  Structured $d$-DNNFs and SDDs are restricted negation normal form circuits
used in knowledge compilation as target languages into which propositional
theories are compiled. Structuredness is imposed by so-called vtrees. By
definition SDDs are restricted structured $d$-DNNFs. Beame and Liew (2015) as
well as Bova and Szeider (2017) mentioned the question whether structured
$d$-DNNFs are really more general than SDDs w.r.t. polynomial-size
representations (w.r.t. the number of Boolean variables the represented
functions are defined on.) The main result in the paper is the proof that a
function can be represented by SDDs of polynomial size if the function and its
complement have polynomial-size structured $d$-DNNFs that respect the same
vtree.
