  An independent dominating set of a graph, also known as a maximal independent
set, is a set $S$ of pairwise non-adjacent vertices such that every vertex not
in $S$ is adjacent to some vertex in $S$. We prove that for $\Delta=4$ or
$\Delta\ge 6$, every connected $n$-vertex graph of maximum degree at most
$\Delta$ has an independent dominating set of size at most
$(1-\frac{\Delta}{\lfloor\Delta^2/4\rfloor+\Delta})(n-1)+1$. In addition, we
characterize all connected graphs having the equality and we show that other
connected graphs have an independent dominating set of size at most
$(1-\frac{\Delta}{ \lfloor\Delta^2/4\rfloor+\Delta})n$.
