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 $$X$$ is a subset of the vertices of the graph, then a convex $$X$$-embedding embeds the graph in such a way that every vertex either belongs to $$X$$ or is placed within the convex hull of its neighbors. A convex embedding into $$d$$-dimensional Euclidean space is said to be in general position if every subset $$S$$ of its vertices spans a subspace of dimension $$\min(d,|S|-1)$$.

Convex embeddings were introduced by W. T. Tutte in 1963. Tutte showed that if the outer face $$F$$ 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 $$F$$-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.

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 $$(k-1)$$-dimensional convex $$S$$-embedding in general position, for some $$S$$ of $$k$$ of its vertices, and that if it is $k$-vertex-connected then such an embedding can be constructed in polynomial time by choosing $$S$$ to be any subset of $$k$$ vertices, and solving Tutte's system of linear equations.

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