http://www.scopus.com/inward/record.url?scp=85189349890&partnerID=8YFLogxK
U2 - 10.1051/e3sconf/202450003046
DO - 10.1051/e3sconf/202450003046
M3 - Conference article
AN - SCOPUS:85189349890
SN - 2555-0403
VL - 500
JO - E3S Web of Conferences
JF - E3S Web of Conferences
M1 - 03046
T2 - 1st International Conference on Environment, Green Technology, and Digital Society, INTERCONNECTS 2023
Y2 - 13 December 2023
ER -