Path NFA-eplison to DFA conversion
Unless I misunderstood your question, in a NFA $N$ that is a path one vertex has in-degree $0$ and outdegree $1$, one vertex (possibly the same) has in-degree $1$ and out-degree $0$, and all the other vertices have in-degree $1$ and out-degree $1$.

Let $N'$ be the NFA obtained by identifying all the vertices in the same component of the (undirected version of the) subgraph induced by all the edges labelled $\varepsilon$ in $N$. All vertices of $N'$ have maximum out-degree $1$, moreover $N'$ contains no edge labelled $\epsilon$. This shows that $N'$ is a DFA. The process can be performed in linear time.

A note: when a set $S$ of vertices from $N$ is identified into a single vertex $v$ in $N'$, you will need to mark $v$ as a final state in $N'$ iff at least one the vertices in $S$ is a final state in $N$.