Convex embedding explained
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
is a subset of the vertices of the graph, then a convex
-embedding embeds the graph in such a way that every vertex either belongs to
or is placed within the convex hull of its neighbors. A convex embedding into
-dimensional Euclidean space is said to be in
general position if every subset
of its vertices spans a subspace of dimension
.
Convex embeddings were introduced by W. T. Tutte in 1963. Tutte showed that if the outer face
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
-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 -vertex-connected if and only if it has a
-dimensional convex
-embedding in general position, for some
of
of its vertices, and that if it is -vertex-connected then such an embedding can be constructed in polynomial time by choosing
to be any subset of
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.
