  In 2000, Dergachev and Kirillov introduced subalgebras of "seaweed type" in
$\mathfrak{gl}_n$ and computed their index using certain graphs, which we call
type-${\sf A}$ meander graphs. Then the subalgebras of seaweed type, or just
"seaweeds", have been defined by Panyushev (2001) for arbitrary reductive Lie
algebras. Recently, a meander graph approach to computing the index in types
${\sf B}$ and ${\sf C}$ has been developed by the authors. In this article, we
consider the most difficult and interesting case of type ${\sf D}$. Some new
phenomena occurring here are related to the fact that the Dynkin diagram has a
branching node.
