  In [M. M. Kant\'e, V. Limouzy, A. Mary, and L. Nourine. On the enumeration of
minimal dominating sets and related notions. SIAM Journal on Discrete
Mathematics, 28(4):1916-1929, 2014] the authors give an $O(n+m)$ delay
algorithm based on neighborhood inclusions for the enumeration of minimal
dominating sets in split and $P_6$-free chordal graphs. In this paper, we
investigate generalizations of this technique to $P_k$-free chordal graphs for
larger integers $k$. In particular, we give $O(n+m)$ and $O(n^3\cdot m)$ delays
algorithms in the classes of $P_7$-free and $P_8$-free chordal graphs. As for
$P_k$-free chordal graphs for $k\geq 9$, we give evidence that such a technique
is inefficient as a key step of the algorithm, namely the irredundant extension
problem, becomes NP-complete.

Discrete Mathematics;Data Structures and Algorithms;Combinatorics