# Finding small-width connected path decompositions in polynomial time

A connected path decomposition of a simple graph G is a path decomposition (X_1,...,X_l) such that the subgraph of G induced by X_1∪...∪ X_i 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.

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