algebraic-graphs-0.5: A library for algebraic graph construction and transformation

Algebra.Graph.Relation

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 the Relation data type, as well as associated operations and algorithms. Relation is an instance of the Graph type class, which can be used for polymorphic graph construction and manipulation.

Synopsis

# Data structure

data Relation a Source #

The Relation data type represents a graph as a binary relation. We define a Num instance as a convenient notation for working with 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 Show instance is defined using basic graph construction primitives:

show (empty     :: Relation Int) == "empty"
show (1         :: Relation Int) == "vertex 1"
show (1 + 2     :: Relation Int) == "vertices [1,2]"
show (1 * 2     :: Relation Int) == "edge 1 2"
show (1 * 2 * 3 :: Relation Int) == "edges [(1,2),(1,3),(2,3)]"
show (1 * 2 + 3 :: Relation Int) == "overlay (vertex 3) (edge 1 2)"

The Eq instance satisfies all axioms of algebraic graphs:

• overlay is commutative and associative:

      x + y == y + x
x + (y + z) == (x + y) + z
• connect is associative and has empty as the identity:

  x * empty == x
empty * x == 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 and m will denote the number of vertices and edges in the graph, respectively.

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

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
 Eq a => Eq (Relation a) Source # Instance detailsDefined in Algebra.Graph.Relation Methods(==) :: Relation a -> Relation a -> Bool #(/=) :: Relation a -> Relation a -> Bool # (Ord a, Num a) => Num (Relation a) Source # Note: this does not satisfy the usual ring laws; see Relation for more details. Instance detailsDefined in Algebra.Graph.Relation Methods(+) :: Relation a -> Relation a -> Relation a #(-) :: Relation a -> Relation a -> Relation a #(*) :: Relation a -> Relation a -> Relation a #negate :: Relation a -> Relation a #abs :: Relation a -> Relation a #signum :: Relation a -> Relation a # Ord a => Ord (Relation a) Source # Instance detailsDefined in Algebra.Graph.Relation Methodscompare :: Relation a -> Relation a -> Ordering #(<) :: Relation a -> Relation a -> Bool #(<=) :: Relation a -> Relation a -> Bool #(>) :: Relation a -> Relation a -> Bool #(>=) :: Relation a -> Relation a -> Bool #max :: Relation a -> Relation a -> Relation a #min :: Relation a -> Relation a -> Relation a # (Ord a, Show a) => Show (Relation a) Source # Instance detailsDefined in Algebra.Graph.Relation MethodsshowsPrec :: Int -> Relation a -> ShowS #show :: Relation a -> String #showList :: [Relation a] -> ShowS # NFData a => NFData (Relation a) Source # Instance detailsDefined in Algebra.Graph.Relation Methodsrnf :: Relation a -> () # Ord a => ToGraph (Relation a) Source # Instance detailsDefined in Algebra.Graph.ToGraph Associated Typestype ToVertex (Relation a) :: Type Source # MethodstoGraph :: Relation a -> Graph (ToVertex (Relation a)) Source #foldg :: r -> (ToVertex (Relation a) -> r) -> (r -> r -> r) -> (r -> r -> r) -> Relation a -> r Source #hasVertex :: ToVertex (Relation a) -> Relation a -> Bool Source #hasEdge :: ToVertex (Relation a) -> ToVertex (Relation a) -> Relation a -> Bool Source #vertexList :: Relation a -> [ToVertex (Relation a)] Source #edgeList :: Relation a -> [(ToVertex (Relation a), ToVertex (Relation a))] Source #vertexSet :: Relation a -> Set (ToVertex (Relation a)) Source #edgeSet :: Relation a -> Set (ToVertex (Relation a), ToVertex (Relation a)) Source #preSet :: ToVertex (Relation a) -> Relation a -> Set (ToVertex (Relation a)) Source #postSet :: ToVertex (Relation a) -> Relation a -> Set (ToVertex (Relation a)) Source #adjacencyList :: Relation a -> [(ToVertex (Relation a), [ToVertex (Relation a)])] Source #dfsForestFrom :: [ToVertex (Relation a)] -> Relation a -> Forest (ToVertex (Relation a)) Source #dfs :: [ToVertex (Relation a)] -> Relation a -> [ToVertex (Relation a)] Source #reachable :: ToVertex (Relation a) -> Relation a -> [ToVertex (Relation a)] Source #topSort :: Relation a -> Either (Cycle (ToVertex (Relation a))) [ToVertex (Relation a)] Source #isDfsForestOf :: Forest (ToVertex (Relation a)) -> Relation a -> Bool Source #isTopSortOf :: [ToVertex (Relation a)] -> Relation a -> Bool Source # Ord a => Graph (Relation a) Source # Instance detailsDefined in Algebra.Graph.Class Associated Typestype Vertex (Relation a) :: Type Source # Methodsvertex :: Vertex (Relation a) -> Relation a Source #overlay :: Relation a -> Relation a -> Relation a Source #connect :: Relation a -> Relation a -> Relation a Source # type ToVertex (Relation a) Source # Instance detailsDefined in Algebra.Graph.ToGraph type ToVertex (Relation a) = a type Vertex (Relation a) Source # Instance detailsDefined in Algebra.Graph.Class type Vertex (Relation a) = a

