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Other graphs that are not polyhedral can also have convex drawings, or strictly convex drawings. Some graphs, such as the
In graph drawing
Graph drawing is an area of mathematics and computer science combining methods from geometric graph theory and information visualization to derive two-dimensional depictions of graphs arising from applications such as social network analysis, car ...
, a convex drawing of a planar graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cro ...
is a drawing that represents the vertices of the graph as points in the Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
and the edges as straight line segments, in such a way that all of the faces of the drawing (including the outer face) have a convex boundary. The boundary of a face may pass straight through one of the vertices of the graph without turning; a strictly convex drawing asks in addition that the face boundary turns at each vertex. That is, in a strictly convex drawing, each vertex of the graph is also a vertex of each convex polygon describing the shape of each incident face.
Every polyhedral graph
In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the 3-vertex-c ...
has a strictly convex drawing, for instance obtained as the Schlegel diagram
In geometry, a Schlegel diagram is a projection of a polytope from \mathbb^d into \mathbb^ through a point just outside one of its facets. The resulting entity is a polytopal subdivision of the facet in \mathbb^ that, together with the ori ...
of a convex polyhedron
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the w ...
representing the graph. For these graphs, a convex (but not necessarily strictly convex) drawing can be found within a grid whose length on each side is linear in the number of vertices of the graph, in linear time. However, strictly convex drawings may require larger grids; for instance, for any polyhedron such as a pyramid
A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrila ...
in which one face has a linear number of vertices, a strictly convex drawing of its graph requires a grid of cubic area. A linear-time algorithm can find strictly convex drawings of polyhedral graphs in a grid whose length on each side is quadratic.
Other graphs that are not polyhedral can also have convex drawings, or strictly convex drawings. Some graphs, such as the complete bipartite graph
In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set..Electronic edition page 17.
Graph theory ...
, have convex drawings but not strictly convex drawings. A combinatorial characterization for the graphs with convex drawings is known, and they can be recognized in linear time, but the grid dimensions needed for their drawings and an efficient algorithm for constructing small convex grid drawings of these graphs are not known in all cases.
Convex drawings should be distinguished from 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 th ...
s, in which each vertex is required to lie within the convex hull of its neighboring vertices. Convex embeddings can exist in dimensions other than two, do not require their graph to be planar, and even for planar embeddings of planar graphs do not necessarily force the outer face to be convex.
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