Testing mutual duality of planar graphs

Abstract

We introduce and study the problem Mutual Planar Duality, which asks for planar graphs G1 and G2 whether G1 can be embedded such that its dual is isomorphic to G2. 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 ∼ G2 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). © 2013 Springer-Verlag.