Finding small-width connected path decompositions in polynomial time

A connected path decomposition of a simple graph G is a path decomposition (X1, . . . , Xl) such that the subgraph of G induced by X1 ∪ · · ·∪Xi is connected for each i ∈ {1, . . . , l}. The connected pathwidth of G is then the minimum width over all connected path decompositions of G. We prove that for each fixed k, the connected pathwidth of any input graph can be computed in polynomial-time. This answers an open question raised by Fedor V. Fomin during the GRASTA 2017 workshop, since connected pathwidth is equivalent to the connected (monotone) node search game.