A *connected* Planar graph with bidirected edges. I.e. the edges (darts) are directed, however, for every directed edge, the edge in the oposite direction is also in the graph.

The types v, e, and f are the are the types of the data associated with the vertices, edges, and faces, respectively.

The orbits in the embedding are assumed to be in counterclockwise order. Therefore, every dart directly bounds the face to its right.

Get the embedding, reprsented as a permutation of the darts, of this graph.

lens to access the Dart Data

edgeData is just an alias for dartData

The world in which the graph lives

We can take the dual of a world. For the Primal this gives us the Dual, for the Dual this gives us the Primal.

An Arc is a directed edge in a planar graph. The type s is used to tie this arc to a particular graph.

Darts have a direction which is either Positive or Negative (shown as +1 or -1, respectively).

Reverse the direcion

A dart represents a bi-directed edge. I.e. a dart has a direction, however the dart of the oposite direction is always present in the planar graph as well.

Get the twin of this dart (edge)

>>> twin (dart 0 "+1")
Dart (Arc 0) -1
>>> twin (dart 0 "-1")
Dart (Arc 0) +1

test if a dart is Positive

A vertex in a planar graph. A vertex is tied to a particular planar graph by the phantom type s, and to a particular world w.

Shorthand for vertices in the primal.

Construct a planar graph, given the darts in cyclic order around each vertex.

running time: \(O(n)\).

Construct a planar graph

running time: \(O(n)\).

Construct a planar graph from a adjacency matrix. For every vertex, all vertices should be given in counter-clockwise order.

pre: No self-loops, and no multi-edges

running time: \(O(n)\).

Produces the adjacencylists for all vertices in the graph. For every vertex, the adjacent vertices are given in counter clockwise order.

Note that in case a vertex u as a self loop, we have that this vertexId occurs twice in the list of neighbours, i.e.: u : [...,u,..,u,...]. Similarly, if there are multiple darts between a pair of edges they occur multiple times.

running time: \(O(n)\)

Read a planar graph, given in JSON format into a planar graph. The adjacencylists should be in counter clockwise order.

running time: \(O(n)\)

Transforms the planar graph into a format that can be easily converted into JSON format. For every vertex, the adjacent vertices are given in counter-clockwise order.

See toAdjacencyLists for notes on how we handle self-loops.

running time: \(O(n)\)

Get the number of vertices

>>> numVertices myGraph
4

Get the number of Darts

>>> numDarts myGraph
12

Get the number of Edges

>>> numEdges myGraph
6

Get the number of faces

>>> numFaces myGraph
4

Enumerate all darts

Get all darts together with their data

>>> mapM_ print $ darts myGraph
(Dart (Arc 0) -1,"a-")
(Dart (Arc 2) +1,"c+")
(Dart (Arc 1) +1,"b+")
(Dart (Arc 0) +1,"a+")
(Dart (Arc 4) -1,"e-")
(Dart (Arc 1) -1,"b-")
(Dart (Arc 3) -1,"d-")
(Dart (Arc 5) +1,"g+")
(Dart (Arc 4) +1,"e+")
(Dart (Arc 3) +1,"d+")
(Dart (Arc 2) -1,"c-")
(Dart (Arc 5) -1,"g-")

Enumerate all edges. We report only the Positive darts

Enumerate all edges with their edge data. We report only the Positive darts.

>>> mapM_ print $ edges myGraph
(Dart (Arc 2) +1,"c+")
(Dart (Arc 1) +1,"b+")
(Dart (Arc 0) +1,"a+")
(Dart (Arc 5) +1,"g+")
(Dart (Arc 4) +1,"e+")
(Dart (Arc 3) +1,"d+")

Enumerate all vertices

>>> vertices' myGraph
[VertexId 0,VertexId 1,VertexId 2,VertexId 3]

Enumerate all vertices, together with their vertex data

Enumerate all faces in the planar graph

All faces with their face data.

Traverse the vertices

(^.vertexData) $ traverseVertices (i x -> Just (i,x)) myGraph Just [(VertexId 0,()),(VertexId 1,()),(VertexId 2,()),(VertexId 3,())] >>> traverseVertices (i x -> print (i,x)) myGraph >> pure () (VertexId 0,()) (VertexId 1,()) (VertexId 2,()) (VertexId 3,())

Traverses the darts

>>> traverseDarts (\d x -> print (d,x)) myGraph >> pure ()
(Dart (Arc 0) +1,"a+")
(Dart (Arc 0) -1,"a-")
(Dart (Arc 1) +1,"b+")
(Dart (Arc 1) -1,"b-")
(Dart (Arc 2) +1,"c+")
(Dart (Arc 2) -1,"c-")
(Dart (Arc 3) +1,"d+")
(Dart (Arc 3) -1,"d-")
(Dart (Arc 4) +1,"e+")
(Dart (Arc 4) -1,"e-")
(Dart (Arc 5) +1,"g+")
(Dart (Arc 5) -1,"g-")

Traverses the faces

>>> traverseFaces (\i x -> print (i,x)) myGraph >> pure ()
(FaceId 0,())
(FaceId 1,())
(FaceId 2,())
(FaceId 3,())

The tail of a dart, i.e. the vertex this dart is leaving from

running time: \(O(1)\)

The vertex this dart is heading in to

running time: \(O(1)\)

endPoints d g = (tailOf d g, headOf d g)

running time: \(O(1)\)

All edges incident to vertex v, in counterclockwise order around v.

running time: \(O(k)\), where \(k\) is the output size

All incoming edges incident to vertex v, in counterclockwise order around v.

All outgoing edges incident to vertex v, in counterclockwise order around v.

Gets the neighbours of a particular vertex, in counterclockwise order around the vertex.

running time: \(O(k)\), where \(k\) is the output size

Given a dart d that points into some vertex v, report the next dart in the cyclic order around v.

running time: \(O(1)\)

Given a dart d that points into some vertex v, report the next dart in the cyclic order around v.

running time: \(O(1)\)

Data corresponding to the endpoints of the dart

Data corresponding to the endpoints of the dart

running time: \(O(1)\)

Get the dual graph of this graph.

The type to reprsent FaceId's

Shorthand for FaceId's in the primal.

The face to the left of the dart

>>> leftFace (dart 1 "+1") myGraph
FaceId 1
>>> leftFace (dart 1 "-1") myGraph
FaceId 2
>>> leftFace (dart 2 "+1") myGraph
FaceId 2
>>> leftFace (dart 0 "+1") myGraph
FaceId 0

running time: \(O(1)\).

The face to the right of the dart

>>> rightFace (dart 1 "+1") myGraph
FaceId 2
>>> rightFace (dart 1 "-1") myGraph
FaceId 1
>>> rightFace (dart 2 "+1") myGraph
FaceId 1
>>> rightFace (dart 0 "+1") myGraph
FaceId 1

running time: \(O(1)\).

Gets a dart bounding this face. I.e. a dart d such that the face lies to the right of the dart.

The darts bounding this face, for internal faces in clockwise order, for the outer face in counter clockwise order.

running time: \(O(k)\), where \(k\) is the output size.

Generates the darts incident to a face, starting with the given dart.

\(O(k)\), where \(k\) is the number of darts reported

The vertices bounding this face, for internal faces in clockwise order, for the outer face in counter clockwise order.

running time: \(O(k)\), where \(k\) is the output size.

Get the next edge along the face

running time: \(O(1)\).

Get the previous edge along the face

running time: \(O(1)\).