Computing a crossing minimum drawing of a given planar graph G augmented by an additional edge e where all crossings involve e, has been a long standing open problem in graph drawing. Alternatively, the problem can be stated as finding a combinatorial embedding of a planar graph G where the given edge e can be inserted with the minimum number of crossings. Many problems concerned with the optimization over the set of all combinatorial embeddings of a planar graph turned out to be NP-hard. Surprisingly, we found a conceptually simple linear time algorithm based on SPQR-trees, that is able to find a solution with the minimum number of crossings.
A clustered graph C = (G, T) consists of an undirected graph G and a rooted tree T in which the leaves of T correspond to the vertices of G = (V, E). Each vertex µ in T corresponds to a subset of the vertices of the graph called "cluster". c-planarity is a natural extension of graph planarity for clustered graphs, and plays an important role in automatic graph drawing. The complexity status of c-planarity testing is unknown. It has been shown in [FCE95,Dah98] that c-planarity can be tested in linear time for c-connected graphs, i.e., graphs in which the cluster induced subgraphs are connected. In this paper, we provide a polynomial time algorithm for c-planarity testing of "almost" c-connected clustered graphs, i.e., graphs for which all nodes corresponding to the non-c-connected clusters lie on the same path in T starting at the root of T , or graphs in which for each nonconnected cluster its super-cluster and all its siblings in T are connected. The algorithm is based on the concepts for the subgraph induced planar connectivity augmentation problem presented in [GJL + 02]. We regard it as a first step towards general c-planarity testing. † Partially supported by the Future and Emerging Technologies programme of the EU under contract number IST-1999-14186 (ALCOM-FT).
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