Algebra.Graph.Undirected

algebraic-graphs-0.7: A library for algebraic graph construction and transformation

Copyright | (c) Andrey Mokhov 2016-2022
License | MIT (see the file LICENSE)
Maintainer | andrey.mokhov@gmail.com
Stability | experimental
Safe Haskell | None
Language | Haskell2010

Algebra.Graph.Undirected

Contents

- Algebraic data type for graphs
- Basic graph construction primitives
- Graph folding
- Relations on graphs
- Graph properties
- Standard families of graphs
- Graph transformation

Description

Alga is a library for algebraic construction and manipulation of graphs in Haskell. See this paper for the motivation behind the library, the underlying theory, and implementation details.

This module defines an undirected version of algebraic graphs. Undirected graphs satisfy all laws of the `Undirected` type class, including the commutativity of `connect`.

To avoid name clashes with Algebra.Graph, this module can be imported qualified:

    import qualified Algebra.Graph.Undirected as Undirected

Synopsis

- data Graph a
- fromUndirected :: Ord a => Graph a -> Graph a
- toUndirected :: Graph a -> Graph a
- empty :: Graph a
- vertex :: a -> Graph a
- edge :: a -> a -> Graph a
- overlay :: Graph a -> Graph a -> Graph a
- connect :: Graph a -> Graph a -> Graph a
- vertices :: [a] -> Graph a
- edges :: [(a, a)] -> Graph a
- overlays :: [Graph a] -> Graph a
- connects :: [Graph a] -> Graph a
- foldg :: b -> (a -> b) -> (b -> b -> b) -> (b -> b -> b) -> Graph a -> b
- isSubgraphOf :: Ord a => Graph a -> Graph a -> Bool
- toRelation :: Ord a => Graph a -> Relation a
- isEmpty :: Graph a -> Bool
- size :: Graph a -> Int
- hasVertex :: Eq a => a -> Graph a -> Bool
- hasEdge :: Eq a => a -> a -> Graph a -> Bool
- vertexCount :: Ord a => Graph a -> Int
- edgeCount :: Ord a => Graph a -> Int
- vertexList :: Ord a => Graph a -> [a]
- edgeList :: Ord a => Graph a -> [(a, a)]
- vertexSet :: Ord a => Graph a -> Set a
- edgeSet :: Ord a => Graph a -> Set (a, a)
- adjacencyList :: Ord a => Graph a -> [(a, [a])]
- neighbours :: Ord a => a -> Graph a -> Set a
- path :: [a] -> Graph a
- circuit :: [a] -> Graph a
- clique :: [a] -> Graph a
- biclique :: [a] -> [a] -> Graph a
- star :: a -> [a] -> Graph a
- stars :: [(a, [a])] -> Graph a
- tree :: Tree a -> Graph a
- forest :: Forest a -> Graph a
- removeVertex :: Eq a => a -> Graph a -> Graph a
- removeEdge :: Eq a => a -> a -> Graph a -> Graph a
- replaceVertex :: Eq a => a -> a -> Graph a -> Graph a
- mergeVertices :: (a -> Bool) -> a -> Graph a -> Graph a
- induce :: (a -> Bool) -> Graph a -> Graph a
- induceJust :: Graph (Maybe a) -> Graph a
- complement :: Ord a => Graph a -> Graph a

# Algebraic data type for graphs

data Graph a #

The `Graph` data type provides the four algebraic graph construction primitives `empty`, `vertex`, `overlay` and `connect`, as well as various derived functions. The only difference compared to the `Graph` data type defined in Algebra.Graph is that the `connect` operation is commutative. We define a `Num` instance as a convenient notation for working with undirected graphs:

    0           == vertex 0
    1 + 2       == overlay (vertex 1) (vertex 2)
    1 * 2       == connect (vertex 1) (vertex 2)
    1 + 2 * 3   == overlay (vertex 1) (connect (vertex 2) (vertex 3))
    1 * (2 + 3) == connect (vertex 1) (overlay (vertex 2) (vertex 3))

Note: the `Num` instance does not satisfy several "customary laws" of `Num`, which dictate that `fromInteger` `0` and `fromInteger` `1` should act as additive and multiplicative identities, and `negate` as additive inverse. Nevertheless, overloading `fromInteger`, `+` and `*` is very convenient when working with algebraic graphs; we hope that in future Haskell's Prelude will provide a more fine-grained class hierarchy for algebraic structures, which we would be able to utilise without violating any laws.

