Maintainer	Ivan.Miljenovic@gmail.com

Data.Graph.Planar

Contents

Graph Information
- Information about the nodes
- Information about the edges
Graph Manipulation
Graph duals and faces
- Constructing the dual
Alternate representations
- Serialisation
- Pretty-Printing

Description

Planar graphs are graphs that can be embedded onto a surface (i.e. they can be drawn on that surface without any edges crossing). As such, it is preferable to use a dedicated data structure for them that has information about how to achieve this embedding rather than a standard graph data structure.

The implementation here is loosely based upon that found in plantri by Gunnar Brinkmann and Brendan McKay: http://cs.anu.edu.au/~bdm/plantri/. The main differences are (if my understanding of the C code is correct):

plantri uses arrays; planar-graph uses Maps (thus making it easier to grow/shrink graphs).
plantri doesn't store the list of edges around a node.
Each edge stores in plantri has the face it is on, but only after they are explicitly calculated. In planar-graph, getFaces instead returns a Map for the faces.
plantri doesn't allow labels.

In particular, all edges - even undirected ones - are stored as two opposing directed edges. As such, care should be taken when dealing with edges. Also, the Node, Edge and Face identifiers are all abstract, and as such cannot be constructed directly.

All returned CLists represent values in a clockwise fashion (relative to the Node or Face in question).

Synopsis

Documentation

data PlanarGraph n e

The overall planar graph data structure.

Instances

Functor (PlanarGraph n)
(Eq n, Eq e) => Eq (PlanarGraph n e)
(Read n, Read e) => Read (PlanarGraph n e)
(Show n, Show e) => Show (PlanarGraph n e)

Graph Information

Information about the nodes

data Node

An abstract representation of a node.

Instances

Bounded Node
Enum Node
Eq Node
Ord Node
Read Node	Note that this instance of `Read` only works when directly applied to a `String`; it is supplied solely to assist with debugging.
Show Node	This instance of `Show` does not produce valid Haskell code; however, the `Node` type is abstract and not designed to be directly accessed.

order :: PlanarGraph n e -> Int

The number of nodes in the graph.

nodes :: PlanarGraph n e -> [Node]

All the nodes in the graph (in some arbitrary order).

labNodes :: PlanarGraph n e -> [(Node, n)]

All the nodes and their labels in the graph (in some arbitrary order).

outgoingEdges :: PlanarGraph n e -> Node -> CList Edge

Returns all outgoing edges for the specified node, travelling clockwise around the node. It assumes the node is indeed in the graph.

incomingEdges :: PlanarGraph n e -> Node -> CList Edge

Returns all incoming edges for the specified node, travelling clockwise around the node. It assumes the node is indeed in the graph.

neighbours :: PlanarGraph n e -> Node -> CList Node

The Nodes that are connected to this Node with an edge (in clockwise order).

nodeLabel :: PlanarGraph n e -> Node -> n

Returns the label for the specified node.

Information about the edges

To be able to embed the required order of edges around a particular Node, we can't rely on just having each node specify which other nodes are adjacent to it as with non-planar graph types; instead, we need a unique identifier (to be able to distinguish between multiple edges between two nodes). Furthermore, each edge has an /inverse edge/ in the opposite direction.

Due to every edge having an inverse, a PlanarGraph implicitly undirected even though each edge is directed. As such, if you require a directed planar graph, use appropriate edge labels to denote whether an edge is the one you want or just its inverse.

data Edge

An abstract representation of an edge. Note that an explicit identifier is used for each edge rather than just using the two nodes that the edge connects. This is required in case more than one edge connects two nodes as we need to be able to distinguish them.

