Algebra.Graph.Fold

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 Fold data type -- the Boehm-Berarducci encoding of algebraic graphs, which is used for generalised graph folding and for the implementation of polymorphic graph construction and transformation algorithms. Fold is an instance of type classes defined in modules Algebra.Graph.Class and Algebra.Graph.HigherKinded.Class, which can be used for polymorphic graph construction and manipulation.

Boehm-Berarducci encoding of algebraic graphs

data Fold a

The Fold datatype is the Boehm-Berarducci encoding of the core graph construction primitives empty, vertex, overlay and connect. We define a law-abiding 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))

The Show instance is defined using basic graph construction primitives:

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

The Eq instance is currently implemented using the AdjacencyMap 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 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 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 Fold then n, m and s can be computed as follows:

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

Note that size is slightly different from the length method of the Foldable type class, as the latter does not count empty leaves of the expression:

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

The size of any graph is positive, and the difference (size g - length g) corresponds to the number of occurrences of empty in an expression g.

Converting a Fold to the corresponding AdjacencyMap 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.

Instances

Monad Fold

Methods

(>>=) :: Fold a -> (a -> Fold b) -> Fold b

(>>) :: Fold a -> Fold b -> Fold b

return :: a -> Fold a

fail :: String -> Fold a

Functor Fold

Methods

fmap :: (a -> b) -> Fold a -> Fold b

(<$) :: a -> Fold b -> Fold a

Applicative Fold

Methods

pure :: a -> Fold a

(<*>) :: Fold (a -> b) -> Fold a -> Fold b

(*>) :: Fold a -> Fold b -> Fold b

(<*) :: Fold a -> Fold b -> Fold a

Foldable Fold

Methods

fold :: Monoid m => Fold m -> m

foldMap :: Monoid m => (a -> m) -> Fold a -> m

foldr :: (a -> b -> b) -> b -> Fold a -> b

foldr' :: (a -> b -> b) -> b -> Fold a -> b

foldl :: (b -> a -> b) -> b -> Fold a -> b

foldl' :: (b -> a -> b) -> b -> Fold a -> b

foldr1 :: (a -> a -> a) -> Fold a -> a

foldl1 :: (a -> a -> a) -> Fold a -> a

toList :: Fold a -> [a]

null :: Fold a -> Bool

length :: Fold a -> Int

elem :: Eq a => a -> Fold a -> Bool

maximum :: Ord a => Fold a -> a

minimum :: Ord a => Fold a -> a

sum :: Num a => Fold a -> a

product :: Num a => Fold a -> a

Traversable Fold

Methods

traverse :: Applicative f => (a -> f b) -> Fold a -> f (Fold b)

sequenceA :: Applicative f => Fold (f a) -> f (Fold a)

mapM :: Monad m => (a -> m b) -> Fold a -> m (Fold b)

sequence :: Monad m => Fold (m a) -> m (Fold a)

Alternative Fold

Methods

empty :: Fold a

(<|>) :: Fold a -> Fold a -> Fold a

some :: Fold a -> Fold [a]

many :: Fold a -> Fold [a]

MonadPlus Fold

Methods

mzero :: Fold a

mplus :: Fold a -> Fold a -> Fold a

ToGraph Fold

Methods

toGraph :: Graph g => Fold a -> g a

Graph Fold

Methods

connect :: Fold a -> Fold a -> Fold a

Ord a => Eq (Fold a)

Methods

(==) :: Fold a -> Fold a -> Bool

(/=) :: Fold a -> Fold a -> Bool

Num a => Num (Fold a)

Methods

(+) :: Fold a -> Fold a -> Fold a

(-) :: Fold a -> Fold a -> Fold a

(*) :: Fold a -> Fold a -> Fold a

negate :: Fold a -> Fold a

abs :: Fold a -> Fold a

signum :: Fold a -> Fold a

fromInteger :: Integer -> Fold a

(Ord a, Show a) => Show (Fold a)

Methods

showsPrec :: Int -> Fold a -> ShowS

show :: Fold a -> String

showList :: [Fold a] -> ShowS

ToGraph (Fold a)

Associated Types

type ToVertex (Fold a) :: *

Methods

toGraph :: (Graph g, (* ~ Vertex g) (ToVertex (Fold a))) => Fold a -> g

Graph (Fold a)

Associated Types

type Vertex (Fold a) :: *

Methods

empty :: Fold a

vertex :: Vertex (Fold a) -> Fold a

overlay :: Fold a -> Fold a -> Fold a

connect :: Fold a -> Fold a -> Fold a

type ToVertex (Fold a)

type ToVertex (Fold a) = a

type Vertex (Fold a)

type Vertex (Fold a) = a

Basic graph construction primitives

empty :: Graph g => g

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

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

vertex :: Graph g => Vertex g -> g

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

isEmpty     (vertex x) == False
hasVertex x (vertex x) == True
hasVertex 1 (vertex 2) == False
vertexCount (vertex x) == 1
edgeCount   (vertex x) == 0
size        (vertex x) == 1

edge :: Graph g => Vertex g -> Vertex g -> g

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 :: Graph g => g -> g -> g

Overlay two graphs. This is an idempotent, commutative and associative 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 g => g -> g -> g

Connect two graphs. This is an associative operation with the identity empty, which distributes over the 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).