The `Eq` instance is currently implemented using the `Relation` as the canonical graph representation and satisfies all axioms of algebraic graphs:

- `overlay` is commutative and associative:

            x + y == y + x
      x + (y + z) == (x + y) + z

- `connect` is associative, commutative and has `empty` as the identity:

        x * empty == x
        empty * x == x
            x * y == y * x
      x * (y * z) == (x * y) * z

- `connect` distributes over `overlay`:

      x * (y + z) == x * y + x * z
      (x + y) * z == x * z + y * z

- `connect` can be decomposed:

      x * y * z == x * y + x * z + y * z

The following useful theorems can be proved from the above set of axioms.

- `overlay` has `empty` as the identity and is idempotent:

        x + empty == x
        empty + x == x
            x + x == x

- Absorption and saturation of `connect`:

      x * y + x + y == x * y
          x * x * x == x * x

When specifying the time and memory complexity of graph algorithms, n will denote the number of vertices in the graph, m will denote the number of edges in the graph, and s will denote the size of the corresponding `Graph` expression. For example, if `g` is a `Graph` then n, m and s can be computed as follows:

    n == vertexCount g
    m == edgeCount g
    s == size g

Note that `size` counts all leaves of the expression:

    vertexCount empty           == 0
    size        empty           == 1
    vertexCount (vertex x)      == 1
    size        (vertex x)      == 1
    vertexCount (empty + empty) == 0
    size        (empty + empty) == 2

Converting an undirected `Graph` to the corresponding `Relation` takes O(s + m * log(m)) time and O(s + m) memory. This is also the complexity of the graph equality test, because it is currently implemented by converting graph expressions to canonical representations based on adjacency maps.

The total order on graphs is defined using size-lexicographic comparison:

- Compare the number of vertices. In case of a tie, continue.
- Compare the sets of vertices. In case of a tie, continue.
- Compare the number of edges. In case of a tie, continue.
- Compare the sets of edges.

Here are a few examples:

    vertex 1 < vertex 2
    vertex 3 < edge 1 2
    vertex 1 < edge 1 1
    edge 1 1 < edge 1 2
    edge 1 2 < edge 1 1 + edge 2 2
    edge 1 2 < edge 1 3
    edge 1 2 == edge 2 1

Note that the resulting order refines the `isSubgraphOf` relation and is compatible with `overlay` and `connect` operations:

