The Lower Irredundance and Domination Parameters are Equal for Inflated Trees

The inflated graph \(G_1\) of a graph \(G\) is obtained by replacing every vertex of degree \(d\) by a clique \(K_d\). We pursue the investigation of domination related parameters of inflated graphs initialized by Dunbar and Haynes. They conjectured that the lower irredundance and domination parameters are equal for inflated graphs. Favaron showed that in general the difference between them can be as large as desired. In this article, we prove that the two parameters are equal for inflated trees.