domain :: Relation a -> Set a Source #

The domain of the relation. Complexity: O(1) time and memory.

relation :: Relation a -> Set (a, a) Source #

The set of pairs of elements that are related. It is guaranteed that each element belongs to the domain. Complexity: O(1) time and memory.

# Basic graph construction primitives

Construct the empty graph. Complexity: O(1) time and memory.

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


vertex :: a -> Relation a Source #

Construct the graph comprising a single isolated vertex. Complexity: O(1) time and memory.

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


edge :: Ord a => a -> a -> Relation a Source #

Construct the graph comprising a single edge. Complexity: O(1) time, memory and size.

edge x y               == connect (vertex x) (vertex y)
hasEdge x y (edge x y) == True
edgeCount   (edge x y) == 1
vertexCount (edge 1 1) == 1
vertexCount (edge 1 2) == 2


overlay :: Ord a => Relation a -> Relation a -> Relation a Source #

Overlay two graphs. This is a commutative, associative and idempotent operation with the identity empty. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

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
vertexCount (overlay 1 2) == 2
edgeCount   (overlay 1 2) == 0


connect :: Ord a => Relation a -> Relation a -> Relation a Source #

Connect two graphs. This is an associative operation with the identity empty, which distributes over overlay and obeys the decomposition axiom. Complexity: O((n + m) * log(n)) time and O(n + m) memory. 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).

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 + edgeCount x + edgeCount y
vertexCount (connect 1 2) == 2
edgeCount   (connect 1 2) == 1


vertices :: Ord a => [a] -> Relation a Source #

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

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


edges :: Ord a => [(a, a)] -> Relation a Source #

Construct the graph from a list of edges. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

edges []          == empty
edges [(x,y)]     == edge x y
edges             == overlays . map (uncurry edge)
edgeCount . edges == length . nub


overlays :: Ord a => [Relation a] -> Relation a Source #

Overlay a given list of graphs. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

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


connects :: Ord a => [Relation a] -> Relation a Source #

Connect a given list of graphs. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

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


# Relations on graphs

isSubgraphOf :: Ord a => Relation a -> Relation a -> Bool Source #

The isSubgraphOf function takes two graphs and returns True if the first graph is a subgraph of the second. Complexity: O((n + m) * log(n)) time.

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 x y                         ==> x <= y


# Graph properties

Check if a relation is empty. Complexity: O(1) 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


hasVertex :: Ord a => a -> Relation a -> Bool Source #

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

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


hasEdge :: Ord a => a -> a -> Relation a -> Bool Source #

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

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


The number of vertices in a graph. Complexity: O(1) time.

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


The number of edges in a graph. Complexity: O(1) time.

