  The scaling of entanglement entropy is computationally studied in several
$1\le d \le 2$ dimensional free fermion systems that are connected by one or
more point contacts (PC). For both the $k$-leg Bethe lattice $(d =1)$ and $d=2$
rectangular lattices with a subsystem of $L^d$ sites, the entanglement entropy
associated with a {\sl single} PC is found to be generically $S \sim L$. We
argue that the $O(L)$ entropy is an expression of the subdominant $O(L)$
entropy of the bulk entropy-area law. For $d=2$ (square) lattices connected by
$m$ PCs, the area law is found to be $S \sim aL^{d-1} + b m \log{L}$ and is
thus consistent with the anomalous area law for free fermions ($S \sim L
\log{L}$) as $m \rightarrow L$. For the Bethe lattice, the relevance of this
result to Density Matrix Renormalization Group (DMRG) schemes for interacting
fermions is discussed.
