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The flip graph for a set $P$ of points in the plane has a vertex for everytriangulation of $P$, and an edge when two triangulations differ by one flipthat replaces one triangulation edge by another. The flip graph is known tohave some connectivity properties: (1) the flip graph is connected; (2)connectivity still holds when restricted to triangulations containing someconstrained edges between the points; (3) for $P$ in general position of size$n$, the flip graph is $\lceil \frac{n}{2} -2 \rceil$-connected, a recentresult of Wagner and Welzl (SODA 2020).  We introduce the study of connectivity properties of the flip graph when someedges between points are forbidden. An edge $e$ between two points is a flipcut edge if eliminating triangulations containing $e$ results in a disconnectedflip graph. More generally, a set $X$ of edges between points of $P$ is a flipcut set if eliminating all triangulations that contain edges of $X$ results ina disconnected flip graph. The flip cut number of $P$ is the minimum size of aflip cut set.  We give a characterization of flip cut edges that leads to an $O(n \log n)$time algorithm to test if an edge is a flip cut edge and, with that aspreprocessing, an $O(n)$ time algorithm to test if two triangulations are inthe same connected component of the flip graph. For a set of $n$ points inconvex position (whose flip graph is the 1-skeleton of the associahedron) weprove that the flip cut number is $n-3$.

Forbidding Edges between Points in the Plane to Disconnect the Triangulation Flip Graph