Suppose points are joined by non-overlapping arrows in such a way that each point is the source of exactly one arrow. We call such a figure a k-doodle. Examples include: (a) a single point with an arrow from itself to itself (a loop); (b) two points with arrows forming a 2-cycle; (c) three points with arrows in cyclic order (a 3-cycle); (d) a 13-doodle with two components that contains two loops.
Does every k-doodle contain a loop? By moving one arrow, we can redraw the 13-doodle in the plane. Can every k-doodle be redrawn in the plane (without crossings) by moving some of its arrows around?
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