  We study the neighborhood polynomial and the complexity of its computation
for chordal graphs. The neighborhood polynomial of a graph is the generating
function of subsets of its vertices that have a common neighbor. We introduce a
parameter for chordal graphs called anchor width and an algorithm to compute
the neighborhood polynomial which runs in polynomial time if the anchor width
is polynomially bounded. The anchor width is the maximal number of different
sub-cliques of a clique which appear as a common neighborhood. Furthermore we
study the anchor width for chordal graphs and some subclasses such as chordal
comparability graphs and chordal graphs with bounded leafage. the leafage of a
chordal graphs is the minimum number of leaves in the host tree of a subtree
representation. We show that the anchor width of a chordal graph is at most
$n^{\ell}$ where $\ell$ denotes the leafage. This shows that for some
subclasses computing the neighborhood polynomial is possible in polynomial time
while it is NP-hard for general chordal graphs.
