On the Pathwidth of Planar Graphs

Description

Fomin and Thilikos in [5] conjectured that there is a constant $c$ such that, for every $2$-connected planar graph $G$, {pw}(G^*) \leq 2\text{pw}(G)+c$ (the same question was asked simutaneously by Coudert, Huc and Sereni in [4]). By the results of Boedlander and Fomin [2] this holds for every outerplanar graph and actually is tight by Coudert, Huc and Sereni [4]. In [5], Fomin and Thilikos proved that there is a constant $c$ such that the pathwidth of every 3-connected graph $G$ satisfies: ${pw}(G^*) \leq 6\text{pw}(G)+c$. In this paper we improve this result by showing that the dual a 3-connected planar graph has pathwidth at most $3$ times the pathwidth of the primal plus two. We prove also that the question can be answered positively for $4$-connected planar graphs.

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