Advances in C-Planarity Testing of Clustered Graphs

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 c 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 for \almost" c-connected clustered graphs, i.e., graphs for which all c-vertices corresponding to the non-cconnected clusters lie on the same path in T starting at the root of T , or graphs in which for each non-connected cluster its super-cluster and all its siblings are connected. The algorithm uses ideas of the algorithm for subgraph induced planar connectivity augmentation presented in [GJL02]. We regard it as a rst step towards general c-planarity testing.