    isSubgraphOf x y ==> x <= y

    empty <= x
    x     <= x + y
    x + y <= x * y

#### Instances

Instances details

Alternative Graph # |
Instance details Defined in Algebra.Graph.Undirected Methods empty :: Graph a # (<|>) :: Graph a -> Graph a -> Graph a # some :: Graph a -> Graph [a] # many :: Graph a -> Graph [a] #
Applicative Graph # |
Instance details Defined in Algebra.Graph.Undirected Methods pure :: a -> Graph a # (<*>) :: Graph (a -> b) -> Graph a -> Graph b # liftA2 :: (a -> b -> c) -> Graph a -> Graph b -> Graph c # (*>) :: Graph a -> Graph b -> Graph b # (<*) :: Graph a -> Graph b -> Graph a #
Functor Graph # |
Instance details Defined in Algebra.Graph.Undirected Methods fmap :: (a -> b) -> Graph a -> Graph b # (<$) :: a -> Graph b -> Graph a #
Monad Graph # |
Instance details Defined in Algebra.Graph.Undirected Methods (>>=) :: Graph a -> (a -> Graph b) -> Graph b # (>>) :: Graph a -> Graph b -> Graph b # return :: a -> Graph a #
MonadPlus Graph # |
Instance details Defined in Algebra.Graph.Undirected Methods mzero :: Graph a # mplus :: Graph a -> Graph a -> Graph a #
Graph (Graph a) # |
Instance details Defined in Algebra.Graph.Class Associated Types
type Vertex (Graph a) |
Instance details Defined in Algebra.Graph.Class type Vertex (Graph a) = a
Methods empty :: Graph a # vertex :: Vertex (Graph a) -> Graph a # overlay :: Graph a -> Graph a -> Graph a # connect :: Graph a -> Graph a -> Graph a #
Undirected (Graph a) # |
Instance details Defined in Algebra.Graph.Class
NFData a => NFData (Graph a) # |
Instance details Defined in Algebra.Graph.Undirected Methods rnf :: Graph a -> () #
Monoid (Graph a) # | Defined via `overlay` and `empty`.
Instance details Defined in Algebra.Graph.Undirected Methods mempty :: Graph a # mappend :: Graph a -> Graph a -> Graph a # mconcat :: [Graph a] -> Graph a #
Semigroup (Graph a) # | Defined via `overlay`.
Instance details Defined in Algebra.Graph.Undirected Methods (<>) :: Graph a -> Graph a -> Graph a # sconcat :: NonEmpty (Graph a) -> Graph a # stimes :: Integral b => b -> Graph a -> Graph a #
IsString a => IsString (Graph a) # |
Instance details Defined in Algebra.Graph.Undirected Methods fromString :: String -> Graph a #
Generic (Graph a) # |
Instance details Defined in Algebra.Graph.Undirected Associated Types
type Rep (Graph a) |
Instance details Defined in Algebra.Graph.Undirected type Rep (Graph a) = D1 ('MetaData "Graph" "Algebra.Graph.Undirected" "algebraic-graphs-0.7-3cBpMvBpglnBK3x0Rk38no" 'True) (C1 ('MetaCons "UG" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Graph a))))
Methods from :: Graph a -> Rep (Graph a) x # to :: Rep (Graph a) x -> Graph a #
Num a => Num (Graph a) # | Note: this does not satisfy the usual ring laws; see `Graph` for more details.
Instance details Defined in Algebra.Graph.Undirected Methods (+) :: Graph a -> Graph a -> Graph a # (-) :: Graph a -> Graph a -> Graph a # (*) :: Graph a -> Graph a -> Graph a # negate :: Graph a -> Graph a # abs :: Graph a -> Graph a # signum :: Graph a -> Graph a # fromInteger :: Integer -> Graph a #
(Show a, Ord a) => Show (Graph a) # |
Instance details Defined in Algebra.Graph.Undirected Methods showsPrec :: Int -> Graph a -> ShowS # show :: Graph a -> String # showList :: [Graph a] -> ShowS #
Ord a => Eq (Graph a) # |
Instance details Defined in Algebra.Graph.Undirected Methods (==) :: Graph a -> Graph a -> Bool # (/=) :: Graph a -> Graph a -> Bool #
Ord a => Ord (Graph a) # |
Instance details Defined in Algebra.Graph.Undirected Methods compare :: Graph a -> Graph a -> Ordering # (<) :: Graph a -> Graph a -> Bool # (<=) :: Graph a -> Graph a -> Bool # (>) :: Graph a -> Graph a -> Bool # (>=) :: Graph a -> Graph a -> Bool # max :: Graph a -> Graph a -> Graph a # min :: Graph a -> Graph a -> Graph a #
type Vertex (Graph a) # |
Instance details Defined in Algebra.Graph.Class type Vertex (Graph a) = a
type Rep (Graph a) # |
Instance details Defined in Algebra.Graph.Undirected type Rep (Graph a) = D1 ('MetaData "Graph" "Algebra.Graph.Undirected" "algebraic-graphs-0.7-3cBpMvBpglnBK3x0Rk38no" 'True) (C1 ('MetaCons "UG" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Graph a))))

fromUndirected :: Ord a => Graph a -> Graph a #

Extract the underlying Algebra.Graph. Complexity: O(n + m) time.

    fromUndirected (edge 1 2)     == edges [(1,2),(2,1)]
    toUndirected . fromUndirected == id
    vertexCount . fromUndirected  == vertexCount
    edgeCount . fromUndirected    <= (*2) . edgeCount

toUndirected :: Graph a -> Graph a #

Construct an undirected graph from a given Algebra.Graph. Complexity: O(1) time.

