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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

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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.)
           Figure 1

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Thanks! This paper seems just what I looked for. –  David Herskovics Jul 7 '12 at 16:53

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