In mathematics, a flip graph is a graph whose vertices are Combinatorics or Geometry objects, and whose edges link two of these objects when they can be obtained from one another by an elementary operation called a flip. Flip graphs are special cases of geometric graphs.
Among notable flip graphs, one finds the N-skeleton of polytopes such as Associahedron
Examples
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
N-skeleton of the
-dimensional
associahedron.
This basic construction can be generalized in a number of ways.
Finite sets of points in Euclidean space
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
Matroid (a minimally
Affine space subset of
). More precisely, if some triangulation
of a circuit
is a subset of
, and if all the cells (faces of maximal dimension) of
have the same link
in
, then one can perform a flip within
by replacing
by
, where
X\mathord{\star}{Y}=\{x\cup{y}:(x,y)\in{X\mathord{\times}{Y}}\},
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 .
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 edges (distinct arcs with the same pair of vertices), 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
N-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 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
Associahedron and
Cyclohedron, a number of
Convex polytope have the property that their
N-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
Convex polytope on
. The subgraph induced by these triangulations in the flip graph of
is the
N-skeleton of a
Convex 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 tesseract 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.
Diameter
The maximum number of flips required to transform a triangulation into another is the diameter of the flip graph. The diameter of the flip graph of a convex
-gon has been obtained by Daniel Sleator,
Robert Tarjan, and
William Thurston[
] when
is sufficiently large and by Lionel Pournin for all
. This diameter is equal to
when
.
The diameter of other flip graphs has been studied. For instance Klaus Wagner provided a quadratic upper bound on the diameter of the flip graph of a set of unmarked points on the sphere.[
] while the best-known lower bound is .[
][
][
]
See also