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


vertexList :: Relation a -> [a] Source #

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

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


edgeList :: Relation a -> [(a, a)] Source #

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

edgeList empty          == []
edgeList (vertex x)     == []
edgeList (edge x y)     == [(x,y)]
edgeList (star 2 [3,1]) == [(2,1), (2,3)]
edgeList . edges        == nub . sort
edgeList . transpose    == sort . map swap . edgeList


adjacencyList :: Eq a => Relation a -> [(a, [a])] Source #

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

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


vertexSet :: Relation a -> Set a Source #

The set of vertices of a given graph. Complexity: O(1) time.

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


edgeSet :: Relation a -> Set (a, a) Source #

The set of edges of a given graph. Complexity: O(1) time.

edgeSet empty      == Set.empty
edgeSet (vertex x) == Set.empty
edgeSet (edge x y) == Set.singleton (x,y)
edgeSet . edges    == Set.fromList


preSet :: Ord a => a -> Relation a -> Set a Source #

The preset of an element x is the set of elements that are related to it on the left, i.e. preSet x == { a | aRx }. In the context of directed graphs, this corresponds to the set of direct predecessors of vertex x. Complexity: O(n + m) time and O(n) memory.

preSet x empty      == Set.empty
preSet x (vertex x) == Set.empty
preSet 1 (edge 1 2) == Set.empty
preSet y (edge x y) == Set.fromList [x]


postSet :: Ord a => a -> Relation a -> Set a Source #

The postset of an element x is the set of elements that are related to it on the right, i.e. postSet x == { a | xRa }. In the context of directed graphs, this corresponds to the set of direct successors of vertex x. Complexity: O(n + m) time and O(n) memory.

postSet x empty      == Set.empty
postSet x (vertex x) == Set.empty
postSet x (edge x y) == Set.fromList [y]
postSet 2 (edge 1 2) == Set.empty


# Standard families of graphs

path :: Ord a => [a] -> Relation a Source #

The path on a list of vertices. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

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


circuit :: Ord a => [a] -> Relation a Source #

The circuit on a list of vertices. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

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


clique :: Ord a => [a] -> Relation a Source #

The clique on a list of vertices. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

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  == transpose . clique


biclique :: Ord a => [a] -> [a] -> Relation a Source #

The biclique on two lists of vertices. Complexity: O(n * log(n) + m) time and O(n + m) memory.

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


star :: Ord a => a -> [a] -> Relation a Source #

The star formed by a centre vertex connected to a list of leaves. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

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 :: Ord a => [(a, [a])] -> Relation a Source #

The stars formed by overlaying a list of stars. An inverse of adjacencyList. Complexity: O(L * log(n)) 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 :: Ord a => Tree a -> Relation a Source #

The tree graph constructed from a given Tree data structure. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

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 :: Ord a => Forest a -> Relation a Source #

The forest graph constructed from a given Forest data structure. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

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 :: Ord a => a -> Relation a -> Relation a Source #

Remove a vertex from a given graph. Complexity: O(n + m) time.

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 :: Ord a => a -> a -> Relation a -> Relation a Source #

Remove an edge from a given graph. Complexity: O(log(m)) time.

removeEdge x y (edge x y)       == vertices [x,y]
removeEdge x y . removeEdge x y == removeEdge x y
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 :: Ord a => a -> a -> Relation a -> Relation a Source #

The function replaceVertex x y replaces vertex x with vertex y in a given AdjacencyMap. If y already exists, x and y will be merged. Complexity: O((n + m) * log(n)) time.

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


mergeVertices :: Ord a => (a -> Bool) -> a -> Relation a -> Relation a Source #

Merge vertices satisfying a given predicate into a given vertex. Complexity: O((n + m) * log(n)) time, assuming that the predicate takes O(1) to be evaluated.

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


transpose :: Ord a => Relation a -> Relation a Source #

Transpose a given graph. Complexity: O(m * log(m)) time.

transpose empty       == empty
transpose (vertex x)  == vertex x
transpose (edge x y)  == edge y x
transpose . transpose == id
edgeList . transpose  == sort . map swap . edgeList


gmap :: Ord b => (a -> b) -> Relation a -> Relation b Source #

Transform a graph by applying a function to each of its vertices. This is similar to Functor's fmap but can be used with non-fully-parametric Relation. Complexity: O((n + m) * log(n)) time.

gmap f empty      == empty
gmap f (vertex x) == vertex (f x)
gmap f (edge x y) == edge (f x) (f y)
gmap id           == id
gmap f . gmap g   == gmap (f . g)


induce :: (a -> Bool) -> Relation a -> Relation a Source #

Construct the induced subgraph of a given graph by removing the vertices that do not satisfy a given predicate. Complexity: O(n + m) time, assuming that the predicate takes O(1) to be evaluated.

