We provide several new algorithmic results for the secluded path problem, specifically approximation and optimality results for the static algorithm of [3,5], and an extension (h-Memory) of it based on de Bruijn graphs, when applied to bounded degree graphs and some other special graph classes which can model wireless communication and line-of-sight settings. Our primary result is that h-Memory is a PTAS for degree-Δ unweighted, undirected graphs, providing a \(\lceil{\sqrt{{\Delta+1}\over{h+1}}}\rceil\)-approximation in time O(n logn); in particular, 0-Memory (i.e., static) provides a \(\sqrt{\Delta+1}\) -approximation (i.e., \(\epsilon=\sqrt{\Delta+1}-1\)), tightening the previous analysis of this algorithm, and Δ-Memory is optimal (i.e., e = 0), and is faster than the known optimal algorithm for this setting [3].