TESTING MUTUAL DUALITY OF PLANAR GRAPHS

Abstract

We introduce and study the problem MUTUAL DUALITY, asking for two planar graphs G1 and G2 whether G1 can be embedded such that its dual is isomorphic to G2. We show NP-completeness for general planar graphs and give a linear-time algorithm for biconnected planar graphs. This algorithm implies an efficient solution to two well-known problems. In fact, it can be used to test whether two biconnected planar graphs are 2-isomorphic, namely whether their graphic matroids are isomorphic, and to test self-duality of any biconnected planar graph, which is a special case of MUTUAL DUALITY with G1 = G2. Further, we show that our NP-hardness proof extends to testing self-duality and map self-duality (which additionally requires to preserve the embedding). In order to obtain our results, we consider the common dual relation ~, where G1 ~ G2 if and only if 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 compact SPQR-tree-like representation of its equivalence classes. Our algorithm for biconnected graphs is based on testing isomorphism for two such representations in linear time.