The dominating partition dimension and locating-chromatic number of graphs

Abstract

For every graph G , the dominating partition dimension of G is either the same as its partition dimension or one higher than its partition dimension. In this paper, we consider some general connections among these three graph parameters: partition dimension, locating-chromatic number, and dominating partition dimension. We will show that β p ( G )≤ η p ( G )≤ χ L ( G ) for any graph G with at least 3 vertices. Therefore, we will derive properties for which graphs G have η p ( G )= β p ( G ) or η p ( G )= β p ( G )+1 .