Shooting Stars in Simple Drawings of K_m,n

09/02/2022

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by   Oswin Aichholzer, et al.

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Simple drawings are drawings of graphs in which two edges have at most one common point (either a common endpoint, or a proper crossing). It has been an open question whether every simple drawing of a complete bipartite graph K_m,n contains a plane spanning tree as a subdrawing. We answer this question to the positive by showing that for every simple drawing of K_m,n and for every vertex v in that drawing, the drawing contains a shooting star rooted at v, that is, a plane spanning tree containing all edges incident to v.