Convex embedding

In geometric graph theory, a convex embedding of a graph is an embedding of the graph into a Euclidean space, with its vertices represented as points and its edges as line segments, so that all of the vertices outside a specified subset belong to the convex hull of their neighbors. More precisely, if ${\displaystyle X}$ is a subset of the vertices of the graph, then a convex ${\displaystyle X}$-embedding embeds the graph in such a way that every vertex either belongs to ${\displaystyle X}$ or is placed within the convex hull of its neighbors. A convex embedding into ${\displaystyle d}$-dimensional Euclidean space is said to be in general position if every subset ${\displaystyle S}$ of its vertices spans a subspace of dimension ${\displaystyle \min(d,|S|-1)}$.^[1]

Convex embeddings were introduced by W. T. Tutte in 1963. Tutte showed that if the outer face ${\displaystyle F}$ of a planar graph is fixed to the shape of a given convex polygon in the plane, and the remaining vertices are placed by solving a system of linear equations describing the behavior of ideal springs on the edges of the graph, then the result will be a convex ${\displaystyle F}$-embedding. More strongly, every face of an embedding constructed in this way will be a convex polygon, resulting in a convex drawing of the graph.^[2]

Beyond planarity, convex embeddings gained interest from a 1988 result of Nati Linial, László Lovász, and Avi Wigderson that a graph is k-vertex-connected if and only if it has a ${\displaystyle (k-1)}$-dimensional convex ${\displaystyle S}$-embedding in general position, for some ${\displaystyle S}$ of ${\displaystyle k}$ of its vertices, and that if it is k-vertex-connected then such an embedding can be constructed in polynomial time by choosing ${\displaystyle S}$ to be any subset of ${\displaystyle k}$ vertices, and solving Tutte's system of linear equations.^[1]

One-dimensional convex embeddings (in general position), for a specified set of two vertices, are equivalent to bipolar orientations of the given graph.^[1]

References

1. Linial, N.; Lovász, L.; Wigderson, A. (1988), "Rubber bands, convex embeddings and graph connectivity", Combinatorica, 8 (1): 91–102, doi:10.1007/BF02122557, MR 0951998, S2CID 6164458
2. Tutte, W. T. (1963), "How to draw a graph", Proceedings of the London Mathematical Society, 13: 743–767, doi:10.1112/plms/s3-13.1.743, MR 0158387.