A Tight Subexponential-time Algorithm for Two-Page Book Embedding

Robert Ganian, Haiko Muller, Sebastian Ordyniak, Giacomo Paesani, Mateus Rychlicki

Abstract: 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 $2^{\mathcal{O}(\sqrt{n})}$ 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.