On the Upward Book Thickness Problem: Combinatorial and Complexity Results
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A long-standing conjecture by Heath, Pemmaraju, and Trenk 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 at-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 FPT in the vertex cover number.
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