The Complexity of Finding

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.