Kevin Cattell, Michael J. Dinneen, Michael R. Fellows. A Simple Linear-Time Algorithm for Finding Path-Decompositions of Small Width. Inf. Process. Lett., 57(4):197-203, 1996. [doi]
We described a simple algorithm running in linear time for each fixed constant $k$, that either establishes that the pathwidth of a graph $G$ is greater than $k$, or finds a path-decomposition of $G$ of width at most $O(2^{k})$.
This provides a simple proof of the result by Bodlaender that many families of graphs of bounded pathwidth can be recognized in linear time.