# Planar Canonical Ordering

    template <typename Graph, typename PlanarEmbedding, typename OutputIterator, typename VertexIndexMap>
    void planar_canonical_ordering(const Graph& g, PlanarEmbedding embedding, OutputIterator ordering, VertexIndexMap vm);

A planar canonical ordering is an ordering v1, v2, ..., vn of the vertices of a maximal planar graph having the property that, for each k, 3 <= k < n, the graph induced by v1, v2, ..., vk

- is biconnected and contains the edge {v1, v2} on its outer face.
- has any vertices in the range v1, v2, ..., vk that are adjacent to v(k+1) on its outer face, and these vertices form a path along the outer face.

Let Gk be the graph induced by the first k vertices in the canonical ordering, along with all edges between any of the first k vertices. After Gk has been drawn, the (k+1)st vertex can be drawn easily without edge crossings, since it's adjacent only to a consecutive sequence of vertices on the outer face of Gk.

A planar canonical ordering exists for every maximal planar graph with at least 2 vertices. `planar_canonical_ordering` expects the input graph to have at least 2 vertices.

The planar canonical ordering is used as an input in some planar graph drawing algorithms, particularly those that create a straight line embedding. de Fraysseix, Pach, and Pollack [72] first proved the existence of such an ordering and showed how to compute one in time O(n) on a maximal planar graph with n vertices.

### Complexity

If the vertex index map provides constant-time access to indices, this function takes time O(n + m) for a planar graph with n vertices and m edges. Note that in a simple planar graph with f faces, m edges, and n vertices, both f and m are O(n).

### Where Defined

`boost/graph/planar_canonical_ordering.hpp`

### Parameters

IN: `Graph& g`

> An undirected graph. The graph type must be a model of VertexAndEdgeListGraph. The graph must:
> - Be maximal planar.
> - Have at least two vertices.
> - Have the edge {v0, v1} on its outer face, where v0 is `*vertices(g).first` and v1 is the first element of `adjacent_vertices(`v0`, g)` distinct from v0.

IN: `PlanarEmbedding`

> A model of PlanarEmbedding.

IN: `OutputIterator`

> An OutputIterator with `value_type` equal to `graph_traits<Graph>::vertex_descriptor`. The canonical ordering will be written to this iterator.

IN: `VertexIndexMap vm`

> A Readable Property Map that maps vertices from `g` to distinct integers in the range `[0, num_vertices(g) )`
> Default: `get(vertex_index,g)`

### Example

`examples/canonical_ordering.cpp`

### See Also

Planar Graphs in the Boost Graph Library

Copyright © 2007 Aaron Windsor ( aaron.windsor@gmail.com)