The AdjacencyMap data type represents a graph by a map of vertices to their adjacency sets. 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 signum method of the type class Num cannot be implemented and will throw an error. Furthermore, 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 (1         :: AdjacencyMap Int) == "vertex 1"
show (1 + 2     :: AdjacencyMap Int) == "vertices1 [1,2]"
show (1 * 2     :: AdjacencyMap Int) == "edge 1 2"
show (1 * 2 * 3 :: AdjacencyMap Int) == "edges1 [(1,2),(1,3),(2,3)]"
show (1 * 2 + 3 :: AdjacencyMap Int) == "overlay (vertex 3) (edge 1 2)"

The Eq instance satisfies the following laws of algebraic graphs:

• overlay is commutative, associative and idempotent:

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

• connect is associative:

  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

• connect satisfies absorption and saturation:

  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

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

Convert a possibly empty AdjacencyMap into NonEmpty.AdjacencyMap. Returns Nothing if the argument is empty. Complexity: O(1) time, memory and size.

toNonEmpty empty          == Nothing
toNonEmpty . fromNonEmpty == Just

Convert a NonEmpty.AdjacencyMap into an AdjacencyMap. The resulting graph is guaranteed to be non-empty. Complexity: O(1) time, memory and size.

isEmpty . fromNonEmpty    == const False
toNonEmpty . fromNonEmpty == Just

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

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

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

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 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.

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 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).

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

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.

vertices1 [x]           == vertex x
hasVertex x . vertices1 == elem x
vertexCount . vertices1 == length . nub
vertexSet   . vertices1 == Set.fromList . toList

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

edges1 [(x,y)]     == edge x y
edges1             == overlays1 . fmap (uncurry edge)
edgeCount . edges1 == length . nub

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

overlays1 [x]   == x
overlays1 [x,y] == overlay x y

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

connects1 [x]   == x
connects1 [x,y] == connect x y

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 x             (overlay x y) ==  True
isSubgraphOf (overlay x y) (connect x y) ==  True
isSubgraphOf (path1 xs)    (circuit1 xs) ==  True
isSubgraphOf x y                         ==> x <= y

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

hasVertex x (vertex y) == (x == y)

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

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

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

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

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

vertexList1 (vertex x)  == [x]
vertexList1 . vertices1 == nub . sort

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

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

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

vertexSet . vertex    == Set.singleton
vertexSet . vertices1 == Set.fromList . toList
vertexSet . clique1   == Set.fromList . toList

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

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

The preset of an element x is the set of its direct predecessors. Complexity: O(n * log(n)) time and O(n) memory.

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

The postset of a vertex is the set of its direct successors. Complexity: O(log(n)) time and O(1) memory.

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

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

path1 [x]       == vertex x
path1 [x,y]     == edge x y
path1 . reverse == transpose . path1

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

circuit1 [x]       == edge x x
circuit1 [x,y]     == edges1 [(x,y), (y,x)]
circuit1 . reverse == transpose . circuit1

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

clique1 [x]        == vertex x
clique1 [x,y]      == edge x y
clique1 [x,y,z]    == edges1 [(x,y), (x,z), (y,z)]
clique1 (xs <> ys) == connect (clique1 xs) (clique1 ys)
clique1 . reverse  == transpose . clique1

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

biclique1 [x1,x2] [y1,y2] == edges1 [(x1,y1), (x1,y2), (x2,y1), (x2,y2)]
biclique1 xs      ys      == connect (vertices1 xs) (vertices1 ys)

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] == edges1 [(x,y), (x,z)]

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.

stars1 [(x, [] )]               == vertex x
stars1 [(x, [y])]               == edge x y
stars1 [(x, ys )]               == star x ys
stars1                          == overlays1 . fmap (uncurry star)
overlay (stars1 xs) (stars1 ys) == stars1 (xs <> ys)

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 []]])                       == path1 [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 []]]) == edges1 [(1,2), (1,3), (3,4), (3,5)]

Remove a vertex from a given graph. Complexity: O(n*log(n)) time.

removeVertex1 x (vertex x)          == Nothing
removeVertex1 1 (vertex 2)          == Just (vertex 2)
removeVertex1 x (edge x x)          == Nothing
removeVertex1 1 (edge 1 2)          == Just (vertex 2)
removeVertex1 x >=> removeVertex1 x == removeVertex1 x

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

removeEdge x y (edge x y)       == vertices1 [x,y]
removeEdge x y . removeEdge x y == removeEdge x y
removeEdge 1 1 (1 * 1 * 2 * 2)  == 1 * 2 * 2
removeEdge 1 2 (1 * 1 * 2 * 2)  == 1 * 1 + 2 * 2

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

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 a given graph. Complexity: O(m * log(n)) time, O(n + m) memory.

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

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 AdjacencyMap. Complexity: O((n + m) * log(n)) time.

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)

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

induce1 (const True ) x == Just x
induce1 (const False) x == Nothing
induce1 (/= x)          == removeVertex1 x
induce1 p >=> induce1 q == induce1 (\x -> p x && q x)

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

induceJust1 (vertex Nothing)                               == Nothing
induceJust1 (edge (Just x) Nothing)                        == Just (vertex x)
induceJust1 . gmap Just                                    == Just
induceJust1 . gmap (\x -> if p x then Just x else Nothing) == induce1 p

Compute the reflexive and transitive closure of a graph. Complexity: O(n * m * log(n)^2) time.

closure (vertex x)       == edge x x
closure (edge x x)       == edge x x
closure (edge x y)       == edges1 [(x,x), (x,y), (y,y)]
closure (path1 $ nub xs) == reflexiveClosure (clique1 $ nub xs)
closure                  == reflexiveClosure . transitiveClosure
closure                  == transitiveClosure . reflexiveClosure
closure . closure        == closure
postSet x (closure y)    == Set.fromList (reachable x y)

Compute the reflexive closure of a graph by adding a self-loop to every vertex. Complexity: O(n * log(n)) time.

reflexiveClosure (vertex x)         == edge x x
reflexiveClosure (edge x x)         == edge x x
reflexiveClosure (edge x y)         == edges1 [(x,x), (x,y), (y,y)]
reflexiveClosure . reflexiveClosure == reflexiveClosure

Compute the symmetric closure of a graph by overlaying it with its own transpose. Complexity: O((n + m) * log(n)) time.

symmetricClosure (vertex x)         == vertex x
symmetricClosure (edge x y)         == edges1 [(x,y), (y,x)]
symmetricClosure x                  == overlay x (transpose x)
symmetricClosure . symmetricClosure == symmetricClosure

Compute the transitive closure of a graph. Complexity: O(n * m * log(n)^2) time.

transitiveClosure (vertex x)          == vertex x
transitiveClosure (edge x y)          == edge x y
transitiveClosure (path1 $ nub xs)    == clique1 (nub xs)
transitiveClosure . transitiveClosure == transitiveClosure

Check that the internal graph representation is consistent, i.e. that all edges refer to existing vertices, and the graph is non-empty. It should be impossible to create an inconsistent adjacency map, and we use this function in testing.

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