The partition dimension of subdivision graph on the Star Amrullah, Syahrul A., Harry S., Turmuzi M., Yunita S. A.
Department of Mathematics Education, Faculty of Teacher Training and Education, Mataram University, Jl. Majapahit 62 Mataram, Indonesia
email: amrullah[at]unram.ac.id, syharulazmi[at]unram.ac.id,harrysoeprianto[at]unram.ac.id, mturmuzi[at]unram.ac.id, na2_math[at]yahoo.com
Abstract
The partition dimension of the graphs is one of the open problems in graph theory. One of the methods which are used researcher is a graph operation, for example, subdivision operations. Let $G$ be a connected graph, the subdivision operation of $G$ is an operation in $G$ with order $n$ that replaces any edge of $G$ by a path $ P_{k_i+2}$ for $k_i\geq 1$ and $i\in [1,n]$. The result graph of this operation is called the subdivision graph, is denoted by $S(G(E;k_1,k_2,\cdots,k_n))$. Furthermore, if each $k_i=1$, then the subdivision graph is denoted by $S(G)$. One of the most recent results on the partition dimension of subdivision operation in the general graphs is published by Amrullah, et al 2016. However, the results show only un upper and lower bounds of the subdivision graph partition dimension. Therefore, this paper is devoted to finding the partition dimension of subdivision graph $S(G)$ and $S(G(E;k_1,k_2,\cdots,k_n))$ on special star graphs $G=K_{1,n}$.
Keywords: Partition dimension; star graph,;resolving partition;subdivision
