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.
This work was initiated at the 6th DACH Workshop on Arrangements and Drawings in Stels, August 2021. We thank all participants, especially Nicolas Grelier and Daniel Perz, for fruitful discussions. O.A., R.P. and A.W. are supported by FWF grant W1230. K.K. is supported by the German Science Foundation (DFG) within the research training group ‘Facets of Complexity’ (GRK 2434). W.M. is partially supported by the German Research Foundation within the collaborative DACH project Arrangements and Drawings as DFG Project MU 3501/3–1, and by ERC StG 757609. J.O. is supported by ERC StG 757609. M.M.R. is supported by the Swiss National Science Foundation within the collaborative DACH project Arrangements and Drawings as SNSF Project 200021E-171681. (Also note that this author’s full last name consists of two words and is Mallik Reddy. However, she consistently refers to herself with the first word of her last name being abbreviated.) B.V. was partially supported by the Austrian Science Fund (FWF) within the collaborative DACH project Arrangements and Drawings as FWF project I 3340-N35.
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Aichholzer, O. et al. (2023). Compatible Spanning Trees in Simple Drawings of \(K_n\). In: Angelini, P., von Hanxleden, R. (eds) Graph Drawing and Network Visualization. GD 2022. Lecture Notes in Computer Science, vol 13764. Springer, Cham. https://doi.org/10.1007/978-3-031-22203-0_2
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