  In this paper we study the problem of existence of a crossing-free acyclic
hamiltonian path completion (for short, HP-completion) set for embedded upward
planar digraphs. In the context of book embeddings, this question becomes:
given an embedded upward planar digraph $G$, determine whether there exists an
upward 2-page book embedding of $G$ preserving the given planar embedding.
  Given an embedded $st$-digraph $G$ which has a crossing-free HP-completion
set, we show that there always exists a crossing-free HP-completion set with at
most two edges per face of $G$. For an embedded $N$-free upward planar digraph
$G$, we show that there always exists a crossing-free acyclic HP-completion set
for $G$ which, moreover, can be computed in linear time. For a width-$k$
embedded planar $st$-digraph $G$, we show that we can be efficiently test
whether $G$ admits a crossing-free acyclic HP-completion set.
