Domination in the hierarchical product and Vizing's conjecture

Given a graph G, a set S⊆V(G) is a dominating set of G if every vertex of G is either in S or adjacent to a vertex in S. The domination number of G, denoted γ(G), is the minimum cardinality of a dominating set of G. Vizing's conjecture states that γ(G□H)≥γ(G)γ(H) for any graphs G and H where G□H denotes the Cartesian product of G and H. In this paper, we continue the work by Anderson et al. (2016) by studying the domination number of the hierarchical product. Specifically, we show that partitioning the vertex set of a graph in a particular way shows a trend in the lower bound of the domination number of the product, providing further evidence that the conjecture is true.