In 1963, William Tutte published a paper ambitiously entitled “How to Draw a Graph”. Let \(\Sigma\) be any planar embedding of any simple planar graph \(G\).
Tutte proved that if the input graph \(G\) is sufficiently well-connected, then this physical system converges to a strictly convex planar embedding of \(G\)!
Let me state the parameters of the theorem more precisely, in slightly more generality than Tutte did.^{1} A Tutte drawing of a planar graph \(G\) is described by a position function \(p\colon V\to \mathbb{R}^2\) mapping the vertices to points in the plane, subject to the two conditions:
(I will use subscript notation \(p_v\) instead of function notation \(p(v)\) throughout this chapter.) The edges of a Tutte drawing are line segments connecting their endpoints. Let me emphasize that that the definition of a Tutte drawing does not require mapping edges to disjoint segments, or even mapping vertices to distinct points. Moreover, the dart coefficients are not required to be symmetric; it is possible that \(\lambda_{u\mathord\to v} \ne \lambda_{v\mathord\to u}\).
A graph \(G\) is 3-connected if we can delete any two vertices without disconnecting the graph, or equivalently (by Menger’s theorem) if every pair of vertices is connected by at least three vertex-disjoint paths.
Finally, a planar embedding is strictly convex if the boundary of every face of the embedding is a convex polygon, and no two edges on any face boundary are collinear.
It is not hard to see that 3-connectivity is required. If \(G\) has an articulation vertex \(v\), that is, a vertex whose deleting leaves a disconnected subgraph, then a Tutte drawing of \(G\) can map an entire component of \(G\setminus v\) to the point \(p_v\). Similarly, if \(G\) has two vertices \(v\) and \(v\) such that \(G\setminus \{u,v\}\) is disconnected, a Tutte drawing of \(G\) can map an entire component of \(G\setminus \{u,v\}\) to the line segment \(p_up_v\). In both cases, the Tutte drawing is not even an embedding, much less a strictly convex embedding.
Whitney (1932) proved that every simple 3-connected graph \(G\) has a unique embedding on the sphere (up to homeomorphism), or equivalently, a unique planar rotation system. I will describe Whitney’s proof later in this note. Thus, in every planar embedding of \(G\), the faces are bounded by the same set of cycles; we can reasonably call these cycles the faces of \(G\).
The definition of a Tutte drawing requires choosing one of the faces of \(G\) to be the outer face \(f_\infty\). We call the vertices of \(f_\infty\) boundary vertices, and the remaining vertices of \(G\) interior vertices. Similarly, we call the edges of \(f_\infty\) boundary edges, and the remaining edges of \(G\) interior edges. This terminology is justified by the following observation:
We prove the lemma by applying Gaussian elimination to the system of linear equations defined by condition (2). Linear system (2) expresses the position \(p_v\) of any vertex \(v\) as a strict convex combination of the positions of its neighbors in \(G\), that is, a weighted average where every neighbor of \(v\) has positive weight. By pivoting on that row, we can remove the variables \(p_v\) from the system.
Such a pivot is equivalent to deleting vertex \(v\) from the graph and adding new edges between the neighbors of \(v\), with appropriate positive coefficients on their darts.^{2} (Of course the resulting graph may not be planar.) Pivoting out one interior vertex does not change which boundary vertices are directly reachable from any other interior vertex. Thus, if we eliminate all but one interior vertex \(w\), the remaining constraint expresses \(w\) as a strict convex combination of at least three boundary vertices.
The same elimination argument implies that every assignment of positive dart coefficients \(\lambda_{u\mathord\to v} > 0\) defines a unique Tutte drawing; the linear system containing the equation \[\sum_{u\mathord\to v} \lambda_{u\mathord\to v} (p_v - p_u) = 0\] for every interior vertex \(v\) always has full rank.
Tutte’s original formulation required that every interior vertex lie at the center of mass of its neighbors; this is equivalent to requiring \(\lambda_{u\mathord\to v} = 1\) for every dart \(u\mathord\to v\).^{3} More generally, the physical interpretation in terms of springs corresponds to the special case where dart coefficients are symmetric.
