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Simple Realizability of Abstract Topological Graphs

Authors Giordano Da Lozzo , Walter Didimo , Fabrizio Montecchiani , Miriam Münch , Maurizio Patrignani , Ignaz Rutter

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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Graph algorithms
  • Human-centered computing → Graph drawings
Keywords
  • Abstract Topological Graphs
  • SPQR-Trees
  • Synchronized PQ-Trees

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Abstract

An abstract topological graph (AT-graph) is a pair A = (G, X), where G = (V,E) is a graph and X ⊆ binom(E,2) is a set of pairs of edges of G. A realization of A is a drawing Γ_A of G in the plane such that any two edges e₁,e₂ of G cross in Γ_A if and only if (e₁,e₂) ∈ X; Γ_A is simple if any two edges intersect at most once (either at a common endpoint or at a proper crossing). The AT-graph Realizability (ATR) problem asks whether an input AT-graph admits a realization. The version of this problem that requires a simple realization is called Simple AT-graph Realizability (SATR). It is a classical result that both ATR and SATR are NP-complete [Kratochvíl, 1991; Kratochvíl and Matoušek, 1989].
In this paper, we study the SATR problem from a new structural perspective. More precisely, we consider the size λ(A) of the largest connected component of the crossing graph of any realization of A, i.e., the graph C(A) = (E, X). This parameter represents a natural way to measure the level of interplay among edge crossings. First, we prove that SATR is NP-complete when λ(A) ≥ 6. On the positive side, we give an optimal linear-time algorithm that solves SATR when λ(A) ≤ 3 and returns a simple realization if one exists. Our algorithm is based on several ingredients, in particular the reduction to a new embedding problem subject to constraints that require certain pairs of edges to alternate (in the rotation system), and a sequence of transformations that exploit the interplay between alternation constraints and the SPQR-tree and PQ-tree data structures to eventually arrive at a simpler embedding problem that can be solved with standard techniques.

Cite As Get BibTex

Giordano Da Lozzo, Walter Didimo, Fabrizio Montecchiani, Miriam Münch, Maurizio Patrignani, and Ignaz Rutter. Simple Realizability of Abstract Topological Graphs. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.23

Author Details

Giordano Da Lozzo
  • Roma Tre University, Italy
Walter Didimo
  • University of Perugia, Italy
Fabrizio Montecchiani
  • University of Perugia, Italy
Miriam Münch
  • University of Passau, Germany
Maurizio Patrignani
  • Roma Tre University, Italy
Ignaz Rutter
  • University of Passau, Germany

Funding

Research by {Da Lozzo}, Didimo, Montecchiani, and {Patrignani} was supported, in part, by MUR of Italy (PRIN Project no. 2022ME9Z78 - NextGRAAL and PRIN Project no. 2022TS4Y3N - EXPAND). Research by Münch and Rutter was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 541433306.

Acknowledgements

Research started at the 1^{st} Summer Workshop on Graph Drawing, September 2021, Castiglione del Lago, Italy.

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