# Disjoint Compatibility via Graph Classes

Oswin Aichholzer, Julia Obmann, Pavel Paták, Daniel Perz*, Josef Tkadlec, Birgit Vogtenhuber

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference paperpeer-review

## Abstract

Two plane drawings of graphs on the same set of points are called disjoint compatible if their union is plane and they do not have an edge in common. Let S be a convex point set of $$2n \ge 10$$ points and let $$\mathcal {H}$$ be a family of plane drawings on S. Two plane perfect matchings $$M:1$$ and $$M:2$$ on S (which do not need to be disjoint nor compatible) are disjoint $$\mathcal {H}$$ -compatible if there exists a drawing in $$\mathcal {H}$$ which is disjoint compatible to both $$M:1$$ and $$M:2$$. In this work, we consider the graph which has all plane perfect matchings as vertices and where two vertices are connected by an edge if the matchings are disjoint $$\mathcal {H}$$ -compatible. We study the diameter of this graph when $$\mathcal {H}$$ is the family of all plane spanning trees, caterpillars or paths. We show that in the first two cases the graph is connected with constant and linear diameter, respectively, while in the third case it is disconnected.

Original language English Graph-Theoretic Concepts in Computer Science - 48th International Workshop, WG 2022, Revised Selected Papers Michael A. Bekos, Michael Kaufmann Springer Science and Business Media Deutschland GmbH 16-28 13 9783031159138 https://doi.org/10.1007/978-3-031-15914-5_2 Published - 2022 48th International Workshop on Graph-Theoretic Concepts in Computer Science: WG 2022 - Tübingen, GermanyDuration: 22 Jun 2022 → 24 Jun 2022

### Publication series

Name Lecture Notes in Computer Science 13453 0302-9743 1611-3349

### Conference

Conference 48th International Workshop on Graph-Theoretic Concepts in Computer Science WG 2022 Germany Tübingen 22/06/22 → 24/06/22

## Keywords

• Compatibility
• Convex Set
• Matchings

## ASJC Scopus subject areas

• Theoretical Computer Science
• Computer Science(all)

## Fields of Expertise

• Information, Communication & Computing

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