On the path-width of planar graphs

We present a result concerning the relation between the path-width of a plane graph and the path-width of its dual. We prove that for a 3-connected planar graph G,pw(G) ≤ 3pw(G*) + 2. For 4-connected planar graphs, and more generally for Hamiltonian planar graphs, we prove a stronger bound pw(G*) ≤ 2pw(G)+ c. The best previously known bound was obtained by Fomin and Thilikos who proved that pw(G*) ≤ 6pw(G) + c. Our proof is based on a transformation which, given a fixed spanning tree of G, sends any given decomposition of G into one of G*.The ratio of the corresponding parameters is bounded by the maximum degree of the spanning tree.