Tutte embedding

In graph drawing and geometric graph theory, a Tutte embedding or barycentric embedding of a 3-vertex-connected planar graph is a crossing-free straight-line embedding with the properties that the outer face is a convex polygon and that each interior vertex is at the average (or barycenter) of its neighbor's positions. If the outer polygon is fixed, this condition on the interior vertices determines their position uniquely as the solution to a system of linear equations. Solving the equations geometrically produces a planar embedding. Tutte's spring theorem, proven by W. T. Tutte (1963), states that this unique solution is always crossing-free, and more strongly that every face of the resulting planar embedding is convex.^[1] It is called the spring theorem because such an embedding can be found as the equilibrium position for a system of springs representing the edges of the graph.

Example

Tutte embedding of the graph of a cube. The outer four vertices are fixed at the corners of a unit square and each remaining vertex is at the average of its neighbors' positions.

Let G be the graph of a cube, and (selecting one of its quadrilateral faces as the outer face) fix the four vertices of the outer face at the four corners of a unit square, the points whose x and y coordinates are all four combinations of zero and one. Then, if the remaining four vertices are placed at the four points whose x and y coordinates are combinations of 1/3 and 2/3, as in the figure, the result will be a Tutte embedding. For, at each interior vertex v of the embedding, and for each of the two coordinates, the three neighbors of v have coordinate values that are equal to v, smaller by 1/3, and larger by 1/3; the average of these values is the same as the coordinate value of v itself.

System of linear equations

The condition that a vertex v be at the average of its neighbors' positions may be expressed as two linear equations, one for the x coordinate of v and another for the y coordinate of v. For a graph with n vertices, h of which are on the outer face, this gives a system of 2(n − h) equations in 2n unknowns; however, fixing the positions of the vertices on the outer face reduces the number of unknowns to 2(n − h). As Tutte (1963) showed, this system is non-degenerate, so it has a unique solution, which can be found in polynomial time by solving the system of equations.^[2]

Polyhedral representation

By Steinitz's theorem, the 3-connected planar graphs to which Tutte's spring theorem applies coincide with the polyhedral graphs, the graphs formed by the vertices and edges of a convex polyhedron. According to the Maxwell–Cremona correspondence, a two-dimensional embedding of a planar graph forms the vertical projection of a three-dimensional convex polyhedron if and only if the embedding has an equilibrium stress, an assignment of forces to each edge (affecting both endpoints in equal and opposite directions parallel to the edge) such that the forces cancel at every vertex. For a Tutte embedding, assigning to each edge an attractive force proportional to its length (like a spring) makes the forces cancel at all interior vertices, but this is not necessarily an equilibrium stress at the vertices of the outer polygon. However, when the outer polygon is a triangle, it is possible to assign repulsive forces to its three edges to make the forces cancel there, too. In this way, Tutte embeddings can be used to find Schlegel diagrams of every convex polyhedron. For every 3-connected planar graph G, either G itself or the dual graph of G has a triangle, so either this gives a polyhedral representation of G or of its dual; in the case that the dual graph is the one with the triangle, polarization gives a polyhedral representation of G itself.^[2]

Related results

A graph is k-vertex-connected, but not necessarily planar, if and only if it has an embedding into (k −1)-dimensional space in which an arbitrary k-tuple of vertices are placed at the vertices of a simplex and, for each remaining vertex v, the convex hull of the neighbors of v is full-dimensional with v in its interior. If such an embedding exists, one can be found by fixing the locations of the chosen k vertices and solving a system of equations that places each vertex at the average of its neighbors, just as in Tutte's planar embedding.^[3]

In finite element mesh generation, Laplacian smoothing is a common method for postprocessing a generated mesh to improve the quality of its elements;^[4] it is particularly popular for quadrilateral meshes, for which other methods such as Lloyd's algorithm for triangular mesh smoothing are less applicable. In this method, each vertex is moved to or towards the average of its neighbors' positions, but this motion is only performed for a small number of iterations, to avoid large distortions of element sizes or (in the case of non-convex mesh domains) tangled non-planar meshes.

Force-directed graph drawing systems continue to be a popular method for visualizing graphs, but these systems typically use more complicated systems of forces that combine attractive forces on graph edges (as in Tutte's embedding) with repulsive forces between arbitrary pairs of vertices. These additional forces may cause the system to have many locally stable configurations rather than, as in Tutte's embedding, a single global solution.^[5]

Gortler, Gotsman & Thurston (2006) proved results analogous to Tutte's spring theorem for graphs embedded on a torus.^[6]

References