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Point Set Embeddability with List Constraints

Authors Thomas Depian , Joseph Dorfer , Boris Klemz , Matthias Pfretzschner , Lena Schlipf

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LIPIcs.GD.2026.35.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Fixed parameter tractability
  • Human-centered computing → Graph drawings
  • Theory of computation → Computational geometry
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Dynamic programming
Keywords
  • Point set embedding
  • List constraints
  • Parameterized algorithms and complexity
  • NP-hardness
  • Dynamic programming
  • Graph drawing
  • Computational geometry

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Abstract

Deciding whether a given graph admits a planar straight-line drawing where each vertex is placed on some point from a given finite point set is known as Point Set Embeddability and is a classical problem in graph drawing. In this paper, we study the more general embeddability question where the placement of each vertex v is restricted to a list L(v) of admissible points.
We first study the case where the given point set is in convex position. We show that this case is NP-hard even if the given graph is a matching and bi-labeled, i.e., each vertex has at most 2 admissible points. On the positive side, we present two efficient algorithms for the case where the given graph G is connected (and not necessarily bi-labeled): if G is equipped with a combinatorial embedding that needs to be respected, we can solve the problem in polynomial time; otherwise we can solve it in FPT-time with regard to the maximum vertex degree. In particular, this answers an open question by Frati, Glisse, Lenhart, Liotta, Mchedlidze, and Nishat [GD'13]. 
We then turn our attention to the more general case where the given point set is not necessarily in convex position. Here, we show NP-hardness for bi-labeled paths; notably these graphs have a unique combinatorial embedding and maximum degree two, thereby creating a stark contrast to our algorithms for the convex case. We also present an FPT-algorithm with respect to the vertex cover number for the special case of bi-labeled graphs. We complement this latter result by establishing paraNP-hardness in the tri-labeled setting for vertex cover number 2 and polynomial-time solvability for vertex cover number 1 and arbitrary L.
Finally, we study optimization and extension variants, where we want to maximize the number of edges or extend a partial drawing, respectively. For the former, we show APX-hardness and for the latter, we provide a parameterized complexity dichotomy under natural extension parameters.

Cite As Get BibTex

Thomas Depian, Joseph Dorfer, Boris Klemz, Matthias Pfretzschner, and Lena Schlipf. Point Set Embeddability with List Constraints. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 35:1-35:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.GD.2026.35

Author Details

Thomas Depian
  • TU Wien, Austria
Joseph Dorfer
  • TU Graz, Austria
Boris Klemz
  • Universität Würzburg, Germany
Matthias Pfretzschner
  • University of Passau, Germany
Lena Schlipf
  • Universität Tübingen, Germany

Funding

  • Depian, Thomas: Supported by the Vienna Science and Technology Fund (WWTF) [10.47379/ICT22029].
  • Dorfer, Joseph: Supported by the Austrian Science Fund (FWF) 10.55776/DOC183.
  • Schlipf, Lena: Supported by the German Research Foundation (DFG grant SCHL 2331/1-1)

Acknowledgements

This work started at the 20th European Research Week on Geometric Graphs (GGWeek'25) in Chorin, Germany. We thank the organizers for fruitful and nice atmosphere.

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