Compatible Spanning Trees in Simple Drawings of $K_n$

Oswin Aichholzer; Kristin Knorr; Wolfgang Mulzer; Nicolas El Maalouly; Johannes Obenaus; Rosna Paul; Meghana M. Reddy; Birgit Vogtenhuber; Alexandra Weinberger

arXiv · 2208.11875

Compatible Spanning Trees in Simple Drawings of $K_n$

Oswin Aichholzer·Kristin Knorr·Wolfgang Mulzer·Nicolas El Maalouly·Johannes Obenaus·Rosna Paul·Meghana M. Reddy·Birgit Vogtenhuber·Alexandra Weinberger

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Abstract

For a simple drawing $D$ of the complete graph $K_n$, two (plane) subdrawings are compatible if their union is plane. Let $\mathcal{T}_D$ be the set of all plane spanning trees on $D$ and $\mathcal{F}(\mathcal{T}_D)$ be the compatibility graph that has a vertex for each element in $\mathcal{T}_D$ and two vertices are adjacent if and only if the corresponding trees are compatible. We show, on the one hand, that $\mathcal{F}(\mathcal{T}_D)$ is connected if $D$ is a cylindrical, monotone, or strongly c-monotone drawing. On the other hand, we show that the subgraph of $\mathcal{F}(\mathcal{T}_D)$ induced by stars, double stars, and twin stars is also connected. In all cases the diameter of the corresponding compatibility graph is at most linear in $n$.

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BibTeX RIS

Oswin Aichholzer, Kristin Knorr, Wolfgang Mulzer, Nicolas El Maalouly, Johannes Obenaus, Rosna Paul, Meghana M. Reddy, Birgit Vogtenhuber, Alexandra Weinberger. 2022-08-25. Compatible Spanning Trees in Simple Drawings of $K_n$. https://arxiv.org/abs/2208.11875

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