Adjacency set representation of directed graphs.

It is assumed that each target of an edge is also explicitly a vertex in the graph.

It is not generally required that graphs are irreflexive, but all concrete graphs that are defined in this module are irreflexive.

Undirected graphs are represented as symmetric directed graphs.

Directed Edge.

The adjacency sets of a graph.

The set of vertices of the graph.

The set edges of the graph.

The set of adjacent pairs of a graph.

The set of incident edges of a graph.

Insert an edge. Returns the graph unmodified if the edge is already in the graph. Non-existing vertices are added.

Construct a graph from a foldable structure of edges.

Insert a vertex. Returns the graph unmodified if the vertex is already in the graph.

Map a function over all vertices of a graph.

The union of two graphs.

Transpose a graph, i.e. reverse all edges of the graph.

Symmetric closure of a graph.

Construct a graph from adjacency lists.

Unsafely construct a graph from adjacency lists.

This function assumes that the input includes a adjacency list of each vertex that appears in a adjacency list of another vertex. Generally, fromList should be preferred.

A predicate that asserts that every target of an edge is also a vertex in the graph. Any graph that is constructed without using unsafe methods is guaranteed to satisfy this predicate.

Return whether two vertices are adjacent in a graph.

Return whether a graph is regular, i.e. whether all vertices have the same out-degree. Note that the latter implies that all vertices also have the same in-degree.

Return whether a graph is symmetric, i.e. whether for each edge \((a,b)\) there is also the edge \((b,a)\) in the graph.

Return whether a graph is irreflexive. A graph is irreflexive if for each edge \((a,b)\) it holds that \(a \neq b\), i.e there are no self-loops in the graph.

Return whether an edge is contained in a graph.

Return whether a vertex is contained in a graph.

The order of a graph is the number of vertices.

Directed Size. This the number of edges of the graph.

Directed Size. This the number of edges of the graph.

Undirected Size of a graph. This is the number of edges of the symmetric closure of the graph.

The number of outgoing edges of vertex in a graph.

The number of incoming edges of vertex in a graph.

The maximum out-degree of the vertices of a graph.

The maximum in-degree of the vertices of a graph.

The minimum out-degree of the vertices of a graph.

The minimum in-degree of the vertices of a graph.

The shortest path matrix of a graph.

The shortest path matrix of a graph can be used to efficiently query the distance and shortest path between any two vertices of the graph. It can also be used to efficiently compute the diameter of the graph.

Computing the shortest path matrix is expensive for larger graphs. The matrix is computed using the Floyd-Warshall algorithm. The space and time complexity is quadratic in the order of the graph. For sparse graphs there are more efficient algorithms for computing distances and shortest paths between the nodes of the graph.

Compute the shortest path matrix for a graph. The result can be used to efficiently query the distance and shortest path between any two vertices of the graph. It can also be used to efficiently compute the diameter of the graph.

Compute the shortest path between two vertices of a graph.

| This is expensive for larger graphs. If more than one path is needed one should use shortestPathCache to cache the result of the search and use shortestPath_ to query paths from the cache.

The algorithm is optimized for dense graphs. For large sparse graphs a more efficient algorithm should be used.

Compute the shortest path between two vertices from the shortest path matrix of a graph.

The algorithm is optimized for dense graphs. For large sparse graphs a more efficient algorithm should be used.

Compute the distance between two vertices of a graph.

| This is expensive for larger graphs. If more than one distance is needed one should use shortestPathCache to cache the result of the search and use distance_ to query paths from the cache.

The algorithm is optimized for dense graphs. For large sparse graphs a more efficient algorithm should be used.

Compute the distance between two vertices from the shortest path matrix of a graph.

The algorithm is optimized for dense graphs. For large sparse graphs a more efficient algorithm should be used.

Compute the Diameter of a graph, i.e. the maximum length of a shortest path between two vertices in the graph.

This is expensive to compute for larger graphs. If also the shortest paths or distances are needed, one should use shortestPathCache to cache the result of the search and use the diameter_, shortestPath_, and distance_ to query the respective results from the cache.

The algorithm is optimized for dense graphs. For large sparse graphs a more efficient algorithm should be used.

Compute the Diameter of a graph from a shortest path matrix. The diameter of a graph is the maximum length of a shortest path between two vertices in the graph.

The empty graph on n nodes. This is the graph of order n and size 0.

The (irreflexive) singleton graph.

Undirected clique.

Undirected pair.

Undirected triangle.

Undirected cycle.

Directed cycle.

Undirected line.

Directed line.

The Peterson graph.

• order: 10
• size: 30
• degree: 3
• diameter: 2

The "twenty chain" graph.

• order: 20
• size: 60
• degree: 3
• diameter: 3

Hoffman-Singleton Graph.

The Hoffman-Singleton graph is a 7-regular graph with 50 vertices and 175 edges. It's the largest graph of max-degree 7 and diameter 2. Cf. [https:/en.wikipedia.orgwiki/Hoffman–Singleton_graph]()