  In this paper we study complexity of an extension of ordered binary decision
diagrams (OBDDs) called $c$-OBDDs on CNFs of bounded (primal graph) treewidth.
In particular, we show that for each $k$ there is a class of CNFs of treewidth
$k \geq 3$ for which the equivalent $c$-OBDDs are of size
$\Omega(n^{k/(8c-4)})$. Moreover, this lower bound holds if $c$-OBDD is
non-deterministic and semantic. Our second result uses the above lower bound to
separate the above model from sentential decision diagrams (SDDs). In order to
obtain the lower bound, we use a structural graph parameter called matching
width. Our third result shows that matching width and pathwidth are linearly
related.