Suppose each edge \(uv\) is to a (first-order linear) spring with spring constant \(\omega_{uv} = \lambda_{u\mathord\to v} = \lambda_{v\mathord\to u}\). For any vertex placement \(p \in (\mathbb{R}^2)^V\), the total potential energy in the network of springs is \[ \Phi(p) := \frac{1}{2} \sum_{u, v} \omega_{uv} \| p_u - p_v \|^2. \] If we fix the positions of the outer vertices, \(\Phi\) becomes a strictly convex^{4} function of the interior vertex coordinates. If we let the interior vertex positions vary, the network of springs will come to rest at a configuration with locally minimal potential energy. The unique minimum of \(\Phi\) can be computed by setting the gradient of \(\Phi\) to the zero vector and solving for the interior coordinates; thus we recover the original linear constraints \[ \sum_v \omega_{uv} (p_u - p_v) = 0 \] for every interior vertex \(u\). The underlying matrix of this linear system is called a weighted Laplacian matrix of \(G\). This matrix is positive definite^{5} and therefore non-singular, so a unique equilibrium configuration always exists.
When the dart coefficients are not symmetric, this physical intuition goes out the window; the linear system of balance equations is no longer the gradient of a convex function. Nevertheless, as we’ve already argued, any choice of positive coefficients \(\lambda_{u\mathord\to v}\) corresponds to a unique straight-line drawing of \(G\). None of the actual proof of Tutte’s theorem relies on any special properties of the coefficients \(\lambda_{u\mathord\to v}\) other than positivity.
Given the graph \(G\), the outer convex polygon, and the dart coefficients, we can compute the corresponding vertex positions in \(O(n^3)\) time via Gaussian elimination. (There are faster algorithms to solve this linear system. In particular, a numerically approximate solution can be computed in \(O(n\log n)\) time in theory, or in \(O(n\,\text{poly}\!\log n)\) time in practice.)
For the rest of this note, fix a simple 3-connected planar graph \(G\) and a Tutte drawing \(p\). At the risk of confusing the reader, I will generally not distinguish between features of the abstract graph \(G\) (vertices, edges, faces, cycles, paths, and so on) and their images under the Tutte drawing (points, line segments, polygons, polygonal chains, and so on). For example, an edge of the Tutte drawing \(p\) is the (possibly degenerate) line segment between the images of the endpoints of an edge of \(H\), and a face of the Tutte drawing \(p\) is the (not necessarily simple) polygon whose vertices are the images of the vertices of a face of \(G\) in cyclic order.
None of the previous lemmas actually require the planar graph \(G\) to be 3-connected. The main technical challenge in proving Tutte’s theorem is showing that if \(G\) is 3-connected, then every Tutte drawing of \(G\) is non-degenerate. The assumption of 3-connectivity is necessary^{6}—if \(G\) is 2-connected but not 3-connected, then some subgraphs of \(G\) can degenerate to line segments in the Tutte drawing, and if \(G\) is connected but not 2-connected, some subgraphs of \(G\) will degenerate to single points.
For the sake of argument, suppose some vertex \(u\) and all of its neighbors lies on a common line \(\ell\), which without loss of generality is horizontal. Let \(V^+\) and \(V^-\) be the subsets of vertices above and below \(\ell\), respectively. Let \(U\) be the set of all vertices that are reachable from \(u\) and whose neighbors all lie on \(\ell\). The halfplane lemma implies that the induced subgraphs \(G[V^+]\) and \(G[V^-]\) are connected, and the induced subgraph \(G[U]\) is connected by definition. Fix arbitrary vertices \(v^+\in V^+\) and \(v^-\in V^-\).
Finally, let \(W\) denote the set of all vertices that lie on line \(\ell\) and are adjacent to vertices in \(U\), but are not in \(U\) themselves. Every vertex in \(W\) has at least one neighbor not in \(\ell\), so by the both-sides lemma, every vertex in \(W\) has neighbors in both \(V^+\) and \(V^-\). Deleting the vertices in \(W\) disconnects \(U\) from the rest of the graph. Thus, because \(G\) is 3-connected, \(W\) contains at least three vertices \(w_1, w_2, w_3\).
