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 graph ${\mathcal{G}}_0(P)$ of noncrossing spanning trees. In this paper, we show that the x-type of P can already be reconstructed from the compatible exchange graph ${\mathcal{G}}_1(P)$, which is a subgraph of ${\mathcal{G}}_0(P)$. 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 ${\mathcal{G}}_1(P)$.

令P为平面中n个点的集合，一行中没有3个点。该订单类型的P指定，对于每个有序三元组，正或负的方向; P的交叉类型（简称为x-type）针对由P跨越的每对无序线段，指定它们是否彼此交叉。Keller和Perles（2016）证明可以从交换图中重建P的x型 ${\mathcal{G}}_0（P）$的跨越树木。在本文中，我们证明了已经可以从兼容交换图中重建P的x类型 ${\mathcal{G}}_{\mathrm{1个}}（P）$，这是的子图 ${\mathcal{G}}_0（P）$。该证明至关重要地依赖于分析两个不相交的平面直线图之间的成对非交叉边缘（m snes）的最大集合。m snes是平面直线图的三角剖分的二分式模拟。它们对应于${\mathcal{G}}_{\mathrm{1个}}（P）$。