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 $k \ge 3$. We show that the problem, for any $k \ge 5$, remains NP-hard for graphs whose domination number is $O(k)$, but it is FPT in the vertex cover number.
翻译:Heath、Pemmaraju和Trenk的长期推测指出,外平面 DAG 外平面的上层书厚度被一个常数捆绑在上面。 在本文中,我们显示,外平面图的下层图的下层图,即其底部图是内部三角外向或仙人掌的图,以及其双连接组件为美元外向平面图的图。在复杂方面,人们知道,对于任何固定的 $k\ge 3 美元来说,要确定一个图是否具有上层书厚度是NP-硬值。我们显示,对于任何美元 $ + Ge 5 的图,其支配值为 $( k) $( k) 的图,问题仍然是NP- 硬值,但在顶层覆盖数中是 FPT。