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Abstract: Mixed graphs have both directed and undirected edges and have received considerable attention in the literature. We study two upward planarity testing problems for embedded mixed graphs, give some NP-hardness results, and describe Integer Linear Programming techniques to solve them. Experiments show the efficiency of our approach. [Copyright &y& Elsevier]

In a fan-planar drawing of a graph an edge can cross only edges with a common end-vertex. Fan-planar drawings have been recently introduced by Kaufmann and Ueckerdt [35] , who proved that every n -vertex fan-planar drawing has at most 5 n − 10 edges, and that this bound is tight for n ≥ 20 . We extend their result from both the combinatorial and the algorithmic point of view. We prove tight bounds on the density of constrained versions of fan-planar drawings and study the relationship between fan-planarity and k -planarity. Also, we prove that testing fan-planarity in the variable embedding setting is NP-complete. [ABSTRACT FROM AUTHOR]