  We show that the digraph of a nondeterministic finite automaton witnessing
the automatic complexity of a word can always be taken to be planar. In the
case of total transition functions studied by Shallit and Wang, planarity can
fail.
  Let $s_q(n)$ be the number of binary words $x$ of length $n$ having
nondeterministic automatic complexity $A_N(x)=q$. We show that $s_q$ is
eventually constant for each $q$ and that the eventual constant value of $s_q$
is computable.
