Testing Mutual Duality of Planar Graphs

TL;DR

This paper investigates the problem of determining if one planar graph can be embedded so that its dual is isomorphic to another, providing complexity results and efficient algorithms for special cases.

Contribution

It introduces the extsc{Mutual Duality} problem, proves NP-completeness in general, and develops a linear-time algorithm for biconnected graphs using a novel succinct representation.

Findings

NP-completeness of the general problem

Linear-time algorithm for biconnected graphs

Characterization of duality via equivalence classes

Abstract

We introduce and study the problem \mpd, which asks for two planar graphs $G_1$ and $G_2$ whether $G_1$ can be embedded such that its dual is isomorphic to $G_2$. Our algorithmic main result is an NP-completeness proof for the general case and a linear-time algorithm for biconnected graphs. To shed light onto the combinatorial structure of the duals of a planar graph, we consider the \emph{common dual relation} $\sim$, where $G_1 \sim G_2$ if and only if they have a common dual. While $\sim$ is generally not transitive, we show that the restriction to biconnected graphs is an equivalence relation. In this case, being dual to each other carries over to the equivalence classes, i.e., two graphs are dual to each other if and only if any two elements of their respective equivalence classes are dual to each other. To achieve the efficient testing algorithm for \mpd on biconnected graphs, we devise a succinct representation of the equivalence class of a biconnected planar graph. It is similar to SPQR-trees and represents exactly the graphs that are contained in the equivalence class. The testing algorithm then works by testing in linear time whether two such representations are isomorphic. We note that a special case of \mpd is testing whether a graph $G$ is self-dual. Our algorithm handles the case where $G$ is biconnected and our NP-hardness proof extends to testing self-duality of general planar graphs and also to testing map self-duality, where a graph $G$ is map self-dual if it admits a planar embedding $\mathcal G$ such that $G^\star$ is isomorphic to $G$, and additionally the embedding induced by $\mathcal G$ on $G^\star$ is $\mathcal G$.