Library for inductive graphs (port of a Haskell library by Martin Erwig).
In this library, graphs are composed and decomposed in an inductive way.
The key idea is as follows:
A graph is either empty
or it consists of node context
and a graph
g'
which
are put together by a constructor (:&)
.
This constructor (:&)
, however, is not a constructor in
the sense of abstract
data type, but more basically a defined constructing funtion.
A context
is a node together withe the edges to and from this node
into the nodes in the graph g'
.
For examples of how to use this library, cf. the module GraphAlgorithms
.
Author: Bernd Brassel
Version: July 2021
(:&)
:: Show a => ([(b,Int)],Int,a,[(b,Int)]) > Graph a b > Graph a b
(:&) takes a nodecontext and a Graph and yields a new graph. 
matchAny
:: Graph a b > (([(b,Int)],Int,a,[(b,Int)]),Graph a b)
decompose a graph into the Context for an arbitrarilychosen Node and the remaining Graph. 
empty
:: Graph a b
An empty Graph. 
mkGraph
:: Show a => [(Int,a)] > [(Int,Int,b)] > Graph a b
Create a Graph from the list of LNodes and LEdges. 
buildGr
:: Show a => [([(b,Int)],Int,a,[(b,Int)])] > Graph a b
Build a Graph from a list of Contexts. 
mkUGraph
:: [Int] > [(Int,Int)] > Graph () ()
Build a quasiunlabeled Graph from the list of Nodes and Edges. 
insNode
:: Show a => (Int,a) > Graph a b > Graph a b
Insert a LNode into the Graph. 
insEdge
:: Show a => (Int,Int,b) > Graph a b > Graph a b
Insert a LEdge into the Graph. 
delNode
:: Int > Graph a b > Graph a b
Remove a Node from the Graph. 
delEdge
:: Show a => (Int,Int) > Graph a b > Graph a b
Remove an Edge from the Graph. 
insNodes
:: Show a => [(Int,a)] > Graph a b > Graph a b
Insert multiple LNodes into the Graph. 
insEdges
:: Show a => [(Int,Int,b)] > Graph a b > Graph a b
Insert multiple LEdges into the Graph. 
delNodes
:: [Int] > Graph a b > Graph a b
Remove multiple Nodes from the Graph. 
delEdges
:: Show a => [(Int,Int)] > Graph a b > Graph a b
Remove multiple Edges from the Graph. 
isEmpty
:: Graph a b > Bool
test if the given Graph is empty. 
match
:: Int > Graph a b > (Maybe ([(b,Int)],Int,a,[(b,Int)]),Graph a b)
match is the complement side of (:&), decomposing a Graph into the MContext found for the given node and the remaining Graph. 
noNodes
:: Graph a b > Int
The number of Nodes in a Graph. 
nodeRange
:: Graph a b > (Int,Int)
The minimum and maximum Node in a Graph. 
context
:: Graph a b > Int > ([(b,Int)],Int,a,[(b,Int)])
Find the context for the given Node. 
lab
:: Graph a b > Int > Maybe a
Find the label for a Node. 
neighbors
:: Graph a b > Int > [Int]
Find the neighbors for a Node. 
suc
:: Graph a b > Int > [Int]
Find all Nodes that have a link from the given Node. 
pre
:: Graph a b > Int > [Int]
Find all Nodes that link to to the given Node. 
lsuc
:: Graph a b > Int > [(Int,b)]
Find all Nodes and their labels, which are linked from the given Node. 
lpre
:: Graph a b > Int > [(Int,b)]
Find all Nodes that link to the given Node and the label of each link. 
out
:: Graph a b > Int > [(Int,Int,b)]
Find all outwardbound LEdges for the given Node. 
inn
:: Graph a b > Int > [(Int,Int,b)]
Find all inwardbound LEdges for the given Node. 
outdeg
:: Graph a b > Int > Int
The outwardbound degree of the Node. 
indeg
:: Graph a b > Int > Int
The inwardbound degree of the Node. 
deg
:: Graph a b > Int > Int
The degree of the Node. 
gelem
:: Int > Graph a b > Bool
True if the Node is present in the Graph. 
equal
:: (Eq a, Eq b) => Graph a b > Graph a b > Bool
graph equality 
node'
:: ([(a,Int)],Int,b,[(a,Int)]) > Int
The Node in a Context. 
lab'
:: ([(a,Int)],Int,b,[(a,Int)]) > b
The label in a Context. 
labNode'
:: ([(a,Int)],Int,b,[(a,Int)]) > (Int,b)
The LNode from a Context. 
neighbors'
:: ([(a,Int)],Int,b,[(a,Int)]) > [Int]
All Nodes linked to or from in a Context. 
suc'
:: ([(a,Int)],Int,b,[(a,Int)]) > [Int]
All Nodes linked to in a Context. 
pre'
:: ([(a,Int)],Int,b,[(a,Int)]) > [Int]
All Nodes linked from in a Context. 
lpre'
:: ([(a,Int)],Int,b,[(a,Int)]) > [(Int,a)]
All Nodes linked from in a Context, and the label of the links. 
lsuc'
:: ([(a,Int)],Int,b,[(a,Int)]) > [(Int,a)]
All Nodes linked from in a Context, and the label of the links. 
out'
:: ([(a,Int)],Int,b,[(a,Int)]) > [(Int,Int,a)]
All outwarddirected LEdges in a Context. 
inn'
:: ([(a,Int)],Int,b,[(a,Int)]) > [(Int,Int,a)]
All inwarddirected LEdges in a Context. 
outdeg'
:: ([(a,Int)],Int,b,[(a,Int)]) > Int
