Abstract
Let P be a set of n points in the plane, no three in a line. The order type of P specifies, for every ordered triple, a positive or negative orientation; and the crossing type (for short, x-type) of P specifies, for every unordered pair of line segments spanned by P, whether they cross each other. Keller and Perles (2016) proved that the x-type of P can be reconstructed from the exchange graphG0(P) of noncrossing spanning trees. In this paper, we show that the x-type of P can already be reconstructed from the compatible exchange graphG1(P), which is a subgraph of G0(P). The proof crucially relies on the analysis of maximal sets of pairwise noncrossing edges (msnes) between two disjoint planar straight-line graphs. msnes are a bipartite analogue of triangulations of planar straight-line graphs; they correspond to maximal cliques in G1(P).
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