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Upward Pointset Embeddings of Planar st-Graphs

Authors Carlos Alegría , Susanna Caroppo , Giordano Da Lozzo , Marco D'Elia , Giuseppe Di Battista , Fabrizio Frati , Fabrizio Grosso , Maurizio Patrignani

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ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Graph algorithms
Keywords
  • Upward pointset embeddings
  • planar st-graphs
  • st-cutset
  • non-crossing monotone Hamiltonian cycles
  • graph drawing enumeration

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Abstract

We study upward pointset embeddings (UPSEs) of planar st-graphs. Let G be a planar st-graph and let S ⊂ ℝ² be a pointset with |S| = |V(G)|. An UPSE of G on S is an upward planar straight-line drawing of G that maps the vertices of G to the points of S. We consider both the problem of testing the existence of an UPSE of G on S (UPSE Testing) and the problem of enumerating all UPSEs of G on S. We prove that UPSE Testing is NP-complete even for st-graphs that consist of a set of directed st-paths sharing only s and t. On the other hand, for n-vertex planar st-graphs whose maximum st-cutset has size k, we prove that it is possible to solve UPSE Testing in 𝒪(n^{4k}) time with 𝒪(n^{3k}) space, and to enumerate all UPSEs of G on S with 𝒪(n) worst-case delay, using 𝒪(k n^{4k} log n) space, after 𝒪(k n^{4k} log n) set-up time. Moreover, for an n-vertex st-graph whose underlying graph is a cycle, we provide a necessary and sufficient condition for the existence of an UPSE on a given poinset, which can be tested in 𝒪(n log n) time. Related to this result, we give an algorithm that, for a set S of n points, enumerates all the non-crossing monotone Hamiltonian cycles on S with 𝒪(n) worst-case delay, using 𝒪(n²) space, after 𝒪(n²) set-up time.

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Carlos Alegría, Susanna Caroppo, Giordano Da Lozzo, Marco D'Elia, Giuseppe Di Battista, Fabrizio Frati, Fabrizio Grosso, and Maurizio Patrignani. Upward Pointset Embeddings of Planar st-Graphs. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.GD.2024.24

Author Details

Carlos Alegría
  • Roma Tre University, Rome, Italy
Susanna Caroppo
  • Roma Tre University, Rome, Italy
Giordano Da Lozzo
  • Roma Tre University, Rome, Italy
Marco D'Elia
  • Roma Tre University, Rome, Italy
Giuseppe Di Battista
  • Roma Tre University, Rome, Italy
Fabrizio Frati
  • Roma Tre University, Rome, Italy
Fabrizio Grosso
  • Roma Tre University, Rome, Italy
Maurizio Patrignani
  • Roma Tre University, Rome, Italy

Funding

This research was supported, in part, by MUR of Italy (PRIN Project no. 2022ME9Z78 - NextGRAAL and PRIN Project no. 2022TS4Y3N - EXPAND).

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