There are some points in the plane and some of them are connected with segments between them. We look at this structure as a graph implemented into the plane where the points are the vertices and the segments are the edges. We search for a maximum matching with no edges intersecting each other.

Is there any results about this question - NP-hardness or algortihm or characterisation even for some cases (the one case I see is when the graph is bipartite and the vertices of the two classes are on two parallel lines respectively)?

Thanks for the answers