Disjoint Compatibility via Graph Classes

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.

Research on this work was initiated at the 6th Austrian-Japanese-Mexican-Spanish Workshop on Discrete Geometry and continued during the 16th European Geometric Graph-Week, both held near Strobl, Austria. We are grateful to the participants for the inspiring atmosphere. We especially thank Alexander Pilz for bringing this class of problems to our attention. D.P. is partially supported by the FWF grant I 3340-N35 (Collaborative DACH project Arrangements and Drawings). The research stay of P.P. at IST Austria is funded by the project CZ.02.2.69/0.0/0.0/17_050/0008466 Improvement of internationalization in the field of research and development at Charles University, through the support of quality projects MSCA-IF.

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 734922.