Optimal Acyclic Hamiltonian Path Completion for Outerplanar Triangulated st-Digraphs (with Application to Upward Topological Book Embeddings)

Given an embedded planar acyclic digraph G, we define the problem of “acyclic hamiltonian path completion with crossing minimization (Acyclic-HPCCM)” to be the problem of determining an hamiltonian path completion set of edges such that, when these edges are embedded on G, they create the smallest possible number of edge crossings and turn G to a hamiltonian digraph. Our results include: –We provide a characterization under which a triangulated st-digraph G is hamiltonian. –For an outerplanar triangulated st-digraph G, we define the st-polygon decomposition of G and, based on its properties, we develop a linear-time algorithm that solves the Acyclic-HPCCM problem with at most one crossing per edge of G. –For the class of st-planar digraphs, we establish an equivalence between the Acyclic-HPCCM problem and the problem of determining an upward 2-page topological book embedding with minimum number of spine crossings. We infer (based on this equivalence) for the class of outerplanar triangulated st-digraphs an upward topological 2-page book embedding with minimum number of spine crossings and at most one spine crossing per edge. To the best of our knowledge, it is the first time that edge-crossing minimization is studied in conjunction with the acyclic hamiltonian completion problem and the first time that an optimal algorithm with respect to spine crossing minimization is presented for upward topological book embeddings.

💡 Research Summary

The paper introduces a novel optimization problem for embedded planar acyclic digraphs, called Acyclic Hamiltonian Path Completion with Crossing Minimization (Acyclic‑HPCCM). Given a planar embedding of an acyclic digraph G, the task is to add a set of edges that makes G Hamiltonian while keeping the number of edge‑edge crossings (including those caused by the added edges) as small as possible. While previous work on Hamiltonian completion has focused either on minimizing the number of added edges or on crossing reduction for already Hamiltonian graphs, this work tackles both objectives simultaneously.

The authors first provide a structural characterization of triangulated st‑digraphs that are Hamiltonian. Central to this characterization is the notion of an st‑polygon, a minimal triangulated sub‑structure consisting of two directed outer paths sharing the source s and sink t and a single interior diagonal. By decomposing an outerplanar triangulated st‑digraph into a sequence of st‑polygons (the st‑polygon decomposition), the global problem can be reduced to a series of local decisions on how to connect consecutive polygons.

For outerplanar triangulated st‑digraphs, the paper presents a linear‑time algorithm that solves Acyclic‑HPCCM optimally. The algorithm proceeds along the st‑polygon decomposition, using dynamic programming to decide, for each polygon, whether to connect it directly to the previous one or to introduce a crossing‑allowing edge. Crucially, the algorithm guarantees that each original edge of G participates in at most one crossing, and the total number of crossings is provably minimal. The time complexity is O(n), where n is the number of vertices, making the approach practical for large graphs.

Beyond the completion problem, the authors establish a deep equivalence between Acyclic‑HPCCM for st‑planar digraphs and the problem of constructing an upward 2‑page topological book embedding with a minimum number of spine crossings. In a book embedding, all vertices lie on a linear spine, and each directed edge is drawn on one of two pages without edge‑edge crossings within a page; the upward constraint forces every edge to be oriented from lower to higher positions on the spine. By mapping the optimal Hamiltonian path obtained from Acyclic‑HPCCM to a spine order, the authors obtain an upward 2‑page embedding whose spine crossing count equals the edge‑crossing count of the completion solution. Consequently, for outerplanar triangulated st‑digraphs, the paper delivers an upward topological book embedding that is optimal with respect to spine crossings and respects the “at most one spine crossing per original edge” bound.

The contributions are twofold. First, the paper formally defines a crossing‑aware Hamiltonian completion problem and supplies the first optimal linear‑time algorithm for a non‑trivial graph class (outerplanar triangulated st‑digraphs). Second, it bridges the Hamiltonian completion literature with the field of upward book embeddings, providing the first optimal algorithm for minimizing spine crossings in upward topological 2‑page embeddings. The results have immediate implications for graph visualization, VLSI layout, and any domain where directed acyclic structures must be displayed compactly while preserving planarity as much as possible.

The authors conclude with several avenues for future work: extending Acyclic‑HPCCM to general planar acyclic digraphs, investigating multi‑page book embeddings under crossing minimization, and developing dynamic algorithms that maintain optimality under incremental graph updates. Such extensions would broaden the applicability of the techniques beyond the outerplanar setting and could lead to new insights in both theoretical graph algorithms and practical layout tools.