# Planar Graphs

A graph is planar if it can be drawn in two-dimensional space with no two of its edges crossing. Such a drawing of a planar graph is called a plane drawing. Every planar graph also admits a straight-line drawing, which is a plane drawing where each edge is represented by a line segment.

##### A planar graph (left), a plane drawing (center), and a straight line drawing (right), all of the same graph

Two examples of non-planar graphs are K5, the complete graph on five vertices, and K3,3, the complete bipartite graph on six vertices with three vertices in each bipartition. No matter how the vertices of either graph are arranged in the plane, at least two edges are forced to cross.

##### K5 (left) and K3,3 (right) - the two Kuratowski subgraphs

The above graphs are both minimal examples of non-planarity within their class of graphs; delete any edge or vertex from either one and the resulting graph is planar. A theorem of Kuratowski singles these two graphs out as fundamental obstructions to planarity within any graph:

> A graph is planar if and only if it does not contain a subgraph that is an expansion[1] of either K5 or K3,3

A subgraph that is an expansion of K5 or K3,3 is called a Kuratowski subgraph. Because of the above theorem, given any graph, one can produce either a plane drawing of a graph, which will certify that the graph is planar, or a minimal set of edges that forms a Kuratowski subgraph, which will certify that the graph is non-planar - in both cases, the certificate of planarity or non-planarity is easy to check.

Any plane drawing separates the plane into distinct regions bordered by graph edges called faces. As a simple example, any embedding of a triangle into the plane separates it into two faces: the region inside the triangle and the (unbounded) region outside the triangle. The unbounded region outside the graph's embedding is called the outer face. Every embedding yields one outer face and zero or more inner faces. A famous result called Euler's formula states that for any planar graph with n vertices, e edges, f faces, and c connected components,

> n + f = e + c + 1

This formula implies that any planar graph with no self-loops or parallel edges has at most 3n - 6 edges and 2n- 4 faces. Because of these bounds, algorithms on planar graphs can run in time O(n) or space O(n) on an n vertex graph even if they have to traverse all edges or faces of the graph.

A convenient way to separate the actual planarity test from algorithms that accept a planar graph as input is through an intermediate structure called a planar embedding. Instead of specifying the absolute positions of the vertices and edges in the plane as a plane drawing would, a planar embedding specifies their positions relative to one another. A planar embedding consists of a sequence, for each vertex in the graph, of all of the edges incident on that vertex in the order in which they are to be drawn around that vertex. The orderings defined by this sequence can either represent a clockwise or counter-clockwise iteration through the neighbors of each vertex, but the orientation must be consistent across the entire embedding.

In the Boost Graph Library, a planar embedding is a model of the PlanarEmbedding concept. A type that models PlanarEmbedding can be passed into the planarity test and populated if the input graph is planar. All other "back end" planar graph algorithms accept this populated PlanarEmbedding as an input. Conceptually, a type that models PlanarEmbedding is a property map that maps each vertex to a sequence of edges, where the sequence of edges has a similar interface to a standard C++ container. The sequence of edges each vertex maps to represents the ordering of edges adjacent to that vertex. This interface is flexible enough to allow storage of the planar embedding independent from the graph in, say, a `std::vector` of `std::vector`s, or to allow for graph implementations that actually store lists of adjacent edges/vertices to internally re-arrange their storage to represent the planar embedding. Currently, only the former approach is supported when using the native graph types (`adjacency_list`, `adjacency_matrix`, etc.) of the Boost Graph Library.

### Tools for working with planar graphs in the Boost Graph Library

The Boost Graph Library planar graph algorithms all work on undirected graphs. Some algorithms require certain degrees of connectivity of the input graph, but all algorithms work on graphs with self-loops and parallel edges.

The function `boyer_myrvold_planarity_test` can be used to test whether or not a graph is planar, but it can also produce two important side-effects: in the case the graph is not planar, it can isolate a Kuratowski subgraph, and in the case the graph is planar, it can compute a planar embedding. The Boyer-Myrvold algorithm works on any undirected graph.