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

vertices :: Graph g => [Vertex g] -> g

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
hasVertex x . vertices == elem x
vertexCount . vertices == length . nub
vertexSet   . vertices == Set.fromList

edges :: Graph g => [(Vertex g, Vertex g)] -> g

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
edgeCount . edges == length . nub

overlays :: Graph g => [g] -> g

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
isEmpty . overlays == all isEmpty

connects :: Graph g => [g] -> g

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
isEmpty . connects == all isEmpty

graph :: Graph g => [Vertex g] -> [(Vertex g, Vertex g)] -> g

Construct the graph from given lists of vertices V and edges E. The resulting graph contains the vertices V as well as all the vertices referred to by the edges E. Complexity: O(|V| + |E|) time, memory and size.

graph []  []      == empty
graph [x] []      == vertex x
graph []  [(x,y)] == edge x y
graph vs  es      == overlay (vertices vs) (edges es)

Graph folding

foldg :: b -> (a -> b) -> (b -> b -> b) -> (b -> b -> b) -> Fold a -> b

Generalised graph folding: recursively collapse a Fold 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 given functions. As an example, the complexity of size is O(s), since all functions have cost O(1).

foldg empty vertex        overlay connect        == id
foldg empty vertex        overlay (flip connect) == transpose
foldg []    return        (++)    (++)           == toList
foldg 0     (const 1)     (+)     (+)            == length
foldg 1     (const 1)     (+)     (+)            == size
foldg True  (const False) (&&)    (&&)           == isEmpty

Relations on graphs

isSubgraphOf :: (Graph g, Eq g) => g -> g -> Bool

The isSubgraphOf function takes two graphs and returns True if the first graph is a subgraph of the second. Here is the current implementation:

isSubgraphOf x y = overlay x y == y

The complexity therefore depends on the complexity of equality testing of the specific graph instance.

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

Graph properties

isEmpty :: Fold a -> Bool

Check if a graph is empty. A convenient alias for null. 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 :: Fold 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 -> Fold a -> Bool

Check if a graph contains a given vertex. A convenient alias for elem. Complexity: O(s) time.

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

hasEdge :: Eq a => a -> a -> Fold 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 . removeEdge x y == const False

vertexCount :: Ord a => Fold 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

edgeCount :: Ord a => Fold 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 => Fold 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 => Fold 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)     == [(x,y)]
edgeList (star 2 [3,1]) == [(2,1), (2,3)]
edgeList . edges        == nub . sort

vertexSet :: Ord a => Fold 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
vertexSet . clique   == Set.fromList

vertexIntSet :: Fold Int -> IntSet

The set of vertices of a given graph. Like vertexSet but specialised for graphs with vertices of type Int. Complexity: O(s * log(n)) time and O(n) memory.

vertexIntSet empty      == IntSet.empty
vertexIntSet . vertex   == IntSet.singleton
vertexIntSet . vertices == IntSet.fromList
vertexIntSet . clique   == IntSet.fromList

edgeSet :: Ord a => Fold 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 (x,y)
edgeSet . edges    == Set.fromList

Standard families of graphs

path :: Graph g => [Vertex g] -> g

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

circuit :: Graph g => [Vertex g] -> g

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

clique :: Graph g => [Vertex g] -> g

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)]

biclique :: Graph g => [Vertex g] -> [Vertex g] -> g

The biclique on a list 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,y1), (x2,y2)]

star :: Graph g => Vertex g -> [Vertex g] -> g

The star formed by a centre vertex and 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)]

tree :: Graph g => Tree (Vertex g) -> g

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).

forest :: Graph g => Forest (Vertex g) -> g

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).

mesh :: (Graph g, Vertex g ~ (a, b)) => [a] -> [b] -> g

Construct a mesh graph from two lists of vertices. Complexity: O(L1 * L2) time, memory and size, where L1 and L2 are the lengths of the given lists.

mesh xs     []   == empty
mesh []     ys   == empty
mesh [x]    [y]  == vertex (x, y)
mesh xs     ys   == box (path xs) (path ys)
mesh [1..3] "ab" == edges [ ((1,'a'),(1,'b')), ((1,'a'),(2,'a')), ((1,'b'),(2,'b')), ((2,'a'),(2,'b'))
                          , ((2,'a'),(3,'a')), ((2,'b'),(3,'b')), ((3,'a'),(3,'b')) ]

torus :: (Graph g, Vertex g ~ (a, b)) => [a] -> [b] -> g

Construct a torus graph from two lists of vertices. Complexity: O(L1 * L2) time, memory and size, where L1 and L2 are the lengths of the given lists.

