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 formula_1 is a subset of the vertices of the graph, then a convex formula_1-embedding embeds the graph in such a way that every vertex either belongs to formula_1 or is placed within the convex hull of its neighbors. A convex embedding into formula_4-dimensional Euclidean space is said to be in general position if every subset formula_5 of its vertices spans a subspace of dimension formula_6.
Convex embeddings were introduced by W. T. Tutte in 1963. Tutte showed that if the outer face formula_7 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 formula_7-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 formula_9-dimensional convex formula_5-embedding in general position, for some formula_5 of formula_12 of its vertices, and that if it is k-vertex-connected then such an embedding can be constructed in polynomial time by choosing formula_5 to be any subset of formula_12 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.
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