Shooting Stars in Simple Drawings of $$K_{m,n}$$

Abstract

AbstractSimple 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.KeywordsSimple drawingSimple topological graphComplete bipartite graphPlane spanning treeShooting star