An undirected graph is connected if, for any two vertices u and v, there's a path from u to v. An undirected graph is biconnected if it is connected and it remains connected even if any single vertex is removed. Finally, a planar graph is maximal planar (also called triangulated) if no additional edge (with the exception of self-loops and parallel edges) can be added to it without creating a non-planar graph. Any maximal planar simple graph on n > 2 vertices has exactly 3n - 6 edges and 2n - 4 faces, a consequence of Euler's formula. If a planar graph isn't connected, isn't biconnected, or isn't maximal planar, there is some set of edges that can be added to the graph to make it satisfy any of those three properties while preserving planarity. Many planar graph drawing algorithms make at least one of these three assumptions about the input graph, so there are functions in the Boost Graph Library that can help:

- `make_connected` adds a minimal set of edges to an undirected graph to make it connected.
- `make_biconnected_planar` adds a set of edges to a connected, undirected planar graph to make it biconnected while preserving planarity.
- `make_maximal_planar` adds a set of edges to a biconnected, undirected planar graph to make it maximal planar.

Some algorithms involve a traversal of the faces of the graph, and the Boost Graph Library has the generic traversal function `planar_face_traversal` for this purpose. This traversal, like other traversals in the Boost Graph Library, can be customized by overriding event points in an appropriately defined visitor class.

An intermediate step in some drawing algorithms for planar graphs is the creation of a canonical ordering of the vertices. A canonical ordering is a permutation of the vertices of a maximal planar graph. It orders the vertices in a way that makes it straightforward to draw the ith vertex once the first (i-1) vertices have been drawn - the only edges connecting the ith vertex to vertices already drawn will be adjacent to a consecutive sequence of vertices along the outer face of the partially embedded graph. The function `planar_canonical_ordering` will create such an ordering, given a maximal planar graph and a planar embedding of that graph.

A straight line drawing can be created using the function `chrobak_payne_straight_line_drawing,` which takes a maximal planar graph, a planar embedding of that graph, and a canonical ordering as input. The resulting drawing maps all of the vertices from a graph with n vertices to integer coordinates on a (2n-4) x (n-2) grid such that when the edges of the graph are drawn as line segments connecting vertices, no two edges cross. Self-loops and parallel edges are ignored by this algorithm.

Finally, there are two functions that can be used to verify the results of the `boyer_myrvold_planarity_test` and `chrobak_payne_straight_line_drawing` functions:

- `is_kuratowski_subgraph` takes the output of `boyer_myrvold_planarity_test` on a nonplanar graph and verifies that it can be contracted into a graph isomorphic to a Kuratowski subgraph.
- `is_straight_line_drawing` takes the output of `chrobak_payne_straight_line_drawing` and uses a planar sweep algorithm to verify that none of the embedded edges intersect.

### Complexity

Most of the algorithms in the Boost Graph Library that deal with planar graphs run in time O(n) on an input graph with n vertices. This achieves a theoretically optimal bound (you must at least iterate over all n vertices in order to embed a graph in the plane.) However, some of the work that goes into achieving these theoretically optimal time bounds may come at the expense of practical performance. For example, since any comparison-based sorting algorithm uses at least on the order of n log n comparisons in the worst case, any time an algorithm dealing with planar graphs needs to sort, a bucket sort is used to sort in O(n) time. Also, computing a planar embedding of a graph involves maintaining an ordered list of edges around a vertex, and this list of edges needs to support an arbitrary sequence of concatenations and reversals. A `std::list` can only guarantee O(n2) for a mixed sequence of n concatenations and reversals (since `reverse` is an O(n) operation.) However, our implementation achieves O(n) for these operations by using a list data structure that implements mixed sequences of concatenations and reversals lazily.

In both of the above cases, it may be preferable to sacrifice the nice theoretical upper bound for performance by using the C++ STL. The bucket sort allocates and populates a vector of vectors; because of the overhead in doing so, `std::stable_sort` may actually be faster in some cases. The custom list also uses more space than `std::list`, and it's not clear that anything other than carefully constructed pathological examples could force a `std::list` to use n2 operations within the planar embedding algorithm. For these reasons, the macro `BOOST_GRAPH_PREFER_STD_LIB` exists, which, when defined, will force the planar graph algorithms to use `std::stable_sort` and `std::list` in the examples above.

See the documentation on individual algorithms for more information about complexity guarantees.

### Examples

1. Testing whether or not a graph is planar.
2. Creating a straight line drawing of a graph in the plane.

### Notes

[1] A graph G' is an expansion of a graph G if G' can be created from G by a series of zero or more edge subdivisions: take any edge {x,y} in the graph, remove it, add a new vertex z, and add the two edges {x,z} and {z,y} to the graph. For example, a path of any length is an expansion of a single edge and a cycle of any length is an expansion of a triangle.

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