Geometric graph theory

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 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, Lรกszlรณ Lovรกsz, and Avi Wigderson that a graph is k-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 k-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. (Wikipedia).

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What is the difference between convex and concave polygons

๐Ÿ‘‰ Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1

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Related pages

Convex drawing | K-vertex-connected graph | System of linear equations | General position | Line segment | Convex hull | Planar graph | Bipolar orientation | Euclidean space | W. T. Tutte | Geometric graph theory