TY - JOUR

T1 - On local metric dimensions of m-neighbourhood corona graphs

AU - Rinurwati,

AU - Wahyudi, S.

AU - Darmaji,

AU - Nabila, R. E.

N1 - Publisher Copyright:
© Published under licence by IOP Publishing Ltd.

PY - 2021/5/14

Y1 - 2021/5/14

N2 - For a nontrivial connected graph G, let VG be a vertex set {v1, v2, ⋯, vn }, J be an index set {1,2, ⋯, n}, and H be any graph. Representation r(x|Z) of a vertex x in VG with respect to an ordered subset Z = {z 1, z 2, ⋯, zk } of VG is k-tuple (dG (x, z 1), dG (x, z 2), ⋯, dG (x, zk )), with dG (x, Zj ) is distance from x to zj for every j in J. If each of two adjacent vertices in VG has different representation, then Z is the local resolving set for G. Local base for G is resolving set Z with minimum number of vertices. Cardinality of base of G, is called the local metric dimension for G, dim l (G). For a positive integer m, an m-neighbourhood-corona of G and H, G ∗ mH, is obtained by taking G and as many as | VG | graph mHj, where j ∈ {1, 2, ⋯, |VG |} and Hj is the jth copy of the graph H, then making each vertex on the mHj graph adjacent to the neighbours of vertex vj in G. This article provides the exact values and characteristics of the local metric dimensions of the graphs generated from the m-neighbourhood-corona operations of G and H graphs, dim l (G∗mH), with G ∈ { Pn, Cn, Kn, Ks,t ; with s + t = n ≥2, and s and t be a positive integer number, and Wn } and H=K 1 and their proofs. We gave dim l (Kn ∗mK 1) = n - 1, and dim l (Cn∗mK 1) = 2; for odd n. Furthermore, dim l (G∗mK 1) = 1 if only if G ∈ { Pn, Cn for even n, Ks,t; s + t = n ≥ 2}. For a graph operating result Wn, we gave dim l (Wn ∗mK 1) = n - 4, n ≥ 7.

AB - For a nontrivial connected graph G, let VG be a vertex set {v1, v2, ⋯, vn }, J be an index set {1,2, ⋯, n}, and H be any graph. Representation r(x|Z) of a vertex x in VG with respect to an ordered subset Z = {z 1, z 2, ⋯, zk } of VG is k-tuple (dG (x, z 1), dG (x, z 2), ⋯, dG (x, zk )), with dG (x, Zj ) is distance from x to zj for every j in J. If each of two adjacent vertices in VG has different representation, then Z is the local resolving set for G. Local base for G is resolving set Z with minimum number of vertices. Cardinality of base of G, is called the local metric dimension for G, dim l (G). For a positive integer m, an m-neighbourhood-corona of G and H, G ∗ mH, is obtained by taking G and as many as | VG | graph mHj, where j ∈ {1, 2, ⋯, |VG |} and Hj is the jth copy of the graph H, then making each vertex on the mHj graph adjacent to the neighbours of vertex vj in G. This article provides the exact values and characteristics of the local metric dimensions of the graphs generated from the m-neighbourhood-corona operations of G and H graphs, dim l (G∗mH), with G ∈ { Pn, Cn, Kn, Ks,t ; with s + t = n ≥2, and s and t be a positive integer number, and Wn } and H=K 1 and their proofs. We gave dim l (Kn ∗mK 1) = n - 1, and dim l (Cn∗mK 1) = 2; for odd n. Furthermore, dim l (G∗mK 1) = 1 if only if G ∈ { Pn, Cn for even n, Ks,t; s + t = n ≥ 2}. For a graph operating result Wn, we gave dim l (Wn ∗mK 1) = n - 4, n ≥ 7.

UR - http://www.scopus.com/inward/record.url?scp=85107394145&partnerID=8YFLogxK

U2 - 10.1088/1742-6596/1872/1/012012

DO - 10.1088/1742-6596/1872/1/012012

M3 - Conference article

AN - SCOPUS:85107394145

SN - 1742-6588

VL - 1872

JO - Journal of Physics: Conference Series

JF - Journal of Physics: Conference Series

IS - 1

M1 - 012012

T2 - 21st IFAC World Congress 2020

Y2 - 12 July 2020 through 17 July 2020

ER -