    toUndirected (edge 1 2)         == edge 1 2
    toUndirected . fromUndirected   == id
    vertexCount . toUndirected      == vertexCount
    (*2) . edgeCount . toUndirected >= edgeCount

# Basic graph construction primitives

empty :: Graph a #

Construct the empty graph.

    isEmpty     empty == True
    hasVertex x empty == False
    vertexCount empty == 0
    edgeCount   empty == 0
    size        empty == 1

vertex :: a -> Graph a #

Construct the graph comprising a single isolated vertex.

    isEmpty     (vertex x) == False
    hasVertex x (vertex y) == (x == y)
    vertexCount (vertex x) == 1
    edgeCount   (vertex x) == 0
    size        (vertex x) == 1

edge :: a -> a -> Graph a #

Construct the graph comprising a single edge.

    edge x y               == connect (vertex x) (vertex y)
    edge x y               == edge y x
    edge x y               == edges [(x,y), (y,x)]
    hasEdge x y (edge x y) == True
    edgeCount   (edge x y) == 1
    vertexCount (edge 1 1) == 1
    vertexCount (edge 1 2) == 2

overlay :: Graph a -> Graph a -> Graph a #

Overlay two graphs. This is a commutative, associative and idempotent operation with the identity `empty`. Complexity: O(1) time and memory, O(s1 + s2) size.

    isEmpty     (overlay x y) == isEmpty   x   && isEmpty   y
    hasVertex z (overlay x y) == hasVertex z x || hasVertex z y
    vertexCount (overlay x y) >= vertexCount x
    vertexCount (overlay x y) <= vertexCount x + vertexCount y
    edgeCount   (overlay x y) >= edgeCount x
    edgeCount   (overlay x y) <= edgeCount x   + edgeCount y
    size        (overlay x y) == size x        + size y
    vertexCount (overlay 1 2) == 2
    edgeCount   (overlay 1 2) == 0

connect :: Graph a -> Graph a -> Graph a #

Connect two graphs. This is a commutative and associative operation with the identity `empty`, which distributes over `overlay` and obeys the decomposition axiom. Complexity: O(1) time and memory, O(s1 + s2) size. Note that the number of edges in the resulting graph is quadratic with respect to the number of vertices of the arguments: m = O(m1 + m2 + n1 * n2).

    connect x y               == connect y x
    isEmpty     (connect x y) == isEmpty   x   && isEmpty   y
    hasVertex z (connect x y) == hasVertex z x || hasVertex z y
    vertexCount (connect x y) >= vertexCount x
    vertexCount (connect x y) <= vertexCount x + vertexCount y
    edgeCount   (connect x y) >= edgeCount x
    edgeCount   (connect x y) >= edgeCount y
    edgeCount   (connect x y) >= vertexCount x * vertexCount y
    edgeCount   (connect x y) >= vertexCount x * vertexCount y div 2
    size        (connect x y) == size x        + size y
    vertexCount (connect 1 2) == 2
    edgeCount   (connect 1 2) == 1

vertices :: [a] -> Graph a #

Construct the graph comprising a given list of isolated vertices. Complexity: O(L) time, memory and size, where L is the length of the given list.

    vertices []            == empty
    vertices [x]           == vertex x
    vertices               == overlays . map vertex
    hasVertex x . vertices == elem x
    vertexCount . vertices == length . nub
    vertexSet   . vertices == Set . fromList

edges :: [(a, a)] -> Graph a #

Construct the graph from a list of edges. Complexity: O(L) time, memory and size, where L is the length of the given list.

    edges []             == empty
    edges [(x,y)]        == edge x y
    edges [(x,y), (y,x)] == edge x y

overlays :: [Graph a] -> Graph a #

Overlay a given list of graphs. Complexity: O(L) time and memory, and O(S) size, where L is the length of the given list, and S is the sum of sizes of the graphs in the list.

    overlays []        == empty
    overlays [x]       == x
    overlays [x,y]     == overlay x y
    overlays           == foldr overlay empty
    isEmpty . overlays == all isEmpty

connects :: [Graph a] -> Graph a #

Connect a given list of graphs. Complexity: O(L) time and memory, and O(S) size, where L is the length of the given list, and S is the sum of sizes of the graphs in the list.

