On the upward book thickness problem: Combinatorial and complexity results

Among the vast literature concerning graph drawing and graph theory, linear layouts of graphs have been the subject of intense research over the years, both from a combinatorial and from an algorithmic perspective. In particular, upward book embeddings of directed acyclic graphs (DAGs) form a popular class of linear layouts with notable applications, and the upward book thickness of a DAG is the minimum number of pages required by any of its upward book embeddings. A long-standing conjecture by Heath, Pemmaraju, and Trenk (1999) states that the upward book thickness of outerplanar DAGs is bounded above by a constant. In this paper, we show that the conjecture holds for subfamilies of upward outerplanar graphs, namely those whose underlying graph is an internally-triangulated outerpath or a cactus, and those whose biconnected components are st-outerplanar graphs. On the complexity side, it is known that deciding whether a graph has upward book thickness k is NP-hard for any fixed k≥3. We show that the problem, for any k≥5, remains NP-hard for graphs whose domination number is O(k), but it is fixed-parameter tractable (FPT) in the vertex cover number.