The partition dimension for a subdivision of a homogeneous firecracker

Abstract

Finding the partition dimension of a graph is one of the interesting (and uncompletely solved) problems of graph theory. For instance, the values of the partition dimensions for most kind of trees are still unknown. Although for several classes of trees such as paths, stars, caterpillars, homogeneous firecrackers and others, we do know their partition dimensions. In this paper, we determine the partition dimension of a subdivision of a particular tree, namely homogeneous firecrackers. Let G be any graph. For any positive integer k and e ∈ E(G), a subdivision of a graph G, denoted by S(G(e; k)), is the graph obtained from G by replacing an edge e with a (k + 1)-path. We show that the partition dimension of S(G(e; k)) is equal to the partition dimension of G if G is a homogeneous firecracker.