Now suppose we contract the induced subgraphs \(G[V^+]\), \(G[V^-]\), and \(G[U]\) to the vertices \(v^+\), \(v^-\), and \(u\), respectively. The resulting minor of \(G\) contains the complete bipartite graph \(\{v^+, v^-, u\}\times \{w_1, w_2, w_3\} = K_{3,3}\). But this is impossible, because \(G\) is planar and therefore every minor of \(G\) is planar. \(\qquad\square\)
It remains to prove that the faces of the Tutte drawing are nondegenerate. First we need a combinatorial lemma, similar to Fáry’s lemma that any simple planar map can be refined into a simple triangulation.
The nondegeneracy lemma also implies that \(u\) and \(v\) have respective neighbors \(u’\) and \(v’\) strictly above \(\ell\), and the halfspace lemma implies that there is a path \(Q’\) from \(u’\) to \(v’\) that lies strictly above \(\ell\). Let \(Q\) be the path from \(u\) to \(v\) consisting of the edge \(uu’\), the path \(Q’\), and the edge \(v’v\).
The edge \(uv\) and the path \(P\) satisfy the conditions of Geelen’s lemma. The path \(Q\) clearly avoids the edge \(uv\), so \(Q\) must cross \(P\). But \(P\) and \(Q\) lie on opposite sides of \(\ell\). We have reached a contradiction, completing the proof.\(\qquad\square\)
Now we are finally ready to prove the main theorem.
For the sake of argument, suppose two edges \(uv\) and \(xy\) intersect in the Tutte drawing. Then any generic point near the intersection \(uv\cap xy\) must lie in two different faces, which we just showed is impossible. We conclude that the Tutte drawing is an embedding; in particular, every face is a simple polygon. We already proved that every face in this embedding is strictly convex. \(\qquad\square\)
Tutte’s theorem is as strong as possible in the following sense: Every planar graph with a strictly convex embedding is 3-connected. This observation follows from an earlier study of 3-connected planar graphs by Hassler Whitney.
A planar map is polyhedral if (1) the boundary of every face is a simple cycle, and (2) the intersection of any two facial cycles is either empty, a single vertex, or a single edge. Every strictly convex planar map is polyhedral.
Lemma: Every planar embedding of a 3-connected planar graph is polyhedral.
Suppose the boundary of some face \(f\) is not a simple cycle. The boundary of \(f\) must have a repeated vertex \(v\). So the radial map \(\Sigma^\diamond\) contains a cycle of length \(2\) through \(v\) and \(f\), which has at least one other vertex of \(\Sigma\) on either side. It follows that \(G\setminus v\) must be disconnected.
Suppose two faces \(f\) and \(g\) have two vertices \(u\) and \(v\) in common, but not the edge \(uv\). Then the radial map \(\Sigma^\diamond\) contains a simple cycle with vertices \(f, u, g, v\), which has at least one other vertex of \(G\) on either side. It follows that \(G\setminus \{u,v\}\) is disconnected.
We conclude that if \(\Sigma\) is not polyhedral, then \(G\) is not 3-connected. \(\qquad\square\)
Lemma: If a graph \(G\) has a polyhedral planar embedding, then \(G\) is 3-connected.
In both cases, we conclude that \(\Sigma\) is not polyhedral. \(\qquad\square\)
Lemma: The dual \(\Sigma^*\) of any polyhedral planar map \(\Sigma\) is polyhedral.
On the other hand, suppose \(\Sigma\) has two faces \(f\) and \(g\) that share two vertices \(v\) and \(w\), but there is no dart with endpoints \(v\) and \(w\) and shores \(f\) and \(g\). It follows that the dual faces \(v^*\) and \(w^*\) in \(\Sigma^*\) share the dual vertices \(f^*\) and \(g^*\), but there is no dart with endpoints \(f^*\) and \(g^*\) and shores \(v^*\) and \(w^*\).
We conclude that if \(\Sigma\) is not polyhedral, then neither is \(\Sigma^*\). \(\qquad\square\)
Lemma (Whitney): Every planar graph has at most one polyhedral embedding.
