Let $G=(V,E)$ be a simple graph with vertex set $V=V(G)$ and edge set $E=E(G)$. A subset $D\subseteq V(G)$ is called an independent dominating set of the graph $G$ if it is both a dominating set and an independent set. The minimum cardinality among all independent dominating sets of $G$ is the independent domination number $i(G)$. An independent dominating set of minimum cardinality is called an $i$-set. For a vertex $v\in V(G)$, the number $i(G-v)$ may be greater than, less than, or equal to $i(G)$. A graph $G$ is independent domination critical, or $i$-critical, if $i(G-v)< i(G)$ for every $v\in V(G)$. More generally, for an integer $t\geq 1$, a graph $G$ is $(i, t)$-critical if $i(G-S)