Graph embedding

The Heawood graph and associated map embedded in the torus.

In topological graph theory, an embedding (also spelled imbedding) of a graph on a surface is a representation of on in which points of are associated with vertices and simple arcs (homeomorphic images of ) are associated with edges in such a way that:

  • the endpoints of the arc associated with an edge are the points associated with the end vertices of
  • no arcs include points associated with other vertices,
  • two arcs never intersect at a point which is interior to either of the arcs.

Here a surface is a connected -manifold.

Informally, an embedding of a graph into a surface is a drawing of the graph on the surface in such a way that its edges may intersect only at their endpoints. It is well known that any finite graph can be embedded in 3-dimensional Euclidean space .[1] A planar graph is one that can be embedded in 2-dimensional Euclidean space

Often, an embedding is regarded as an equivalence class (under homeomorphisms of ) of representations of the kind just described.

Some authors define a weaker version of the definition of "graph embedding" by omitting the non-intersection condition for edges. In such contexts the stricter definition is described as "non-crossing graph embedding".[2]

This article deals only with the strict definition of graph embedding. The weaker definition is discussed in the articles "graph drawing" and "crossing number".

  1. ^ Cohen, Robert F.; Eades, Peter; Lin, Tao; Ruskey, Frank (1995), "Three-dimensional graph drawing", in Tamassia, Roberto; Tollis, Ioannis G. (eds.), Graph Drawing: DIMACS International Workshop, GD '94 Princeton, New Jersey, USA, October 10–12, 1994, Proceedings, Lecture Notes in Computer Science, vol. 894, Springer, pp. 1–11, doi:10.1007/3-540-58950-3_351, ISBN 978-3-540-58950-1.
  2. ^ Katoh, Naoki; Tanigawa, Shin-ichi (2007), "Enumerating Constrained Non-crossing Geometric Spanning Trees", Computing and Combinatorics, 13th Annual International Conference, COCOON 2007, Banff, Canada, July 16-19, 2007, Proceedings, Lecture Notes in Computer Science, vol. 4598, Springer-Verlag, pp. 243–253, CiteSeerX 10.1.1.483.874, doi:10.1007/978-3-540-73545-8_25, ISBN 978-3-540-73544-1.