    connects []        == empty
    connects [x]       == x
    connects [x,y]     == connect x y
    connects           == foldr connect empty
    isEmpty . connects == all isEmpty
    connects           == connects . reverse

# Graph folding

foldg :: b -> (a -> b) -> (b -> b -> b) -> (b -> b -> b) -> Graph a -> b #

Generalised `Graph` folding: recursively collapse a `Graph` by applying the provided functions to the leaves and internal nodes of the expression. The order of arguments is: empty, vertex, overlay and connect. Complexity: O(s) applications of the given functions. As an example, the complexity of `size` is O(s), since `const` and `+` have constant costs.

    foldg empty vertex        overlay connect        == id
    foldg empty vertex        overlay (flip connect) == id
    foldg 1     (const 1)     (+)     (+)            == size
    foldg True  (const False) (&&)    (&&)           == isEmpty
    foldg False (== x)        (||)    (||)           == hasVertex x

# Relations on graphs

isSubgraphOf :: Ord a => Graph a -> Graph a -> Bool #

The `isSubgraphOf` function takes two graphs and returns `True` if the first graph is a subgraph of the second. Complexity: O(s + m * log(m)) time. Note that the number of edges m of a graph can be quadratic with respect to the expression size s.

    isSubgraphOf empty         x             ==  True
    isSubgraphOf (vertex x)    empty         ==  False
    isSubgraphOf x             (overlay x y) ==  True
    isSubgraphOf (overlay x y) (connect x y) ==  True
    isSubgraphOf (path xs)     (circuit xs)  ==  True
    isSubgraphOf (edge x y)    (edge y x)    ==  True
    isSubgraphOf x y                         ==> x <= y

toRelation :: Ord a => Graph a -> Relation a #

Convert an undirected graph to a symmetric `Relation`.

# Graph properties

isEmpty :: Graph a -> Bool #

Check if a graph is empty. Complexity: O(s) time.

    isEmpty empty                       == True
    isEmpty (overlay empty empty)       == True
    isEmpty (vertex x)                  == False
    isEmpty (removeVertex x $ vertex x) == True
    isEmpty (removeEdge x y $ edge x y) == False

size :: Graph a -> Int #

The size of a graph, i.e. the number of leaves of the expression including `empty` leaves. Complexity: O(s) time.

    size empty         == 1
    size (vertex x)    == 1
    size (overlay x y) == size x + size y
    size (connect x y) == size x + size y
    size x             >= 1
    size x             >= vertexCount x

hasVertex :: Eq a => a -> Graph a -> Bool #

Check if a graph contains a given vertex. Complexity: O(s) time.

    hasVertex x empty            == False
    hasVertex x (vertex y)       == (x == y)
    hasVertex x . removeVertex x == const False

hasEdge :: Eq a => a -> a -> Graph a -> Bool #

Check if a graph contains a given edge. Complexity: O(s) time.

    hasEdge x y empty            == False
    hasEdge x y (vertex z)       == False
    hasEdge x y (edge x y)       == True
    hasEdge x y (edge y x)       == True
    hasEdge x y . removeEdge x y == const False
    hasEdge x y                  == elem (min x y, max x y) . edgeList

vertexCount :: Ord a => Graph a -> Int #

The number of vertices in a graph. Complexity: O(s * log(n)) time.

    vertexCount empty             ==  0
    vertexCount (vertex x)        ==  1
    vertexCount                   ==  length . vertexList
    vertexCount x < vertexCount y ==> x < y

edgeCount :: Ord a => Graph a -> Int #

The number of edges in a graph. Complexity: O(s + m * log(m)) time. Note that the number of edges m of a graph can be quadratic with respect to the expression size s.

    edgeCount empty      == 0
    edgeCount (vertex x) == 0
    edgeCount (edge x y) == 1
    edgeCount            == length . edgeList

vertexList :: Ord a => Graph a -> [a] #

The sorted list of vertices of a given graph. Complexity: O(s * log(n)) time and O(n) memory.

