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In a drawing of a clustered graph vertices and edges are drawn as points and curves, respectively, while clusters are represented by simple closed regions. A drawing of a clustered graph is c-planar if it has no edge–edge, edge–region, or region–region crossings. Determining the complexity of testing whether a clustered graph admits a c-planar drawing is a long-standing open problem in the Graph Drawing research area. An obvious necessary condition for c-planarity is the planarity of the graph underlying the clustered graph. However, this condition is not sufficient and the consequences on the problem due to the requirement of not having edge–region and region–region crossings are not yet fully understood. In order to shed light on the c-planarity problem, we consider a relaxed version of it, where some kinds of crossings (either edge–edge, edge–region, or region–region) are allowed even if the underlying graph is planar. We investigate the relationships among the minimum number of edge–edge, edge–region, and region–region crossings for drawings of the same clustered graph. Also, we consider drawings in which only crossings of one kind are admitted. In this setting, we prove that drawings with only edge–edge or with only edge–region crossings always exist, while drawings with only region–region crossings may not. Further, we provide upper and lower bounds for the number of such crossings. Finally, we give a polynomial-time algorithm to test whether a drawing with only region–region crossings exists for biconnected graphs, hence identifying a first non-trivial necessary condition for c-planarity that can be tested in polynomial time for a noticeable class of graphs. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we analyze the relationships among different planar embeddings of the same graph and study how two planar embeddings can be morphed one into the other with the minimum number of elementary changes, while preserving the mental map of the user. We call this problem Topological Morphing, in analogy with the well-known Geometric Morphing problem, in which it is studied how two planar drawings can be morphed one into the other with the minimum number of elementary changes. First, we define two operations, called flip and skip, describing natural transformations of a graph embedding that preserve the mental map of the user. Then, we study the problem of performing a morph while minimizing the number of these operations. We show that the Topological Morphing problem is NP-hard for biconnected planar graphs, we give polynomial-time algorithms for some restricted versions of the problem, and, based on such polynomial-time algorithms, we give a fixed-parameter tractable algorithm for the general case. [Copyright &y& Elsevier]