Graph immersed into the plane with segments as edges and we search for matching with no edges intersecting

Question

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 immersed 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.

Are there any results about this question — NP-hardness or algorithm 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)?

Answer 1

Perhaps the in-press 2012 paper, "Non-crossing matchings of points with geometric objects," by Aloupis et al. (Elsevier link; Prelim. PDF link) will help, either directly or through its references. (I have only passing familiarity with the contents of this paper myself.)
          
_{(source: els-cdn.com)}