Structural similarity between polyhedral embeddings and their duals and its application to self-duality of pathwidth

Let $G$ be a graph embedded on a closed surface. We call $G$ a \emph{polyhedral embedding} if all facial walks are cycles, and any two of them are either disjoint or intersect in a single vertex or a single edge. In this paper, we present a new bound on the relation between the pathwidth of a polyhedral embedding and its dual. More precisely, we prove that for a polyhedral embedding $G$ on a closed surface with Euler characteristic $\chi$, $\mathsf{pw}(G^*) \leq 3\ \mathsf{pw}(G)+c$, where $c$ is a constant depending only on $\chi$. This result improves the coefficient of $\mathsf{pw}(G)$ in the previously known bound by Fomin and Thilikos (2007) and extends that of Amini, Huc, and P\'erennes (2009) for plane graphs. Furthermore, we obtain analogous bounds on the treewidth and pathwidth of the face subdivision of a polyhedral embedding. Our approach is based on a new quantitative estimate which demonstrates the structural similarity between a polyhedral embedding and its dual.