Unique irredundance, domination and independent domination in graphs

In this paper we prove that any graph with equal irredundance and domination numbers has a unique minimum irredundant set if and only if it has a unique minimum dominating set. Using a result by Zverovich and Zverovich [An induced subgraph characterization of domination perfect graphs, J. Graph Theory 20(3) (1995) 375-395], we characterize the hereditary class of graphs G such that for every induced subgraph H of G, H has a unique ι-set if and only if H has a unique γ-set. Furthermore, for trees with equal domination and independent domination numbers we present a characterization of unique minimum independent dominating sets, which leads to a linear time algorithm to decide whether such trees have unique minimum independent dominating sets.