# The partition dimension of the vertex amalgamation of some cycles
> Hasmawati
URL kanonis: https://discover.unhas.ac.id/publications/the-partition-dimension-of-the-vertex-amalgamation-of-some-cycles
Jurnal / Konferensi: Heliyon
Tahun terbit: 2022
DOI: https://doi.org/10.1016/j.heliyon.2022.e09596
ISSN: 24058440
Kuartil SJR: Q1
Citations: 4
## Authors
- Hasmawati
## Abstract
Let G = ( V ( G ) , E ( G ) ) be a connected, finite, simple, and undirected graph. The distance between two vertices u , w ∈ V ( G ) , denoted by d ( u , w ) , is the shortest length of ( u , w ) -path in G. The distance between a vertex v ∈ V ( G ) is defined as min { d ( v , x ) : x ∈ S } where S ⊂ V ( G ) , denoted by d ( v , S ) . For an ordered partition Π = { S 1 , S 2 , … , S k } of the vertices of a graph G, the partition representation of a vertex v ∈ V ( G ) with respect to Π is defined as the k-vektor r ( v | Π ) = ( d ( v , S 1 ) , d ( v , S 1 ) , … , d ( v , S 1 ) . The partition set Π is called a resolving partition of G, if r ( u | Π ) ≠ r ( v | Π ) , for all u ≠ v , u , v ∈ V ( G ) . The partition dimension of G is the minimum number of sets in any resolving partition of G. In this paper we study the partition dimension of the vertex amalgamation of some cycles. Specifically, we present the vertex amalgamation of m copies of the cycle C n at a fixed vertex v ∈ V ( C n ) , for n ≥ 6 and k 2 - 3 k + 4 2 ≤ m ≤ k 2 - k 2 , k ≥ 3 .
## Keywords
- Partition (number theory)
- Dimension (graph theory)
- Vertex (graph theory)
- Mathematics
- Combinatorics
- Graph
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Sumber: Discover Unhas — RIMS Universitas Hasanuddin.
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