First, suppose there is a dart \(d\) such that \(\textsf{succ}'(d)\) is not equal to either \(\textsf{succ}(d)\) or \(\textsf{succ}^{-1}(d)\). In other words, suppose there is a vertex \(v = \textsf{head}(d)\) where the cyclic orders of darts into \(v\) in the two embeddings are different. The darts \(d\) and \(\textsf{succ}'(d)\) split the cycle of darts around \(v\) into two non-empty intervals; color the darts in one interval red and the other interval blue. In particular, color \(\textsf{succ}(d)\) red and color \(\textsf{succ}^{-1}(d)\) blue. There must be another dart \(d'\) that is red or blue, whose successor \(\textsf{succ}'(d)\) in \(\Sigma'\) is either blue or red, respectively.
Let \(C\) be the simple cycle in \(G\) that bounds face \(f = \textsf{left}'(d) = \textsf{right}'(\textsf{succ}'(d))\) in \(\Sigma'\). (If the boundary of \(f\) is not a simple cycle, then \(\Sigma'\) is not polyhedral and we are done.) Similarly, let Let \(C'\) be the cycle in \(G\) that bounds \(f' = \textsf{left}'(d') = \textsf{right}'(\textsf{succ}'(d'))\) in \(\Sigma'\). The images of \(C\) and \(C'\) in the polyhedral embedding \(\Sigma\) cross each other at \(v\), and therefore (by the Jordan curve theorem) share at least one other vertex \(w\). It follows that faces \(f\) and \(f'\) in \(\Sigma'\) share vertices \(v\) and \(w\), but do not share the edge \(vw\) (if that edge exists). We conclude that \(\Sigma'\) is not polyhedral.
On the other hand, suppose are two darts \(d\) and \(d'\) such that \(\textsf{succ}'(d) = \textsc{succ}(d)\) and \(\textsf{succ}'(d')\) or \(\textsf{succ}^{-1}(d')\). In other words, suppose the dart order around \(v = \textsf{head}(d)\) is the same in both embeddings, but the dart order around \(w = \textsf{head}(d)\) is reversed from one embedding to the other. Without loss of generality, \(v\) and \(w\) are adjacent, and we can assume \(d = v\mathord\to w\) and \(d' = \textsf{rev}(d) = w\mathord\to v\).
Let \(C\) and \(C'\) be the cycles in \(G\) that bound faces \(f = \textsf{left}'(d) = \textsf{right}'(d')\) and \(f' = \textsf{left}'(d') = \textsf{right}'(d)\) in \(\Sigma'\), respectively. After an arbitrarily small perturbation, the images of \(C\) and \(C'\) in the polyhedral embedding \(\Sigma\) cross each other at the midpoint \(vw\), and therefore share at least one other vertex \(x\). It follows that the faces \(f\) and \(f'\) in \(\Sigma'\) have disconnected intersection, and therefore \(\Sigma'\) is not polyhedral. \(\qquad\square\)
Together, the previous lemmas now imply Whitney’s unique-embedding theorem.
Theorem (Whitney): Every 3-connected planar graph has a unique planar embedding (up to homeomorphism), which is polyhedral.
In light of Whitney’s observation, Tutte’s spring-embedding theorem immediately implies the following corollary:
Convex Embedding Theorem: For every polyhedral planar embedding, there is an equivalent strictly convex embedding.
The formulation and proof of Tutte’s theorem that I’m presenting here follows a lecture note by Dan Spielman (2018), which is based on papers by Michael Floater (1997); László Lovász (1999); Steven Gortler, Craig Gotsman, and Dylan Thurston (2006); and Jim Geelen (2012).↩︎
This modification is called a star-mesh transformation; the special case of removing a vertex of degree \(3\) is called a Y-\(\Delta\) transformation.↩︎
It is sometimes more convenient to formalize Tutte’s description as \(\lambda_{u\mathord\to v} = 1/\deg(v)\), so that the weights of all darts into each vertex \(v\) sum to \(1\). This formalization is inconsistent with the physical spring analogy, but instead treats weights as transition probabilities of a (backward) random walk. Both formalizations lead to the same Tutte drawing.↩︎
The Hessian of \(\Phi\) is positive definite, meaning all of its eigenvalues are positive.↩︎
The Laplacian matrix is just the Hessian of \(\Phi\).↩︎
In fact, we only need the weaker assumption that \(G\) is internally 3-connected, meaning each interior vertex has three vertex-disjoint paths to the outer face.↩︎