Upward Book Embeddings of Partitioned Digraphs

2026-03-17

AI Summary

This study addresses the problem of determining whether a directed graph, whose edge set is partitioned into two subsets, admits a two-page upward book embedding such that vertices are placed along the spine in a topological order and edges on the same page do not cross. By leveraging structural graph-theoretic analysis, SPQR-tree decomposition, and constraints from planar embeddings, the work provides the first characterization of digraphs admitting such embeddings. The main contributions include proving that the problem is NP-complete in general, thereby resolving the complexity gap for the case k=2; presenting an O(n log³n)-time decision algorithm for graphs with a fixed planar embedding; and designing a cubic-time algorithm for biconnected directed partial 2-trees.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• upward book embedding
• partitioned digraph
• NP-completeness
• topological ordering
• edge partition

Innovation

• upward book embedding
• partitioned digraph
• NP-completeness
• planar embedding
• SPQR-tree