  A book embedding of a graph is a drawing that maps vertices onto a line and
edges to simple pairwise non-crossing curves drawn into pages, which are
half-planes bounded by that line. Two-page book embeddings, i.e., book
embeddings into 2 pages, are of special importance as they are both NP-hard to
compute and have specific applications. We obtain a 2^(O(\sqrt{n})) algorithm
for computing a book embedding of an n-vertex graph on two pages -- a result
which is asymptotically tight under the Exponential Time Hypothesis. As a key
tool in our approach, we obtain a single-exponential fixed-parameter algorithm
for the same problem when parameterized by the treewidth of the input graph. We
conclude by establishing the fixed-parameter tractability of computing
minimum-page book embeddings when parameterized by the feedback edge number,
settling an open question arising from previous work on the problem.