Instances

Bounded Edge
Enum Edge
Eq Edge
Ord Edge
Read Edge	Note that this instance of `Read` only works when directly applied to a `String`; it is supplied solely to assist with debugging.
Show Edge	This instance of `Show` does not produce valid Haskell code; however, the `Edge` type is abstract and not designed to be directly accessed.

size :: PlanarGraph n e -> Int

The number of edges in the graph.

edges :: PlanarGraph n e -> [Edge]

All the edges in the graph (in some arbitrary order). Note that inverses are also included.

labEdges :: PlanarGraph n e -> [(Edge, e)]

All the edges and their labels in the graph (in some arbitrary order).

edgesBetween :: PlanarGraph n e -> [(Node, Node)]

A variant of edges that returns the pair of nodes that form an edge rather than its unique identifier (again including inverse edges).

labEdgesBetween :: PlanarGraph n e -> [((Node, Node), e)]

As with edgesBetween, but including the labels.

fromNode :: PlanarGraph n e -> Edge -> Node

The Node which this Edge is coming from.

toNode :: PlanarGraph n e -> Edge -> Node

The Node which this Edge is going to.

prevEdge :: PlanarGraph n e -> Edge -> Edge

The previous Edge going clockwise around the fromNode.

nextEdge :: PlanarGraph n e -> Edge -> Edge

The next Edge going clockwise around the fromNode.

inverseEdge :: PlanarGraph n e -> Edge -> Edge

The Edge that is an inverse to this one; i.e.:

 fromNode pg e == toNode pg $ inverseEdge pg e
 toNode pg e == fromNode pg $ inverseEdge pg e

edgeLabel :: PlanarGraph n e -> Edge -> e

Return the label for the specified edge.

Graph Manipulation

mergeGraphs :: PlanarGraph n e -> PlanarGraph n e -> (PlanarGraph n e, Node -> Node, Edge -> Edge)

mergeGraphs pg1 pg2 creates a disjoint union between pg1 and pg2 (i.e. puts them into the same graph but disconnected). This is used when they were created independently and thus probably have clashing Node and Edge values. For best performance, pg1 should be larger than pg2.

Along with the merged graph, two functions are returned: they respectively convert Node and Edge values from pg2 to those found in the merged graph.

Graph Construction

empty :: PlanarGraph n e

Constructs an empty planar graph.

addNode :: n -> PlanarGraph n e -> (Node, PlanarGraph n e)

Add a node with the provided label to the graph, returning the updated graph and the node identifier.

addUNode :: Monoid n => PlanarGraph n e -> (Node, PlanarGraph n e)

As with addNode, but uses mempty as the label.

data EdgePos

Specification of where to place a new edge on a node in clockwise order.

Constructors

Anywhere	The new edge can be placed anywhere.
BeforeEdge Edge	The new edge should be placed before the specified edge.
AfterEdge Edge	The new edge should be placed after the specified edge.

Instances

Eq EdgePos
Ord EdgePos
Read EdgePos
Show EdgePos

addEdge

Arguments

:: Node	The node `f` at which the main edge starts.
-> EdgePos	Positioning information at `f`.
-> Node	The node `t` at which the main edge ends.
-> EdgePos	Positioning information at `t` for the inverse edge (i.e. refers to `outgoingEdges t`).
-> e	The label for the main edge `f -> t`.
-> e	The label for the inverse edge `t -> f`.
-> PlanarGraph n e	The graph at which to add the edge.
-> ((Edge, Edge), PlanarGraph n e)	The main and inverse edge identifiers, and the updated graph.

Add an edge between two nodes f and t. In reality, since all edges are duplicated (see inverseEdge), two edges are inserted.

If either node does not currently have any edges, then its corresponding EdgePos value is ignored.

For example, let g refer to the following graph (where n1, etc. are both the labels and the variable names):

     ====                    ====
    ( n1 )                  ( n2 )
     ====                    ====





                             ====
                            ( n3 )
                             ====

We can add an edge between n1 and n2 (using Anywhere as the EdgePos since there are currently no edges on either node):

 ((e1,e2),g') = addEdge n1 Anywhere n2 Anywhere "e1" "e2" g

This will result in the following graph:

                  e2
     ====  <---------------  ====
    ( n1 )                  ( n2 )
     ====  --------------->  ====
                  e1




                             ====
                            ( n3 )
                             ====

If we want to add edges between n2 and n3, we have three options for the location on n2:

Use Anywhere: since there is only one other edge, it makes no difference in terms of the embedding where the second edge goes.
Put the new edge BeforeEdge e2 (going clockwise around n2).
Put the new edge AfterEdge e2 (going clockwise around n2).