    vertexList empty      == []
    vertexList (vertex x) == [x]
    vertexList . vertices == nub . sort

edgeList :: Ord a => Graph a -> [(a, a)] #

The sorted list of edges of a graph. Complexity: O(s + m * log(m)) time and O(m) memory. Note that the number of edges m of a graph can be quadratic with respect to the expression size s.

    edgeList empty          == []
    edgeList (vertex x)     == []
    edgeList (edge x y)     == [(min x y, max y x)]
    edgeList (star 2 [3,1]) == [(1,2), (2,3)]

vertexSet :: Ord a => Graph a -> Set a #

The set of vertices of a given graph. Complexity: O(s * log(n)) time and O(n) memory.

    vertexSet empty      == Set.empty
    vertexSet . vertex   == Set.singleton
    vertexSet . vertices == Set.fromList

edgeSet :: Ord a => Graph a -> Set (a, a) #

The set of edges of a given graph. Complexity: O(s * log(m)) time and O(m) memory.

    edgeSet empty      == Set.empty
    edgeSet (vertex x) == Set.empty
    edgeSet (edge x y) == Set.singleton (min x y, max x y)

adjacencyList :: Ord a => Graph a -> [(a, [a])] #

The sorted adjacency list of a graph. Complexity: O(n + m) time and memory.

    adjacencyList empty          == []
    adjacencyList (vertex x)     == [(x, [])]
    adjacencyList (edge 1 2)     == [(1, [2]), (2, [1])]
    adjacencyList (star 2 [3,1]) == [(1, [2]), (2, [1,3]), (3, [2])]
    stars . adjacencyList        == id

neighbours :: Ord a => a -> Graph a -> Set a #

The set of vertices adjacent to a given vertex.

    neighbours x empty      == Set.empty
    neighbours x (vertex x) == Set.empty
    neighbours x (edge x y) == Set.fromList [y]
    neighbours y (edge x y) == Set.fromList [x]

# Standard families of graphs

path :: [a] -> Graph a #

The path on a list of vertices. Complexity: O(L) time, memory and size, where L is the length of the given list.

    path []        == empty
    path [x]       == vertex x
    path [x,y]     == edge x y
    path . reverse == path

circuit :: [a] -> Graph a #

The circuit on a list of vertices. Complexity: O(L) time, memory and size, where L is the length of the given list.

    circuit []        == empty
    circuit [x]       == edge x x
    circuit [x,y]     == edge (x,y)
    circuit . reverse == circuit

clique :: [a] -> Graph a #

The clique on a list of vertices. Complexity: O(L) time, memory and size, where L is the length of the given list.

    clique []         == empty
    clique [x]        == vertex x
    clique [x,y]      == edge x y
    clique [x,y,z]    == edges [(x,y), (x,z), (y,z)]
    clique (xs ++ ys) == connect (clique xs) (clique ys)
    clique . reverse  == clique

biclique :: [a] -> [a] -> Graph a #

The biclique on two lists of vertices. Complexity: O(L1 + L2) time, memory and size, where L1 and L2 are the lengths of the given lists.

    biclique []      []      == empty
    biclique [x]     []      == vertex x
    biclique []      [y]     == vertex y
    biclique [x1,x2] [y1,y2] == edges [(x1,y1), (x1,y2), (x2,x2), (x2,y2)]
    biclique xs      ys      == connect (vertices xs) (vertices ys)

star :: a -> [a] -> Graph a #

The star formed by a centre vertex connected to a list of leaves. Complexity: O(L) time, memory and size, where L is the length of the given list.

    star x []    == vertex x
    star x [y]   == edge x y
    star x [y,z] == edges [(x,y), (x,z)]
    star x ys    == connect (vertex x) (vertices ys)

stars :: [(a, [a])] -> Graph a #

The stars formed by overlaying a list of `star`s. An inverse of `adjacencyList`. Complexity: O(L) time, memory and size, where L is the total size of the input.

    stars []                      == empty
    stars [(x, [])]               == vertex x
    stars [(x, [y])]              == edge x y
    stars [(x, ys)]               == star x ys
    stars                         == overlays . map (uncurry star)
    stars . adjacencyList         == id
    overlay (stars xs) (stars ys) == stars (xs ++ ys)

tree :: Tree a -> Graph a #

The tree graph constructed from a given `Tree` data structure. Complexity: O(T) time, memory and size, where T is the size of the given tree (i.e. the number of vertices in the tree).

