A prototypical flip graph is that of a convex -gon. The vertices of this graph are the triangulations of, and two triangulations are adjacent in it whenever they differ by a single interior edge. In this case, the flip operation consists in exchanging the diagonals of a convex quadrilateral. These diagonals are the interior edges by which two triangulations adjacent in the flip graph differ. The resulting flip graph is both the Hasse diagram of the Tamari lattice and the 1-skeleton of the -dimensional associahedron. This basic construction can be generalized in a number of ways.
Let be a triangulation of a finite set of points. Under some conditions, one may transform into another triangulation of by a flip. This operation consists in modifying the way triangulates a circuit. More precisely, if some triangulation of a circuit is a subset of, and if all the cells of have the same link in, then one can perform a flip within by replacing by, where and is, by Radon's partition theorem, the unique other triangulation of. The conditions just stated, under which a flip is possible, make sure that this operation results in a triangulation of. The corresponding flip graph, whose vertices are the triangulations of and whose edges correspond to flips between them, is a natural generalization of the flip graph of a convex polygon, as the two flip graphs coincide when is the set of the vertices of a convex -gon.
Topological surfaces
Another kind of flip graphs is obtained by considering the triangulations of a topological surface: consider such a surface, place a finite number of points on it, and connect them by arcs in such a way that any two arcs never cross. When this set of arcs is maximal, it decomposes into triangles. If in addition there are no multiple arcs, nor loops, this set of arcs defines a triangulation of. In this setting, two triangulations of that can be obtained from one another by a continuous transformation are identical. Two triangulations are related by a flip when they differ by exactly one of the arcs they are composed of. Note that, these two triangulations necessarily have the same number of vertices. As in the Euclidean case, the flip graph of is the graph whose vertices are the triangulations of with vertices and whose edges correspond to flips between them. This definition can be straightforwardly extended to bordered topological surfaces. The flip graph of a surface generalises that of a -gon, as the two coincide when the surface is a topological disk with points placed on its boundary.
Other flip graphs
A number of other flip graphs can be defined using alternative definitions of a triangulation. For instance, the flip graph whose vertices are the centrally-symmetric triangulations of a -gon and whose edges correspond to the operation of doing two centrally-symmetric flips is the 1-skeleton of the -dimensional cyclohedron. One can also consider an alternative flip graph of a topological surface, defined by allowing multiple arcs and loops in the triangulations of this surface. Flip graphs may also be defined using combinatorial objects other than triangulations. An example of such combinatorial objects are the domino tilings of a given region in the plane. In this case, a flip can be performed when two adjacent dominos cover a square: it consists in rotating these dominos by 90 degrees around the center of the square, resulting in a different domino tiling of the same region.
Properties
Polytopality
Apart from associahedra and cyclohedra, a number of polytopes have the property that their 1-skeleton is a flip graph. For instance, if is a finite set of points in, the regular triangulations of are the ones that can be obtained by projecting some faces of a -dimensional polytope on. The subgraph induced by these triangulations in the flip graph of is the 1-skeleton of a polytope, the secondary polytope of.
Connectedness
Polytopal flip graphs are, by this property, connected. As shown by Klaus Wagner in the 1930s, the flip graph of the topological sphere is connected. Among the connected flip graphs, one also finds the flip graphs of any finite 2-dimensional set of points. In higher dimensional Euclidean spaces, the situation is much more complicated. Finite sets of points of with disconnected flip graphs have been found whenever is at least 5. The flip graph of the vertex set of the 4-dimensional hypercube is known to be connected. However, it is yet unknown whether the flip graphs of finite 3- and 4-dimensional sets of points are always connected or not.