Since n2 currently only has one edge, all three EdgePos values will result in the same graph, so we can arbitrarily pick one:

 ((e3,e4),g'') = addEdge n2 (BeforeEdge e2) n3 Anywhere "e3" "e4" g'

However, with more edges care must be taken on which EdgePos value is used. The resulting graph is:

                  e2
     ====  <---------------  ====
    ( n1 )                  ( n2 )
     ====  --------------->  ====
                  e1         |  ^
                             |  |
                          e3 |  | e4
                             |  |
                             v  |
                             ====
                            ( n3 )
                             ====

The same graph (up to the actual Edge values; so it won't satisfy ==) would have been obtained with:

 ((e4,e3), g'') = addEdge n3 Anywhere n2 (BeforeEdge e2) "e4" "e3" g'

addEdgeUndirected :: Node -> EdgePos -> Node -> EdgePos -> e -> PlanarGraph n e -> (Edge, PlanarGraph n e)

As with addEdge, but the edges are meant to be undirected so use the same label for both.

addUEdge :: Monoid e => Node -> EdgePos -> Node -> EdgePos -> PlanarGraph n e -> ((Edge, Edge), PlanarGraph n e)

As with addEdge, but both labels are set to mempty.

Graph Deconstruction

isEmpty :: PlanarGraph n e -> Bool

Determines if the graph is empty.

deleteNode :: Node -> PlanarGraph n e -> PlanarGraph n e

Delete the node and all adjacent edges from the graph.

deleteEdge :: Edge -> PlanarGraph n e -> PlanarGraph n e

Delete the edge and its inverse from the graph.

contractEdge :: Edge -> (n -> n -> n) -> PlanarGraph n e -> PlanarGraph n e

Merges the two nodes adjoined by this edge, and delete all edges between them. The provided function is to decide what the label for the resulting node should be (if the edge goes from f to t, then the function is fLabel -> tLabel -> newLabel). The Node value for the merged node is fromEdge pg e.

Note that this may result in multiple edges between the new node and another node if it is adjacent to both nodes being merged.

Other

mapNodes :: (n -> n') -> PlanarGraph n e -> PlanarGraph n' e

Apply a mapping function over the node labels.

adjustNodeLabel :: (n -> n) -> Node -> PlanarGraph n e -> PlanarGraph n e

Apply a function to the label of the specified node.

setNodeLabel :: n -> Node -> PlanarGraph n e -> PlanarGraph n e

Set the label of the specified node.

mapEdges :: (e -> e') -> PlanarGraph n e -> PlanarGraph n e'

Apply a mapping function over the edge labels.

adjustEdgeLabel :: (e -> e) -> Edge -> PlanarGraph n e -> PlanarGraph n e

Apply a function to the label of the specified edge.

setEdgeLabel :: e -> Edge -> PlanarGraph n e -> PlanarGraph n e

Set the label of the specified edge.

Graph duals and faces

The dual of a planar graph G is another planar graph H such that H has an node for every face in G, and an edge between two nodes if the corresponding faces in G are adjacent. For example, the graph (drawn as an undirected graph for simplicity):

                o---------o---------o
                |         |         |
                |   f1    |   f2    |
                |         |         |
                o---------o---------o
                 \                 /
                  \               /
                   \     f3      /
                    \           /
        outer        \         /
         face         \       /
                       \     /
                        \   /
                         \ /
                          o

has a dual graph of:

                 ......
            .....      .....
         ...                ..
       ..      ......        ..
      .       .      .         .
     .       .     =====     ===== .....
     .      .   ..( f1  )...( f2  )    ....
     .     .   ..  =====     =====         ..
     .    .   .       .      .               .
     .   .   .          .   .                 .
     .  =====           =====                  .
     . /     \.........( f3  )...               .
      /       \         =====   ....             .
      | outer |                     .            .
      \  face /                      .           .
       \     / .                      .          .
        =====   .                     .          .
           .      .                  .           .
            .       .               .           .
              .       .............            .
                .                             .
                  ..                         .
                     .                      .
                       .               ....
                        ................

