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, ..., v_n of the vertices of a maximal planar graph having the property that, for each k, 3 <= k < n, the graph induced by v₁, v₂, ..., v_k

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

Let G_k 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 G_k 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 G_k.

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.

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

Planar Canonical Ordering

Complexity

Where Defined

Parameters

Example

See Also