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 $Δ=4$ or $Δ\ge 6$, every connected $n$-vertex graph of maximum degree at most $Δ$ has an independent dominating set of size at most $(1-\fracΔ{\lfloorΔ^2/4\rfloor+Δ})(n-1)+1$. 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 $(1-\fracΔ{ \lfloorΔ^2/4\rfloor+Δ})n$.