Upward Book Embeddings of Partitioned Digraphs

Abstract

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.

Figures (6)

• Figure 1: (a) A book embedding of the octahedron in $2$ pages. (b) An orientation of the octahedron. (c) An upward book embedding of the directed octahedron in (b) in $3$ pages, which is optimal.
• Figure 2: (a) An upward planar drawing $\Gamma$ and its big, small, and flat angles depicted as red, green, and yellow sectors, respectively. (b) The upward-consistent angle assignmentas $\lambda_\Gamma$ defined by $\Gamma$.
• Figure 3: (a) An upward planar drawing $\Gamma$ of a biconnected partitioned directed partial $2$-tree $G$. The labeling $\lambda_\Gamma$ of the angles determined by $\Gamma$ is shown; the missing labels are equal to $-1$. (b) The SPQ-tree of $G$ rooted at the Q-node corresponding to the edge $e^*$ of $G$.
• Figure 4: Impossible faces in (a) a plane $st$-graph and (b) an upward plane digraph.
• Figure 7: (a) An upward planar drawing $\Gamma$ of an upward plane digraph $G$; only large and small switch angles at switches of $G$ are depicted (as red and green sectors of discs centered at switch vertices, respectively). (b) The network $\cal N$ constructed from $G$ and its planar embedding; for convenience, the edges of $G$, which are not part of $\mathcal{N}$, are drawn as gray dashed curves. Arcs of $\cal N$ traversed by the flow are red, whereas arcs with no flow are green.

Theorems & Definitions (18)

• Theorem 1: DBLP:journals/algorithmica/BertolazziBLM94DBLP:journals/siamdm/DidimoGL09
• Theorem 2
• Theorem 3
• Theorem 4
• Corollary 5
• Theorem 6
• Lemma 7
• Lemma 8
• Lemma 9
• Lemma 10