    tree (Node x [])                                         == vertex x
    tree (Node x [Node y [Node z []]])                       == path [x,y,z]
    tree (Node x [Node y [], Node z []])                     == star x [y,z]
    tree (Node 1 [Node 2 [], Node 3 [Node 4 [], Node 5 []]]) == edges [(1,2), (1,3), (3,4), (3,5)]

forest :: Forest a -> Graph a #

The forest graph constructed from a given `Forest` data structure. Complexity: O(F) time, memory and size, where F is the size of the given forest (i.e. the number of vertices in the forest).

    forest []                                                  == empty
    forest [x]                                                 == tree x
    forest [Node 1 [Node 2 [], Node 3 []], Node 4 [Node 5 []]] == edges [(1,2), (1,3), (4,5)]
    forest                                                     == overlays . map tree

# Graph transformation

removeVertex :: Eq a => a -> Graph a -> Graph a #

Remove a vertex from a given graph. Complexity: O(s) time, memory and size.

    removeVertex x (vertex x)       == empty
    removeVertex 1 (vertex 2)       == vertex 2
    removeVertex x (edge x x)       == empty
    removeVertex 1 (edge 1 2)       == vertex 2
    removeVertex x . removeVertex x == removeVertex x

removeEdge :: Eq a => a -> a -> Graph a -> Graph a #

Remove an edge from a given graph. Complexity: O(s) time, memory and size.

    removeEdge x y (edge x y)       == vertices [x,y]
    removeEdge x y . removeEdge x y == removeEdge x y
    removeEdge x y                  == removeEdge y x
    removeEdge x y . removeVertex x == removeVertex x
    removeEdge 1 1 (1 * 1 * 2 * 2)  == 1 * 2 * 2
    removeEdge 1 2 (1 * 1 * 2 * 2)  == 1 * 1 + 2 * 2

replaceVertex :: Eq a => a -> a -> Graph a -> Graph a #

The function ``replaceVertex` x y` replaces vertex `x` with vertex `y` in a given `Graph`. If `y` already exists, `x` and `y` will be merged. Complexity: O(s) time, memory and size.

    replaceVertex x x            == id
    replaceVertex x y (vertex x) == vertex y
    replaceVertex x y            == mergeVertices (== x) y

mergeVertices :: (a -> Bool) -> a -> Graph a -> Graph a #

Merge vertices satisfying a given predicate into a given vertex. Complexity: O(s) time, memory and size, assuming that the predicate takes constant time.

    mergeVertices (const False) x    == id
    mergeVertices (== x) y           == replaceVertex x y
    mergeVertices even 1 (0 * 2)     == 1 * 1
    mergeVertices odd  1 (3 + 4 * 5) == 4 * 1

induce :: (a -> Bool) -> Graph a -> Graph a #

Construct the induced subgraph of a given graph by removing the vertices that do not satisfy a given predicate. Complexity: O(s) time, memory and size, assuming that the predicate takes constant time.

    induce (const True ) x      == x
    induce (const False) x      == empty
    induce (/= x)               == removeVertex x
    induce p . induce q         == induce (\x -> p x && q x)
    isSubgraphOf (induce p x) x == True

induceJust :: Graph (Maybe a) -> Graph a #

Construct the induced subgraph of a given graph by removing the vertices that are `Nothing`. Complexity: O(s) time, memory and size.

    induceJust (vertex Nothing)                               == empty
    induceJust (edge (Just x) Nothing)                        == vertex x
    induceJust . fmap Just                                    == id
    induceJust . fmap (\x -> if p x then Just x else Nothing) == induce p

complement :: Ord a => Graph a -> Graph a #

The edge complement of a graph. Note that, as can be seen from the examples below, this operation ignores self-loops. Complexity: O(n^2 * log n) time, O(n^2) memory.

    complement empty           == empty
    complement (vertex x)      == (vertex x)
    complement (edge 1 2)      == (vertices [1, 2])
    complement (edge 0 0)      == (edge 0 0)
    complement (star 1 [2, 3]) == (overlay (vertex 1) (edge 2 3))
    complement . complement    == id