Dimensi Matriks Dan Dimensi Partisi Pada Graf Hasil Operasi Korona

Yuni Listiana

Sari


Let𝐺(𝑉,𝐸)is a connected graph.For an ordered set π‘Š={𝑀1,𝑀2,…,π‘€π‘˜} of vertices, π‘ŠβŠ†π‘‰(𝐺), and a vertex π‘£βˆˆπ‘‰(𝐺), the representation of 𝑣 with respect to π‘Š is the ordered k-tuple π‘Ÿ(𝑣|π‘Š)={𝑑(𝑣,𝑀1),𝑑(𝑣,𝑀2),…,𝑑(𝑣,π‘€π‘˜)|βˆ€π‘£βˆˆπ‘‰(𝐺)}. The set W is called a resolving set of G if every vertex of G has a distinct representation. A resolving set containing a minimum number of vertices is called a basis for 𝐺. The metric dimension of 𝐺, denoted by π‘‘π‘–π‘š(𝐺), is the number of vertices in a basis of 𝐺. Then, for a subset S of V(G), the distance between u and S is 𝑑(𝑣,𝑆)=π‘šπ‘–π‘›{𝑑(𝑣,π‘₯)|βˆ€π‘₯βˆˆπ‘†,βˆ€π‘£βˆˆπ‘‰(𝐺)}. Let Ξ =(𝑆1,𝑆2,…,𝑆𝑙)be an ordered l-partition of V(G), forβˆ€π‘†π‘™βŠ‚π‘‰(𝐺) danπ‘£βˆˆπ‘‰(𝐺), the representation of v with respect to Ξ  is the l-vector π‘Ÿ(𝑣|Ξ )=(𝑑(𝑣,𝑆1),𝑑(𝑣,𝑆2),…,𝑑(𝑣,𝑆𝑙)). The set Ξ  is called a resolving partition for G if the π‘™βˆ’vector π‘Ÿ(𝑣|Ξ ),βˆ€π‘£βˆˆπ‘‰(𝐺)are distinct. The minimum l for which there is a resolving l-partition of V(G) is the partition dimension of G, denoted by 𝑝𝑑(𝐺). In this paper, we determine the metric dimension and the partition dimension of corona product graphs πΎπ‘›β¨€πΎπ‘›βˆ’1, and we get some result that the metric dimension and partition dimension of πΎπ‘›β¨€πΎπ‘›βˆ’1respectively is𝑛(π‘›βˆ’2) and 2π‘›βˆ’1, for𝑛β‰₯3.
Keyword: Metric dimention, partition dimenstion,corona product graphs


Kata Kunci


Metric dimention, partition dimenstion,corona product graphs

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DOI: http://dx.doi.org/10.25139/sm.v4i5.235

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ISSN:Β 2337-9421 (cetak)
ISSN: 2581-1290 (online)Β 

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