torus xs    []   == empty
torus []    ys   == empty
torus [x]   [y]  == edge (x, y) (x, y)
torus xs    ys   == box (circuit xs) (circuit ys)
torus [1,2] "ab" == edges [ ((1,'a'),(1,'b')), ((1,'a'),(2,'a')), ((1,'b'),(1,'a')), ((1,'b'),(2,'b'))
                          , ((2,'a'),(1,'a')), ((2,'a'),(2,'b')), ((2,'b'),(1,'b')), ((2,'b'),(2,'a')) ]

deBruijn :: (Graph g, Vertex g ~ [a]) => Int -> [a] -> g

Construct a De Bruijn graph of given dimension and symbols of a given alphabet. Complexity: O(A * D^A) time, memory and size, where A is the size of the alphabet and D is the dimention of the graph.

deBruijn k []    == empty
deBruijn 1 [0,1] == edges [ ([0],[0]), ([0],[1]), ([1],[0]), ([1],[1]) ]
deBruijn 2 "0"   == edge "00" "00"
deBruijn 2 "01"  == edges [ ("00","00"), ("00","01"), ("01","10"), ("01","11")
                          , ("10","00"), ("10","01"), ("11","10"), ("11","11") ]

Graph transformation

removeVertex :: (Eq (Vertex g), Graph g) => Vertex g -> Fold (Vertex g) -> g

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

removeVertex x (vertex x)       == empty
removeVertex x . removeVertex x == removeVertex x

removeEdge :: (Eq (Vertex g), Graph g) => Vertex g -> Vertex g -> Fold (Vertex g) -> g

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

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 :: (Eq (Vertex g), Graph g) => Vertex g -> Vertex g -> Fold (Vertex g) -> g

The function replaceVertex x y replaces vertex x with vertex y in a given graph expression. 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 :: Graph g => (Vertex g -> Bool) -> Vertex g -> Fold (Vertex g) -> g

Merge vertices satisfying a given predicate with a given vertex. Complexity: O(s) time, memory and size, 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

splitVertex :: (Eq (Vertex g), Graph g) => Vertex g -> [Vertex g] -> Fold (Vertex g) -> g

Split a vertex into a list of vertices with the same connectivity. Complexity: O(s + k * L) time, memory and size, where k is the number of occurrences of the vertex in the expression and L is the length of the given list.

splitVertex x []                  == removeVertex x
splitVertex x [x]                 == id
splitVertex x [y]                 == replaceVertex x y
splitVertex 1 [0,1] $ 1 * (2 + 3) == (0 + 1) * (2 + 3)

transpose :: Graph g => Fold (Vertex g) -> g

Transpose a given graph. Complexity: O(s) time, memory and size.

transpose empty       == empty
transpose (vertex x)  == vertex x
transpose (edge x y)  == edge y x
transpose . transpose == id

gmap :: Graph g => (a -> Vertex g) -> Fold a -> g

Transform a given graph by applying a function to each of its vertices. This is similar to fmap but can be used with non-fully-parametric graphs.

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)

bind :: Graph g => Fold a -> (a -> g) -> g

Transform a given graph by substituting each of its vertices with a subgraph. This is similar to Monad's bind >>= but can be used with non-fully-parametric graphs.

bind empty f         == empty
bind (vertex x) f    == f x
bind (edge x y) f    == connect (f x) (f y)
bind (vertices xs) f == overlays (map f xs)
bind x (const empty) == empty
bind x vertex        == x
bind (bind x f) g    == bind x (\y -> bind (f y) g)

induce :: Graph g => (Vertex g -> Bool) -> Fold (Vertex g) -> g

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 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

simplify :: (Eq g, Graph g) => Fold (Vertex g) -> g

Simplify a graph expression. Semantically, this is the identity function, but it simplifies a given polymorphic graph expression according to the laws of the algebra. The function does not compute the simplest possible expression, but uses heuristics to obtain useful simplifications in reasonable time. Complexity: the function performs O(s) graph comparisons. It is guaranteed that the size of the result does not exceed the size of the given expression. Below the operator ~> denotes the is simplified to relation.

simplify             == id
size (simplify x)    <= size x
simplify empty       ~> empty
simplify 1           ~> 1
simplify (1 + 1)     ~> 1
simplify (1 + 2 + 1) ~> 1 + 2
simplify (1 * 1 * 1) ~> 1 * 1

Graph composition

box :: (Graph g, Vertex g ~ (a, b)) => Fold a -> Fold b -> g

Compute the Cartesian product of graphs. Complexity: O(s1 * s2) time, memory and size, where s1 and s2 are the sizes of the given graphs.

box (path [0,1]) (path "ab") == edges [ ((0,'a'), (0,'b'))
                                      , ((0,'a'), (1,'a'))
                                      , ((0,'b'), (1,'b'))
                                      , ((1,'a'), (1,'b')) ]

Up to an isomorphism between the resulting vertex types, this operation is commutative, associative, distributes over overlay, has singleton graphs as identities and empty as the annihilating zero. Below ~~ stands for the equality up to an isomorphism, e.g. (x, ()) ~~ x.

box x y             ~~ box y x
box x (box y z)     ~~ box (box x y) z
box x (overlay y z) == overlay (box x y) (box x z)
box x (vertex ())   ~~ x
box x empty         ~~ empty