A dual graph is a planar multigraph: it will still be a planar graph, but may have loops and multiple edges. However, the dual of a dual graph will be the original graph (though no guarantees are made that g == makeDual (makeDual g) due to differing Node and Edge values).

Note that the functions here assume that the graph is connected; in effect multiple connected components will be treated individually with no notion of relative embeddings.

data Face

An abstract representation of a face.

Instances

Bounded Face
Enum Face
Eq Face
Ord Face
Read Face	Note that this instance of `Read` only works when directly applied to a `String`; it is supplied solely to assist with debugging.
Show Face	This instance of `Show` does not produce valid Haskell code; however, the `Face` type is abstract and not designed to be directly accessed.

type FaceMap = Map Face FaceInfo

Information about the faces in a planar graph.

data FaceInfo

Information about a particular Face.

Constructors

FInfo
Fields faceNodes :: CList Node The `Node`s that make up the face. edgeCrossings :: CList ((Edge, Edge), Face) The `Edge`s that make up the face, its inverse and the `Face` on the other side of that `Edge`.

Instances

Eq FaceInfo
Read FaceInfo
Show FaceInfo

faceEdges :: FaceInfo -> CList Edge

The Edges that make up the face.

adjoiningFaces :: FaceInfo -> CList Face

The adjoining Faces. Will have repeats if the Faces are adjacent over more than one Edge.

getFaces :: PlanarGraph n e -> FaceMap

Finds all faces in the planar graph. A face is defined by traversing along the right-hand-side of edges, e.g.:

           o----------------------------->o
           ^..............................|
           |..............................|
           |..............FACE............|
           |..............................|
           |..............................v
           o<-----------------------------o

(with the inverse edges all being on the outside of the edges shown).

getFace :: PlanarGraph n e -> Edge -> ([Node], [Edge])

Returns all nodes and edges in the same face as the provided edge (including that edge); assumes the edge is part of the graph.

Constructing the dual

makeDual :: PlanarGraph n e -> PlanarGraph () ()

Create the dual of a planar graph. If actual node and edge labels are required, use toDual.

toDual :: (Face -> n) -> (Face -> Edge -> Face -> e) -> FaceMap -> PlanarGraph n e

Create the planar graph corresponding to the dual of the face relationships. The usage of FaceMap rather than PlanarGraph is to allow you to use the FaceMap for constructing the label-creation functions if you so wish.

The function edgeLabel for edge labels takes the Face that the edge comes from, the Edge belonging to that Face that it is crossing and then the Face that it is going to. For example:

                  ....              ....>
                      ...> =====....
                          (#####)
                           =====
                            | ^  e2
                            | |
                            | |
              face1         | |      face2
                            | |
                            | |
                            | |
                        e1  v |
                           =====
                          (#####)
                        ...===== <..
                    <...            ....
                                        ...

Here, the edge in the dual graph going from face1 to face2 will have a label of "edgeLabel face1 e1 face2", and the edge going from face2 to face1 will have a label of "edgeLabel face2 e2 face1".

Alternate representations

Serialisation

Serialisation support can be found here to aid in converting a PlanarGraph to alternate formats. Care should be taken when constructing the SerialisedGraph, and these functions should not be abused just to edit an existing PlanarGraph.

type SerialisedGraph n e = [(Int, n, [(Int, Int, e, Int)])]

The definition of a more compact, serialised form of a planar graph. The various fields correspond to:

 [( node index
  , node label
  , [( edge index
     , node index that this edge points to
     , edge label
     , inverse edge index
    )]
 )]

The list of edges should be in clockwise order around the node.

serialise :: PlanarGraph n e -> SerialisedGraph n e

deserialise :: SerialisedGraph n e -> PlanarGraph n e

Pretty-Printing

prettify :: (Show n, Show e) => PlanarGraph n e -> String

Pretty-print the graph. Note that this loses a lot of information, such as edge inverses, etc.

prettyPrint :: (Show n, Show e) => PlanarGraph n e -> IO ()

Pretty-print the graph to stdout.