The outward degree of a Context. 
indeg'
:: ([(a,Int)],Int,b,[(a,Int)]) > Int
The inward degree of a Context. 
deg'
:: ([(a,Int)],Int,b,[(a,Int)]) > Int
The degree of a Context. 
labNodes
:: Graph a b > [(Int,a)]
A list of all LNodes in the Graph. 
labEdges
:: Graph a b > [(Int,Int,b)]
A list of all LEdges in the Graph. 
nodes
:: Graph a b > [Int]
List all Nodes in the Graph. 
edges
:: Graph a b > [(Int,Int)]
List all Edges in the Graph. 
newNodes
:: Int > Graph a b > [Int]
List N available Nodes, ie Nodes that are not used in the Graph. 
ufold
:: (([(a,Int)],Int,b,[(a,Int)]) > c > c) > c > Graph b a > c
Fold a function over the graph. 
gmap
:: Show a => (([(b,Int)],Int,c,[(b,Int)]) > ([(d,Int)],Int,a,[(d,Int)])) > Graph c b > Graph a d
Map a function over the graph. 
nmap
:: Show a => (b > a) > Graph b c > Graph a c
Map a function over the Node labels in a graph. 
emap
:: Show a => (b > c) > Graph a b > Graph a c
Map a function over the Edge labels in a graph. 
labUEdges
:: [(a,b)] > [(a,b,())]
add label () to list of edges (node,node) 
labUNodes
:: [a] > [(a,())]
add label () to list of nodes 
showGraph
:: (Show a, Show b) => Graph a b > String
Represent Graph as String 
The type variables of Graph are i
nodeLabel/i
and i
edgeLabel/i
.
The internal representation of Graph is hidden.
Constructors:
Nodes and edges themselves (in contrast to their labels) are coded as integers.
For both of them, there are variants as labeled, unlabelwd and quasi unlabeled (labeled with ()).
Unlabeled node
Type synonym: Node = Int
Labeled node
Type synonym: LNode a = (Node,a)
Quasiunlabeled node
Type synonym: UNode = LNode ()
Unlabeled edge
Type synonym: Edge = (Node,Node)
Labeled edge
Type synonym: LEdge a = (Node,Node,a)
Quasiunlabeled edge
Type synonym: UEdge = LEdge ()
The context of a node is the node itself (along with label) and its adjacent nodes. Thus, a context is a quadrupel, for node n it is of the form (edges to n,node n,n's label,edges from n)
Type synonym: Context a b = (Adj b,Node,a,Adj b)
maybe context
Type synonym: MContext a b = Maybe (Context a b)
context with edges and node label only, without the node identifier itself
Type synonym: Context' a b = (Adj b,a,Adj b)
Unlabeled context.
Type synonym: UContext = ([Node],Node,[Node])
A graph decompostion is a context for a node n and the remaining graph without that node.
Type synonym: GDecomp a b = (Context a b,Graph a b)
a decomposition with a maybe context
Type synonym: Decomp a b = (MContext a b,Graph a b)
Unlabeled decomposition.
Type synonym: UDecomp a = (Maybe UContext,a)
Unlabeled path
Type synonym: Path = [Node]
Labeled path
Type synonym: LPath a = [LNode a]
Quasiunlabeled path
Type synonym: UPath = [UNode]
a graph without any labels
Type synonym: UGr = Graph () ()
(:&) takes a nodecontext and a Graph and yields a new graph. The according key idea is detailed at the beginning. nl is the type of the node labels and el the edge labels. Note that it is an error to induce a context for a node already contained in the graph.

decompose a graph into the Context for an arbitrarilychosen Node and the remaining Graph. In order to use graphs as abstract data structures, we also need means to decompose a graph. This decompostion should work as much like pattern matching as possible. The normal matching is done by the function matchAny, which takes a graph and yields a graph decompostion. According to the main idea, matchAny . (:&) should be an identity.

An empty Graph.

Build a quasiunlabeled Graph from the list of Nodes and Edges. 
match is the complement side of (:&), decomposing a Graph into the MContext found for the given node and the remaining Graph. 
The number of Nodes in a Graph.

Find the context for the given Node. In contrast to "match", "context" causes an error if the Node is not present in the Graph. 
Find all Nodes and their labels, which are linked from the given Node. 
Find all Nodes that link to the given Node and the label of each link. 

The label in a Context.



All Nodes linked from in a Context, and the label of the links. 
All Nodes linked from in a Context, and the label of the links. 
List N available Nodes, ie Nodes that are not used in the Graph. 
Fold a function over the graph. 
Map a function over the graph. 
Map a function over the Node labels in a graph. 
Map a function over the Edge labels in a graph. 
add label () to list of edges (node,node) 
add label () to list of nodes 