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In this paper we settle the computational complexity of two open problems related to the extension of the notion of level planarity to surfaces different from the plane. Namely, we show that the problems of testing the existence of a level embedding of a level graph on the surface of the rolling cylinder or on the surface of the torus, respectively known by the name of Cyclic Level Planarity and Torus Level Planarity , are polynomial-time solvable. Moreover, we show a complexity dichotomy for testing the Simultaneous Level Planarity of a set of level graphs, with respect to both the number of level graphs and the number of levels. [ABSTRACT FROM AUTHOR]

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]