A simple linear-time algorithm for finding path-decompositions of small   width
  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. 
Kevin Cattell, Michael J. Dinneen, Michael R. Fellows