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 ugraphs.pl -- Graph manipulation library

The S-representation of a graph is a list of (vertex-neighbours) pairs, where the pairs are in standard order (as produced by keysort) and the neighbours of each vertex are also in standard order (as produced by sort). This form is convenient for many calculations.

A new UGraph from raw data can be created using vertices_edges_to_ugraph/3.

Adapted to support some of the functionality of the SICStus ugraphs library by Vitor Santos Costa.

Ported from YAP 5.0.1 to SWI-Prolog by Jan Wielemaker.

author
- R.A.O'Keefe
- Vitor Santos Costa
- Jan Wielemaker
- BSD-2 or Artistic 2.0
vertices(+Graph, -Vertices)
Unify Vertices with all vertices appearing in Graph. Example:
```?- vertices([1-[3,5],2-[4],3-[],4-[5],5-[]], L).
L = [1, 2, 3, 4, 5]```
vertices_edges_to_ugraph(+Vertices, +Edges, -UGraph) is det
Create a UGraph from Vertices and edges. Given a graph with a set of Vertices and a set of Edges, Graph must unify with the corresponding S-representation. Note that the vertices without edges will appear in Vertices but not in Edges. Moreover, it is sufficient for a vertice to appear in Edges.
```?- vertices_edges_to_ugraph([],[1-3,2-4,4-5,1-5], L).
L = [1-[3,5], 2-[4], 3-[], 4-[5], 5-[]]```

In this case all vertices are defined implicitly. The next example shows three unconnected vertices:

```?- vertices_edges_to_ugraph([6,7,8],[1-3,2-4,4-5,1-5], L).
L = [1-[3,5], 2-[4], 3-[], 4-[5], 5-[], 6-[], 7-[], 8-[]]```
Unify NewGraph with a new graph obtained by adding the list of Vertices to Graph. Example:
```?- add_vertices([1-[3,5],2-[]], [0,1,2,9], NG).
NG = [0-[], 1-[3,5], 2-[], 9-[]]```
del_vertices(+Graph, +Vertices, -NewGraph) is det
Unify NewGraph with a new graph obtained by deleting the list of Vertices and all the edges that start from or go to a vertex in Vertices to the Graph. Example:
```?- del_vertices([1-[3,5],2-[4],3-[],4-[5],5-[],6-[],7-[2,6],8-[]],
[2,1],
NL).
NL = [3-[],4-[5],5-[],6-[],7-[6],8-[]]```
Compatibility
- Upto 5.6.48 the argument order was (+Vertices, +Graph, -NewGraph). Both YAP and SWI-Prolog have changed the argument order for compatibility with recent SICStus as well as consistency with del_edges/3.
Unify NewGraph with a new graph obtained by adding the list of Edges to Graph. Example:
```?- add_edges([1-[3,5],2-[4],3-[],4-[5],
5-[],6-[],7-[],8-[]],
[1-6,2-3,3-2,5-7,3-2,4-5],
NL).
NL = [1-[3,5,6], 2-[3,4], 3-[2], 4-[5],
5-[7], 6-[], 7-[], 8-[]]```
ugraph_union(+Graph1, +Graph2, -NewGraph)
NewGraph is the union of Graph1 and Graph2. Example:
```?- ugraph_union([1-[2],2-[3]],[2-[4],3-[1,2,4]],L).
L = [1-[2], 2-[3,4], 3-[1,2,4]]```
del_edges(+Graph, +Edges, -NewGraph)
Unify NewGraph with a new graph obtained by removing the list of Edges from Graph. Notice that no vertices are deleted. Example:
```?- del_edges([1-[3,5],2-[4],3-[],4-[5],5-[],6-[],7-[],8-[]],
[1-6,2-3,3-2,5-7,3-2,4-5,1-3],
NL).
NL = [1-[5],2-[4],3-[],4-[],5-[],6-[],7-[],8-[]]```
edges(+Graph, -Edges)
Unify Edges with all edges appearing in Graph. Example:
```?- edges([1-[3,5],2-[4],3-[],4-[5],5-[]], L).
L = [1-3, 1-5, 2-4, 4-5]```
transitive_closure(+Graph, -Closure)
Generate the graph Closure as the transitive closure of Graph. Example:
```?- transitive_closure([1-[2,3],2-[4,5],4-[6]],L).
L = [1-[2,3,4,5,6], 2-[4,5,6], 4-[6]]```
transpose_ugraph(Graph, NewGraph) is det
Unify NewGraph with a new graph obtained from Graph by replacing all edges of the form V1-V2 by edges of the form V2-V1. The cost is O(|V|*log(|V|)). Notice that an undirected graph is its own transpose. Example:
```?- transpose([1-[3,5],2-[4],3-[],4-[5],
5-[],6-[],7-[],8-[]], NL).
NL = [1-[],2-[],3-[1],4-[2],5-[1,4],6-[],7-[],8-[]]```
Compatibility
- This predicate used to be known as transpose/2. Following SICStus 4, we reserve transpose/2 for matrix transposition and renamed ugraph transposition to transpose_ugraph/2.
compose(+LeftGraph, +RightGraph, -NewGraph)
Compose NewGraph by connecting the drains of LeftGraph to the sources of RightGraph. Example:
```?- compose([1-[2],2-[3]],[2-[4],3-[1,2,4]],L).
L = [1-[4], 2-[1,2,4], 3-[]]```
ugraph_layers(Graph, -Layers) is semidet
top_sort(+Graph, -Sorted) is semidet
Sort vertices topologically. Layers is a list of lists of vertices where there are no edges from a layer to an earlier layer. The predicate top_sort/2 flattens the layers using append/2.

