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If we are given a set of disjoint line segments in the plane, can we prove (or disprove) that we can always join the line segments to make a simple polygon where the vertices of the polygon are the endpoints of the segments and every segment is an edge on the boundary of the polygon.

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Here is a simple counter example :

You most join the two leftmost vertices to the two middle vertices, and both ways to do that don't work.

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