On The Partition Dimension of a Lollipop Graph and a Generalized Jahangir Graph

[摘要] Let G be a connected graph with vertex set V (G), such that V (G) can be divided into any partition set S. The set Π with S ∈ Π is a resolving partition of G if each vertex in G has a distinct representation with respect to Π, and Π is an ordered k-partition. The minimum cardinality of resolving k-partitions of V (G) is called a partition dimension of G, denoted by pd(G). The lollipop graph Lm,nis a graph obtained by joining a complete graph Kmto a path Pnwith a bridge. A generalized Jahangir graph is a graph consisting of a cycle Cmnand one additional vertex which is adjacent to n vertices of Cmnat m distance to each other on Cmn. Many researchers have conducted research in determining the partition dimension for speci c graph classes. They are as references to determine some of the graph classes that haven't been studied previously. In this paper, we determine the partition dimension of a lollipop graph Lm,nand a generalized Jahangir graph Jm,n. The research methods in this paper is a book study. The results of this paper are as follows. We obtain the partition dimension of a lollipop graph pd(Lm,n) = m for m ≥ 3 and n ≥ 1. The partition dimension of a generalized Jahangir graph consists of two cases. We showed that pd(Jm,n) = 3 for n = 3, 4, 5 and we prove that pd(Jm,n) = ⌊n/2⌋ +1 for n ≥ 6.

[发布机构] Department of Mathematics, Faculty of Mathematics and Natural Sciences, Sebelas Maret University, Surakarta, Central Java, Indonesia^1

[效力级别] 教育 [学科分类] 发展心理学和教育心理学

[关键词] Cardinalities;Complete graphs;Connected graph;Graph class;K-partition;Partition dimensions;Resolving partitions;Vertex set