The upward planarity testing problem consists of testing if a digraph admits a drawing Γ such that all edges in Γ are monotonically increasing in the vertical direction and no edges in Γ cross. In this paper we reduce the problem of testing a digraph for upward planarity to the problem of testing if its blocks admit upward planar drawings with certain properties. We also show how to test if a block of a digraph admits an upward planar drawing with the aforementioned properties.

Upward planarity is a widely investigated topic in graph theory and graph drawing. A planar digraph is said to be upward planar if it admits a planar drawing where all edges are drawn as curves monotonically increasing in the upward direction. It is known that testing whether a planar digraph is upward planar is NP-Complete in general. However, the upward planarity testing problem can be solved in polynomial time if the planar embedding of the digraph is fixed. This motivates the following question: Given an embedded planar digraph G, how is difficult to find an embedding preserving subgraph of G that is upward planar and whose number of edges is maximum? We prove that this problem is NP-Hard. This negative result motivates the study of polynomial-time algorithms for the computation of maximal (not necessarily maximum) upward planar subgraphs as well as the design of exact algorithms that perform well in practice. As a consequence of our proof technique, we also show that the problem of extracting a maximum bimodal subgraph from an embedded planar digraph is NP-Hard.

In this paper we present a new characterization of switch-regular upward embeddings, a concept introduced by Di Battista and Liotta in 1998. This characterization allows us to define a new efficient algorithm for computing upward planar drawings of embedded planar digraphs. If compared with a popular approach described by Bertolazzi, Di Battista, Liotta, and Mannino, our algorithm computes drawings that are significantly better in terms of total edge length and aspect ratio, especially for low-density digraphs. Also, we experimentally prove that the running time of the drawing process is reduced in most cases.