In 1999, Heath, Pemmaraju, and Trenk [SIAM J. Comput. 28(4), 1999] extended the classic notion of book embeddings to digraphs, introducing the concept of upward book embeddings, in which the vertices must appear along the spine in a topological order and the edges are partitioned into pages, so that no two edges in the same page cross. For a partitioned digraph $G=(V,\bigcup^k_{i=1} E_i)$, that is, a digraph whose edge set is partitioned into $k$ subsets, an upward book embedding is required to assign edges to pages as prescribed by the given partition. In a companion paper, Heath and Pemmaraju [SIAM J. Comput 28(5), 1999] proved that the problem of testing the existence of an upward book embedding of a partitioned digraph is linear-time solvable for $k=1$ and recently Akitaya, Demaine, Hesterberg, and Liu [GD, 2017] have shown the problem NP-complete for $k\geq 3$. In this paper, we study upward book embeddings of partitioned digraphs and focus on the unsolved case $k=2$. Our first main result is a novel characterization of the upward embeddings that support an upward book embedding in two pages. We exploit this characterization in several ways, and obtain a rich picture of the complexity landscape of the problem. First, we show that the problem remains NP-complete when $k=2$, thus closing the complexity gap for the problem. Second, we show that, for an $n$-vertex partitioned digraph $G$ with a prescribed planar embedding, the existence of an upward book embedding of $G$ that respects the given planar embedding can be tested in $O(n \log^3 n)$ time. Finally, leveraging the SPQ(R)-tree decomposition of biconnected graphs into triconnected components, we present a cubic-time testing algorithm for biconnected directed partial $2$-trees.

翻译：1999年，Heath、Pemmaraju和Trenk [SIAM J. Comput. 28(4), 1999] 将经典的书本嵌入概念扩展到有向图，提出了向上书本嵌入的概念，其中顶点必须沿书脊按拓扑序排列，边被划分到不同页面，使得同一页面中的边互不交叉。对于划分有向图 $G=(V,\bigcup^k_{i=1} E_i)$，即边集被划分为 $k$ 个子集的有向图，向上书本嵌入需要按照给定划分将边分配到指定页面。在一篇相关论文中，Heath和Pemmaraju [SIAM J. Comput 28(5), 1999] 证明了当 $k=1$ 时，测试划分有向图是否存在向上书本嵌入的问题可在线性时间内求解；而最近Akitaya、Demaine、Hesterberg和Liu [GD, 2017] 证明了当 $k\geq 3$ 时该问题是NP完全的。本文研究划分有向图的向上书本嵌入，重点关注未解决的 $k=2$ 情形。我们的第一个主要成果是提出了支持两页向上书本嵌入的向上嵌入的新颖特征刻画。我们通过多种方式利用这一特征刻画，获得了该问题复杂度格局的丰富图景。首先，我们证明当 $k=2$ 时该问题仍然是NP完全的，从而填补了该问题的复杂度空白。其次，我们证明对于具有给定平面嵌入的 $n$ 顶点划分有向图 $G$，可在 $O(n \log^3 n)$ 时间内测试是否存在尊重该平面嵌入的向上书本嵌入。最后，利用双连通图到三连通分量的SPQ(R)-树分解，我们提出了针对双连通有向部分 $2$-树的立方时间测试算法。