These predicates fail if Graph is cyclic. If Graph is not connected, the sub-graphs are individually sorted, where the root of each subgraph is in the first layer, the nodes connected to the roots in the second, etc.

```?- top_sort([1-[2], 2-[3], 3-[]], L).
L = [1, 2, 3]```
Compatibility
- The original version of this library provided top_sort/3 as a difference list version of top_sort/2. We removed this because the argument order was non-standard. Fixing causes hard to debug compatibility issues while we expect top_sort/3 was rarely used. A backward compatible top_sort/3 can be defined as
```top_sort(Graph, Tail, Sorted) :-
top_sort(Graph, Sorted0),
append(Sorted0, Tail, Sorted).```

The original version returned all vertices in a layer in reverse order. The current one returns them in standard order of terms, i.e., each layer is an ordered set.

- ugraph_layers/2 is a SWI-Prolog specific addition to this library.
neighbors(+Vertex, +Graph, -Neigbours) is det
neighbours(+Vertex, +Graph, -Neigbours) is det
Neigbours is a sorted list of the neighbours of Vertex in Graph. Example:
```?- neighbours(4,[1-[3,5],2-[4],3-[],
4-[1,2,7,5],5-[],6-[],7-[],8-[]], NL).
NL = [1,2,7,5]```
connect_ugraph(+UGraphIn, -Start, -UGraphOut) is det
Adds Start as an additional vertex that is connected to all vertices in UGraphIn. This can be used to create an topological sort for a not connected graph. Start is before any vertex in UGraphIn in the standard order of terms. No vertex in UGraphIn can be a variable.

Can be used to order a not-connected graph as follows:

```top_sort_unconnected(Graph, Vertices) :-
(   top_sort(Graph, Vertices)
->  true
;   connect_ugraph(Graph, Start, Connected),
top_sort(Connected, Ordered0),
Ordered0 = [Start|Vertices]
).```
complement(+UGraphIn, -UGraphOut)
UGraphOut is a ugraph with an edge between all vertices that are not connected in UGraphIn and all edges from UGraphIn removed. Example:
```?- complement([1-[3,5],2-[4],3-[],
4-[1,2,7,5],5-[],6-[],7-[],8-[]], NL).
NL = [1-[2,4,6,7,8],2-[1,3,5,6,7,8],3-[1,2,4,5,6,7,8],
4-[3,5,6,8],5-[1,2,3,4,6,7,8],6-[1,2,3,4,5,7,8],
7-[1,2,3,4,5,6,8],8-[1,2,3,4,5,6,7]]```
To be done
- Simple two-step algorithm. You could be smarter, I suppose.
reachable(+Vertex, +UGraph, -Vertices)
True when Vertices is an ordered set of vertices reachable in UGraph, including Vertex. Example:
```?- reachable(1,[1-[3,5],2-[4],3-[],4-[5],5-[]],V).
V = [1, 3, 5]```

## Re-exported predicates

The following predicates are exported from this file while their implementation is defined in imported modules or non-module files loaded by this module.

ugraph_layers(Graph, -Layers) is semidet
top_sort(+Graph, -Sorted) is semidet
Sort vertices topologically. Layers is a list of lists of vertices where there are no edges from a layer to an earlier layer. The predicate top_sort/2 flattens the layers using append/2.

These predicates fail if Graph is cyclic. If Graph is not connected, the sub-graphs are individually sorted, where the root of each subgraph is in the first layer, the nodes connected to the roots in the second, etc.

```?- top_sort([1-[2], 2-[3], 3-[]], L).
L = [1, 2, 3]```
Compatibility
- The original version of this library provided top_sort/3 as a difference list version of top_sort/2. We removed this because the argument order was non-standard. Fixing causes hard to debug compatibility issues while we expect top_sort/3 was rarely used. A backward compatible top_sort/3 can be defined as
```top_sort(Graph, Tail, Sorted) :-
top_sort(Graph, Sorted0),
append(Sorted0, Tail, Sorted).```

The original version returned all vertices in a layer in reverse order. The current one returns them in standard order of terms, i.e., each layer is an ordered set.

- ugraph_layers/2 is a SWI-Prolog specific addition to this library.
neighbors(+Vertex, +Graph, -Neigbours) is det
neighbours(+Vertex, +Graph, -Neigbours) is det
Neigbours is a sorted list of the neighbours of Vertex in Graph. Example:
```?- neighbours(4,[1-[3,5],2-[4],3-[],
4-[1,2,7,5],5-[],6-[],7-[],8-[]], NL).
NL = [1,2,7,5]```