TY - JOUR

T1 - MINIMAL DOUBLY RESOLVING SETS OF ANTIPRISM GRAPHS AND MOBIUS LADDERS

AU - Sultan, Saba

AU - Baca, Martin

AU - Ahmad, Ali

AU - Imran, Muhammad

N1 - Publisher Copyright:
© 2022. Miskolc University Press

PY - 2022

Y1 - 2022

N2 - Consider a simple connected graph G = (V(G),E(G)), where V(G) represents the vertex set and E(G) represents the edge set respectively. A subset W of V(G) is called a resolving set for a graph G if for every two distinct vertices x, y ∈V(G), there exist some vertex w ∈W such that d(x,w) 6≠ d(y,w), where d(u, v) denotes the distance between vertices u and v. A resolving set of minimal cardinality is called a metric basis for G and its cardinality is called the metric dimension of G, which is denoted by β(G). A subset D of V(G) is called a doubly resolving set of G if for every two distinct vertices x, y of G, there are two vertices u, v ∈ D such that d(u, x) − d(u, y) 6= d(v, x) − d(v, y). A doubly resolving set with minimum cardinality is called minimal doubly resolving set. This minimum cardinality is denoted by ψ(G). In this paper, we determine the minimal doubly resolving sets for antiprism graphs denoted by An with n ≥ 3 and for Möbius ladders denoted by Mn, for every even positive integer n ≥ 8.

AB - Consider a simple connected graph G = (V(G),E(G)), where V(G) represents the vertex set and E(G) represents the edge set respectively. A subset W of V(G) is called a resolving set for a graph G if for every two distinct vertices x, y ∈V(G), there exist some vertex w ∈W such that d(x,w) 6≠ d(y,w), where d(u, v) denotes the distance between vertices u and v. A resolving set of minimal cardinality is called a metric basis for G and its cardinality is called the metric dimension of G, which is denoted by β(G). A subset D of V(G) is called a doubly resolving set of G if for every two distinct vertices x, y of G, there are two vertices u, v ∈ D such that d(u, x) − d(u, y) 6= d(v, x) − d(v, y). A doubly resolving set with minimum cardinality is called minimal doubly resolving set. This minimum cardinality is denoted by ψ(G). In this paper, we determine the minimal doubly resolving sets for antiprism graphs denoted by An with n ≥ 3 and for Möbius ladders denoted by Mn, for every even positive integer n ≥ 8.

KW - Antiprism graph

KW - Metric dimension

KW - Minimal doubly resolving set

KW - Mobius ladder.

KW - Resolving set

UR - http://www.scopus.com/inward/record.url?scp=85131731725&partnerID=8YFLogxK

U2 - 10.18514/MMN.2022.1950

DO - 10.18514/MMN.2022.1950

M3 - Artículo

AN - SCOPUS:85131731725

SN - 1787-2405

VL - 23

SP - 457

EP - 469

JO - Miskolc Mathematical Notes

JF - Miskolc Mathematical Notes

IS - 1

ER -