### 10.29 Unweighted Graph Operations—`library(ugraphs)`

This library module provides operations on directed graphs. An unweighted directed graph (ugraph) is represented as a list of (vertex-neighbors) pairs, where the pairs are in standard order (as produced by `keysort/2` with unique keys) and the neighbors of each vertex are also in standard order (as produced by `sort/2`), and every neighbor appears as a vertex even if it has no neighbors itself.

An undirected graph is represented as a directed graph where for each edge (U,V) there is a symmetric edge (V,U).

An edge (U,V) is represented as the term U-V.

A vertex can be any term. Two vertices are distinct iff they are not identical (`==`).

A path is represented as a list of vertices. No vertex can appear twice in a path.

Exported predicates:

`vertices_edges_to_ugraph(`+Vertices`, `+Edges`, `-Graph`)`
is true if Vertices is a list of vertices, Edges is a list of edges, and Graph is a graph built from Vertices and Edges. Vertices and Edges may be in any order. The vertices mentioned in Edges do not have to occur explicitly in Vertices. Vertices may be used to specify vertices that are not connected to any edges.
`vertices(`+Graph`, `-Vertices`)`
unifies Vertices with the vertices in Graph. Could be defined as:
```          vertices(Graph, Vertices) :-
(   foreach(V-_,Graph),
foreach(V,Vertices)
do  true
).
```

`edges(`+Graph`, `-Edges`)`
unifies Edges with the edges in Graph. Could be defined as:
```          edges(Graph, Edges) :-
(   foreach(V1-Neibs,Graph),
fromto(Edges,S0,S,[])
do  (   foreach(V2,Neibs),
param(V1),
fromto(S0,[V1-V2|S1],S1,S)
do  true
)
).
```

`add_vertices(`+Graph1`, `+Vertices`, `-Graph2`)`
is true if Graph2 is Graph1 with Vertices added to it.
`del_vertices(`+Graph1`, `+Vertices`, `-Graph2`)`
is true if Graph2 is Graph1 with Vertices and all edges to and from Vertices removed from it.
`add_edges(`+Graph1`, `+Edges`, `-Graph2`)`
is true if Graph2 is Graph1 with Edges and their "to" and "from" vertices added to it.
`del_edges(`+Graph1`, `+Edges`, `-Graph2`)`
is true if Graph2 is Graph1 with Edges removed from it.
`transpose_ugraph(`+Graph`, `-Transpose`)`
is true if Transpose is the graph computed by replacing each edge (u,v) in Graph by its symmetric edge (v,u). It can only be used one way around. The cost is O(N log N).
`neighbors(`+Vertex`, `+Graph`, `-Neighbors`)`
`neighbours(`+Vertex`, `+Graph`, `-Neighbors`)`
is true if Vertex is a vertex in Graph and Neighbors are its neighbors.
`complement(`+Graph`, `-Complement`)`
Complement is the complement graph of Graph, i.e. the graph that has the same vertices as Graph but only the edges that are not in Graph.
`compose(`+G1`, `+G2`, `-Composition`)`
computes Composition as the composition of two graphs, which need not have the same set of vertices.
`transitive_closure(`+Graph`, `-Closure`)`
computes Closure as the transitive closure of Graph in O(N^3) time.
`symmetric_closure(`+Graph`, `-Closure`)`
computes Closure as the symmetric closure of Graph, i.e. for each edge (u,v) in Graph, add its symmetric edge (v,u). Approx. O(N log N) time. This is useful for making a directed graph undirected. Could be defined as:
```          symmetric_closure(Graph, Closure) :-
transpose_ugraph(Graph, Transpose),
(   foreach(V-Neibs1,Graph),
foreach(V-Neibs2,Transpose),
foreach(V-Neibs,Closure)
do  ord_union(Neibs1, Neibs2, Neibs)
).
```

`top_sort(`+Graph`, `-Sorted`)`
finds a topological ordering of Graph and returns the ordering as a list of Sorted vertices. Fails iff no ordering exists, i.e. iff the graph contains cycles. Approx. O(N log N) time.
`max_path(`+V1`, `+V2`, `+Graph`, `-Path`, `-Cost`)`
is true if Path is a list of vertices constituting a longest path of cost Cost from V1 to V2 in Graph, there being no cyclic paths from V1 to V2. Takes O(N^2) time.
`min_path(`+V1`, `+V2`, `+Graph`, `-Path`, `-Length`)`
is true if Path is a list of vertices constituting a shortest path of length Length from V1 to V2 in Graph. Takes O(N^2) time.
`min_paths(`+Vertex`, `+Graph`, `-Tree`)`
is true if Tree is a tree of all the shortest paths from Vertex to every other vertex in Graph. This is the single-source shortest paths problem. The algorithm is straightforward.
`path(`+Vertex`, `+Graph`, `-Path`)`
is given a Graph and a Vertex of that Graph, and returns a maximal Path rooted at Vertex, enumerating more Paths on backtracking.
`reduce(`+Graph`, `-Reduced`)`
is true if Reduced is the reduced graph for Graph. The vertices of the reduced graph are the strongly connected components of Graph. There is an edge in Reduced from u to v iff there is an edge in Graph from one of the vertices in u to one of the vertices in v. A strongly connected component is a maximal set of vertices where each vertex has a path to every other vertex. Algorithm from "Algorithms" by Sedgewick, page 482, Tarjan's algorithm.
`reachable(`+Vertex`, `+Graph`, `-Reachable`)`
is given a Graph and a Vertex of that Graph, and returns the set of vertices that are Reachable from that Vertex. Takes O(N^2) time.
`random_ugraph(`+P`, `+N`, `-Graph`)`
where P is a probability, unifies Graph with a random graph of N vertices where each possible edge is included with probability P.
`min_tree(`+Graph`, `-Tree`, `-Cost`)`
is true if Tree is a spanning tree of an undirected Graph with cost Cost, if it exists. Using a version of Prim's algorithm.

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