Testing Mutual Duality of Planar Graphs

Abstract

We introduce and study the problem Mutual Planar Duality, which asks for planar graphs G 1 and G 2 whether G 1 can be embedded such that its dual is isomorphic to G 2. We show NP-completeness for general graphs and give a linear-time algorithm for biconnected graphs. We consider the common dual relation ~, where G 1 ~G 2 if and only they admit embeddings that result in the same dual graph. We show that ~ is an equivalence relation on the set of biconnected graphs and devise a succinct, SPQR-tree-like representation of its equivalence classes. To solve Mutual Planar Duality for biconnected graphs, we show how to do isomorphism testing for two such representations in linear time. A special case of Mutual Planar Duality is testing whether a graph is self-dual. Our algorithm can handle the case of biconnected graphs in linear time and our NP-hardness proof extends to self-duality and also to map self-duality testing (which additionally requires to preserve the embedding).