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Augmenting Plane Straight-Line Graphs to Meet Parity Constraints

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Graph-Theoretic Concepts in Computer Science (WG 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14760))

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Abstract

Given a plane geometric graph G on n vertices, we want to augment it so that given parity constraints of the vertex degrees are met. In other words, given a subset R of the vertices, we are interested in a plane geometric supergraph \(G'\) such that exactly the vertices of R have odd degree in \(G'\setminus G\). We show that the question of whether such a supergraph exists can be decided in polynomial time for two interesting cases. First, when the vertices are in convex position, we present a linear-time algorithm. Building on this insight, we solve the case when G is a plane geometric path in \(O(n \log n)\) time. This solves an open problem posed by Catana, Olaverri, Tejel, and Urrutia (Appl. Math. Comput. 2020).

A.B.G.C. supported by the VILLUM Foundation grant 37507 “Efficient Recomputations for Changeful Problems”. I.P. is a Serra Húnter fellow and acknowledges the support of Independent Research Fund Denmark grant 2020-2023 (9131-00044B) “Dynamic Network Analysis”, the Margarita Salas Fellowship funded by the Ministry of Universities of Spain and the European Union (NextGenerationEU), and grant PID2019-104129GB-I00 funded by MICIU/AEI/10.13039/501100011033. E.R. partially supported by the VILLUM Foundation grant 37507 “Efficient Recomputations for Changeful Problems” and the Independent Research Fund Denmark grant 2020-2023 (9131-00044B) “Dynamic Network Analysis”.

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Author information

Authors and Affiliations

  1. Technical University of Denmark, Lyngby, Denmark

    Aleksander Bjørn Grodt Christiansen & Eva Rotenberg

  2. Universität Potsdam, Potsdam, Germany

    Linda Kleist

  3. Universitat Politècnica de Catalunya, Barcelona, Spain

    Irene Parada

Authors
  1. Aleksander Bjørn Grodt Christiansen
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  2. Linda Kleist
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  3. Irene Parada
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  4. Eva Rotenberg
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Corresponding author

Correspondence to Irene Parada .

Editor information

Editors and Affiliations

  1. Masaryk University, Brno, Czech Republic

    Daniel Kráľ

  2. University of Primorska, Koper, Slovenia

    Martin Milanič

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Christiansen, A.B.G., Kleist, L., Parada, I., Rotenberg, E. (2025). Augmenting Plane Straight-Line Graphs to Meet Parity Constraints. In: Kráľ, D., Milanič, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2024. Lecture Notes in Computer Science, vol 14760. Springer, Cham. https://doi.org/10.1007/978-3-031-75409-8_9

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  • DOI: https://doi.org/10.1007/978-3-031-75409-8_9

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