Math.Combinatorics.Graph

A module defining a polymorphic data type for (simple, undirected) graphs, together with constructions of some common families of graphs, new from old constructions, and calculation of simple properties of graphs.

Documentation

combinationsOf :: Integral t => t -> [a] -> [[a]]

combinationsOf k xs returns the subsets of xs of size k. If xs is in ascending order, then the returned list is in ascending order

data Graph a

Datatype for graphs, represented as a list of vertices and a list of edges. For most purposes, graphs are required to be in normal form. A graph G vs es is in normal form if (i) vs is in ascending order without duplicates, (ii) es is in ascending order without duplicates, (iii) each e in es is a 2-element list [x,y], x<y

Constructors

G [a] [[a]]

Instances

Functor Graph

Eq a => Eq (Graph a)

Ord a => Ord (Graph a)

Show a => Show (Graph a)

nf :: Ord a => Graph a -> Graph a

Convert a graph to normal form. The input is assumed to be a valid graph apart from order

graph :: Ord t => ([t], [[t]]) -> Graph t

Safe constructor for graph from lists of vertices and edges. graph (vs,es) checks that vs and es are valid before returning the graph.

nullGraph :: Integral t => t -> Graph t

The null graph on n vertices is the graph with no edges

nullGraph' :: Graph Int

The null graph, with no vertices or edges

c :: Integral t => t -> Graph t

c n is the cyclic graph on n vertices

k :: Integral t => t -> Graph t

k n is the complete graph on n vertices

kb :: Integral t => t -> t -> Graph t

kb m n is the complete bipartite graph on m and n vertices

kb' :: Integral t => t -> t -> Graph (Either t t)

kb' m n is the complete bipartite graph on m left and n right vertices

q :: Integral t => Int -> Graph t

q k is the graph of the k-cube

q' :: Integral t => Int -> Graph [t]

fromDigits :: Integral a => Graph [a] -> Graph a

Given a graph with vertices which are lists of small integers, eg [1,2,3], return a graph with vertices which are the numbers obtained by interpreting these as digits, eg 123. The caller is responsible for ensuring that this makes sense (eg that the small integers are all < 10)

fromBinary :: Integral a => Graph [a] -> Graph a

Given a graph with vertices which are lists of 0s and 1s, return a graph with vertices which are the numbers obtained by interpreting these as binary digits. For example, [1,1,0] -> 6.

petersen :: Graph [Integer]

complement :: Ord t => Graph t -> Graph t

restriction :: Eq a => Graph a -> [a] -> Graph a

The restriction of a graph to a subset of the vertices

inducedSubgraph :: Eq a => Graph a -> [a] -> Graph a

isRegular :: Eq t => Graph t -> Bool

A graph is regular if all vertices have the same valency (degree)

isCubic :: Eq t => Graph t -> Bool

A 3-regular graph is called a cubic graph

distance :: Eq a => Graph a -> a -> a -> Int

Within a graph G, the distance d(u,v) between vertices u, v is length of the shortest path from u to v

diameter :: Ord t => Graph t -> Int

The diameter of a graph is maximum distance between two distinct vertices

girth :: Eq t => Graph t -> Int

The girth of a graph is the size of the smallest cycle that it contains. Note: If the graph contains no cycles, we return -1, representing infinity.

isConnected :: Ord t => Graph t -> Bool

Is the graph connected?

kneser :: Integral t => t -> t -> Graph [t]

kneser n k returns the kneser graph KG n,k - whose vertices are the k-element subsets of [1..n], with edges joining disjoint subsets