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 :: Ord a => Relation (Maybe a) -> Relation a Source #

Construct the induced subgraph of a given graph by removing the vertices that are Nothing. Complexity: O(n + m) time.

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


# Relational operations

compose :: Ord a => Relation a -> Relation a -> Relation a Source #

Left-to-right relational composition of graphs: vertices x and z are connected in the resulting graph if there is a vertex y, such that x is connected to y in the first graph, and y is connected to z in the second graph. There are no isolated vertices in the result. This operation is associative, has empty and single-vertex graphs as annihilating zeroes, and distributes over overlay. Complexity: O(n * m * log(m)) time and O(n + m) memory.

compose empty            x                == empty
compose x                empty            == empty
compose (vertex x)       y                == empty
compose x                (vertex y)       == empty
compose x                (compose y z)    == compose (compose x y) z
compose x                (overlay y z)    == overlay (compose x y) (compose x z)
compose (overlay x y)    z                == overlay (compose x z) (compose y z)
compose (edge x y)       (edge y z)       == edge x z
compose (path    [1..5]) (path    [1..5]) == edges [(1,3), (2,4), (3,5)]
compose (circuit [1..5]) (circuit [1..5]) == circuit [1,3,5,2,4]


closure :: Ord a => Relation a -> Relation a Source #

Compute the reflexive and transitive closure of a graph. Complexity: O(n * m * log(n) * log(m)) time.

closure empty           == empty
closure (vertex x)      == edge x x
closure (edge x x)      == edge x x
closure (edge x y)      == edges [(x,x), (x,y), (y,y)]
closure (path $nub xs) == reflexiveClosure (clique$ nub xs)
closure                 == reflexiveClosure . transitiveClosure
closure                 == transitiveClosure . reflexiveClosure
closure . closure       == closure
postSet x (closure y)   == Set.fromList (reachable x y)


reflexiveClosure :: Ord a => Relation a -> Relation a Source #

Compute the reflexive closure of a graph. Complexity: O(n * log(m)) time.

reflexiveClosure empty              == empty
reflexiveClosure (vertex x)         == edge x x
reflexiveClosure (edge x x)         == edge x x
reflexiveClosure (edge x y)         == edges [(x,x), (x,y), (y,y)]
reflexiveClosure . reflexiveClosure == reflexiveClosure


symmetricClosure :: Ord a => Relation a -> Relation a Source #

Compute the symmetric closure of a graph. Complexity: O(m * log(m)) time.

symmetricClosure empty              == empty
symmetricClosure (vertex x)         == vertex x
symmetricClosure (edge x y)         == edges [(x,y), (y,x)]
symmetricClosure x                  == overlay x (transpose x)
symmetricClosure . symmetricClosure == symmetricClosure


transitiveClosure :: Ord a => Relation a -> Relation a Source #

Compute the transitive closure of a graph. Complexity: O(n * m * log(n) * log(m)) time.

transitiveClosure empty               == empty
transitiveClosure (vertex x)          == vertex x
transitiveClosure (edge x y)          == edge x y
transitiveClosure (path \$ nub xs)     == clique (nub xs)
transitiveClosure . transitiveClosure == transitiveClosure


# Miscellaneous

consistent :: Ord a => Relation a -> Bool Source #

Check that the internal representation of a relation is consistent, i.e. if all pairs of elements in the relation refer to existing elements in the domain. It should be impossible to create an inconsistent Relation, and we use this function in testing.

consistent empty         == True
consistent (vertex x)    == True
consistent (overlay x y) == True
consistent (connect x y) == True
consistent (edge x y)    == True
consistent (edges xs)    == True
consistent (stars xs)    == True