Implementation and documentation are copied from YAP 5.0.1. The`library(ugraph)`

library is based on code originally written by Richard O'Keefe. The code was then extended to be compatible with the SICStus Prolog ugraphs library. Code and documentation have been cleaned and style has been changed to be more in line with the rest of SWI-Prolog.

The ugraphs library was originally released in the public domain. YAP is convered by the Perl artistic license, which does not imply further restrictions on the SWI-Prolog LGPL license.

The routines assume directed graphs, undirected graphs may be implemented by using two edges.

Originally graphs where represented in two formats. The SICStus
library and this version of `library(ugraphs.pl)`

only uses
the
*S-representation*. The S-representation of a graph is a list of
(vertex-neighbors) pairs, where the pairs are in standard order (as
produced by keysort) and the neighbors of each vertex are also in
standard order (as produced by sort). This form is convenient for many
calculations. Each vertex appears in the S-representation, also if it
has no neighbors.

**vertices_edges_to_ugraph**(`+Vertices, +Edges, -Graph`)-
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-[]] ?

**vertices**(`+Graph, -Vertices`)-
Unify
`Vertices`with all vertices appearing in graph`Graph`. Example:?- vertices([1-[3,5],2-[4],3-[],4-[5],5-[]], L). L = [1, 2, 3, 4, 5]

**edges**(`+Graph, -Edges`)-
Unify
`Edges`with all edges appearing in`Graph`. In the next example:?- edges([1-[3,5],2-[4],3-[],4-[5],5-[]], L). L = [1-3, 1-5, 2-4, 4-5]

**add_vertices**(`+Graph, +Vertices, -NewGraph`)-
Unify
`NewGraph`with a new graph obtained by adding the list of`Vertices`to the`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`)-
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([2,1], [1-[3,5],2-[4],3-[],4-[5],5-[],6-[],7-[2,6],8-[]], NL). NL = [3-[],4-[5],5-[],6-[],7-[6],8-[]]

**add_edges**(`+Graph, +Edges, -NewGraph`)-
Unify
`NewGraph`with a new graph obtained by adding the list of edges`Edges`to the graph`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-[]]

**del_edges**(`+Graph, +Edges, -NewGraph`)-
Unify
`NewGraph`with a new graph obtained by removing the list of`Edges`from the`Graph`. Notice that no vertices are deleted. In the next 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-[]]

**transpose**(`+Graph, -NewGraph`)-
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|^2)`. 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-[]]

**neighbours**(`+Vertex, +Graph, -Vertices`)-
Unify
`Vertices`with the list of neighbours of vertex`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]

**neighbors**(`+Vertex, +Graph, -Vertices`)- American version of neighbours/3.
**complement**(`+Graph, -NewGraph`)-
Unify
`NewGraph`with the graph complementary to`Graph`. 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]]

**compose**(`+LeftGraph, +RightGraph, -NewGraph`)-
Compose, 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_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]]

**top_sort**(`+Graph, -Sort`)-
Generate the set of nodes
`Sort`as a topological sorting of graph`Graph`, if one is possible. A toplogical sort is possible if the graph is connected and acyclic. In the example we show how topological sorting works for a linear graph:?- top_sort([1-[2], 2-[3], 3-[]], L). L = [1, 2, 3]

**top_sort**(`+Graph, -Sort0, -Sort`)-
Generate the difference list Sort-Sort0 as a topological sorting of
graph
`Graph`, if one is possible. **transitive_closure**(`+Graph, -Closure`)-
Generate the graph Closure as the transitive closure of graph
`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]]

**reachable**(`+Vertex, +Graph, -Vertices`)-
Unify
`Vertices`with the set of all vertices in graph`Graph`that are reachable from`Vertex`. Example:?- reachable(1,[1-[3,5],2-[4],3-[],4-[5],5-[]],V). V = [1, 3, 5]