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GHC 9.14.1 · lts/ghc-9.14.x · d82dee8 · 2026-10-12

Modulealgebraic-graphs-0.8Haskell2010

Algebra.Graph.Relation.Symmetric

An abstract implementation of symmetric binary relations. To avoid name clashes with Algebra.Graph.Relation, this module can be imported qualified:

import qualified Algebra.Graph.Relation.Symmetric as Symmetric

Relation is an instance of the Graph type class, which can be used for polymorphic graph construction and manipulation.

  • 1 type
  • 39 values

Data structure

3 declarations
newtypenewtype Relation a
#

This data type represents a symmetric binary relation over a set of elements of type a. Symmetric relations satisfy all laws of the Undirected type class, including the commutativity of connect:

connect x y == connect y x

The Show instance lists edge vertices in non-decreasing order:

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

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 2 1 < 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
Instances13Eq, Num, Ord, Show, IsString, Semigroup, …

Basic graph construction primitives

9 declarations
valueoverlay :: Ord a => Relation a -> Relation a -> Relation a
#

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
valueconnect :: Ord a => Relation a -> Relation a -> Relation 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((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).

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 `div` 2
vertexCount (connect 1 2) == 2
edgeCount   (connect 1 2) == 1
valueedges :: Ord a => [(a, a)] -> Relation a
#

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

Relations on graphs

1 declaration

Graph properties

11 declarations
valueedgeList :: Ord a => Relation a -> [(a, a)]
#

The sorted list of edges of a graph, where edge vertices appear in the non-decreasing order. Complexity: O(n + m) time and O(m) memory.

Note: If you need the sorted list of edges where an edge appears in both directions, use edgeList . fromSymmetric.

edgeList empty          == []
edgeList (vertex x)     == []
edgeList (edge x y)     == [(min x y, max y x)]
edgeList (star 2 [3,1]) == [(1,2), (2,3)]
valueadjacencyList :: Eq a => Relation 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
valueedgeSet :: Ord a => Relation a -> Set (a, a)
#

The set of edges of a given graph, where edge vertices appear in the non-decreasing order. Complexity: O(m) time.

Note: If you need the set of edges where an edge appears in both directions, use relation . fromSymmetric. The latter is much faster than this function, and takes only O(1) time and memory.

edgeSet empty      == Set.empty
edgeSet (vertex x) == Set.empty
edgeSet (edge x y) == Set.singleton (min x y, max x y)
valueneighbours :: Ord a => a -> Relation a -> Set a
#

The set of neighbours of an element x is the set of elements that are related to it, i.e. neighbours x == { a | aRx }. In the context of undirected graphs, this corresponds to the set of adjacent vertices of vertex x.

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

8 declarations
valuepath :: Ord a => [a] -> Relation a
#

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       == path . reverse
valuecircuit :: Ord a => [a] -> Relation a
#

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] == edge x y
circuit       == circuit . reverse
valueclique :: Ord a => [a] -> Relation a
#

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            == clique . reverse
valuebiclique :: Ord a => [a] -> [a] -> Relation a
#

The biclique on two lists of vertices. Complexity: O((n + m) * log(n)) 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,x2), (x2,y2)]
biclique xs      ys      == connect (vertices xs) (vertices ys)
valuestar :: Ord a => a -> [a] -> Relation a
#

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)
valuetree :: Ord a => Tree a -> Relation a
#

The tree graph constructed from a given Tree.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)]
valueforest :: Ord a => Forest a -> Relation a
#

The forest graph constructed from a given Tree.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

7 declarations
valueremoveEdge :: Ord a => a -> a -> Relation a -> Relation a
#

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                  == 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
valuemergeVertices :: Ord a => (a -> Bool) -> a -> Relation a -> Relation a
#

Merge vertices satisfying a given predicate into a given vertex. Complexity: O((n + m) * log(n)) time, 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
valuegmap :: Ord b => (a -> b) -> Relation a -> Relation b
#

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)
valueinduce :: (a -> Bool) -> Relation a -> Relation a
#

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 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

Miscellaneous

1 declaration
valueconsistent :: Ord a => Relation a -> Bool
#

Check that the internal representation of a symmetric relation is consistent, i.e. that (i) that all edges refer to existing vertices, and (ii) all edges have their symmetric counterparts. 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