# libgraph v0.13.3 Graph

This module defines a graph data structure, which supports directed and undirected graphs, in both acyclic and cyclic forms. It also defines the API for creating, manipulating, and querying that structure.

As far as memory usage is concerned, `Graph`

should be fairly compact in memory, but if you want to do a rough
comparison between the memory usage for a graph between `libgraph`

and `digraph`

, use `:digraph.info/1`

and
`Graph.info/1`

on the two graphs, and both results will contain memory usage information. Keep in mind we don’t have a precise
way to measure the memory usage of a term in memory, whereas ETS is able to give a more precise answer, but we do have
a fairly good way to estimate the usage of a term, and we use that method within `libgraph`

.

The Graph struct is structured like so:

- A map of vertex ids to vertices (
`vertices`

) - A map of vertex ids to their out neighbors (
`out_edges`

), - A map of vertex ids to their in neighbors (
`in_edges`

), effectively the transposition of`out_edges`

- A map of vertex ids to vertex labels (
`vertex_labels`

), (labels are only stored if a non-nil label was provided) - A map of edge ids (where an edge id is simply a tuple of
`{vertex_id, vertex_id}`

) to a map of edge metadata (`edges`

) - Edge metadata is a map of
`label => weight`

, and each entry in that map represents a distinct edge. This allows us to support multiple edges in the same direction between the same pair of vertices, but for many purposes simply treat them as a single logical edge.

This structure is designed to be as efficient as possible once a graph is built, but it turned out that it is also quite efficient for manipulating the graph as well. For example, splitting an edge and introducing a new vertex on that edge can be done with very little effort. We use vertex ids everywhere because we can generate them without any lookups, we don’t incur any copies of the vertex structure, and they are very efficient as keys in a map.

# Link to this section Summary

## Functions

Gets the shortest path between `a`

and `b`

Like `add_edge/3`

or `add_edge/4`

, but takes a `Graph.Edge`

struct created with
`Graph.Edge.new/2`

or `Graph.Edge.new/3`

Adds an edge connecting `v1`

to `v2`

. If either `v1`

or `v2`

do not exist in the graph,
they are automatically added. Adding the same edge more than once does not create multiple edges,
each edge is only ever stored once

This function is like `add_edge/3`

, but for multiple edges at once, it also accepts edge specifications
in a few different ways to make it easy to generate graphs succinctly

Adds a new vertex to the graph. If the vertex is already present in the graph, the add is a no-op

Like `add_vertex/2`

, but takes a list of vertices to add to the graph

Returns the root vertex of the arborescence, if one exists, otherwise nil

Detects all maximal cliques in the provided graph

Returns a list of connected components, where each component is a list of vertices

Calculates the k-coreness of vertex `v`

in graph `g`

Determines the k-degeneracy of the given graph

Calculates the degeneracy core of a given graph

Returns the degree of vertex `v`

of graph `g`

Removes all edges connecting `v1`

to `v2`

, regardless of label

Removes an edge connecting `v1`

to `v2`

. A label can be specified to disambiguate the
specific edge you wish to delete, if not provided, the unlabelled edge, if one exists,
will be removed

Like `delete_edge/3`

, but takes a list of edge specifications, and deletes the corresponding
edges from the graph, if they exist

This function can be used to remove all edges between `v1`

and `v2`

. This is useful if
you are defining multiple edges between vertices to represent different relationships, but
want to remove them all as if they are a single unit

Removes a vertex from the graph, as well as any edges which refer to that vertex. If the vertex does not exist in the graph, it is a no-op

Like `delete_vertex/2`

, but takes a list of vertices to delete from the graph

Gets the shortest path between `a`

and `b`

Get an Edge struct for a specific vertex pair, or vertex pair + label

Return a list of all the edges, where each edge is expressed as a tuple
of `{A, B}`

, where the elements are the vertices involved, and implying the
direction of the edge to be from `A`

to `B`

Returns a list of all edges inbound or outbound from vertex `v`

Returns a list of all edges between `v1`

and `v2`

Builds a list of paths between vertex `a`

and vertex `b`

See `dijkstra/1`

Returns true if the given vertex exists in the graph. Otherwise false

Returns the in-degree of vertex `v`

of graph `g`

Returns a list of `Graph.Edge`

structs representing the in edges to vertex `v`

Returns a list of vertices which all have edges coming in to the given vertex `v`

Returns a map of summary information about this graph

Returns true if and only if the graph `g`

is acyclic

Returns true if the graph is an aborescence, a directed acyclic graph,
where the *root*, a vertex, of the arborescence has a unique path from itself
to every other vertex in the graph

Returns true if the graph `g`

is not acyclic

Returns true if graph `g1`

is a subgraph of `g2`

Returns true if and only if the graph `g`

is a tree

Detects all maximal cliques of degree `k`

Calculates the k-core for a given graph and value of `k`

Groups all vertices by their k-coreness into a single map

Updates the labels for the given vertex

Returns a list of vertices from graph `g`

which are included in a loop, where a loop is a cycle of length 1

Return all neighboring vertices of the given vertex

Creates a new directed graph

Creates a new graph using the provided options

Returns the number of edges in the graph

Returns the number of vertices in the graph

Returns the out-degree of vertex `v`

of graph `g`

Returns a list of `Graph.Edge`

structs representing the out edges from vertex `v`

Returns a list of vertices which the given vertex `v`

has edges going to

Returns all vertices of graph `g`

. The order is given by a depth-first traversal of the graph,
collecting visited vertices in postorder. More precisely, the vertices visited while searching from an
arbitrarily chosen vertex are collected in postorder, and all those collected vertices are placed before
the subsequently visited vertices

Returns all vertices of graph `g`

. The order is given by a depth-first traversal of the graph,
collecting visited vertices in preorder

Returns an unsorted list of vertices from the graph, such that for each vertex in the list (call it `v`

),
there is a path in the graph from some vertex of `vs`

to `v`

Returns an unsorted list of vertices from the graph, such that for each vertex in the list (call it `v`

),
there is a path in the graph of length one or more from some vertex of `vs`

to `v`

Returns an unsorted list of vertices from the graph, such that for each vertex in the list (call it `v`

),
there is a path from `v`

to some vertex of `vs`

Returns an unsorted list of vertices from the graph, such that for each vertex in the list (call it `v`

),
there is a path of length one or more from `v`

to some vertex of `vs`

Replaces `vertex`

with `new_vertex`

in the graph

Splits the edges between `v1`

and `v2`

by inserting a new vertex, `v3`

, deleting
the edges between `v1`

and `v2`

, and inserting new edges from `v1`

to `v3`

and from
`v3`

to `v2`

Returns a list of strongly connected components, where each component is a list of vertices

Builds a maximal subgraph of `g`

which includes all of the vertices in `vs`

and the edges which connect them

Converts the given Graph to DOT format, which can then be converted to
a number of other formats via Graphviz, e.g. `dot -Tpng out.dot > out.png`

Returns a topological ordering of the vertices of graph `g`

, if such an ordering exists, otherwise it returns false.
For each vertex in the returned list, no out-neighbors occur earlier in the list

The transposition of a graph is another graph with the direction of all the edges reversed

Given two vertices, this function updates the metadata (weight/label) for the unlabelled edge between those two vertices. If no unlabelled edge exists between them, an error tuple is returned. If you set a label, the unlabelled edge will be replaced with a new labelled edge

Like `update_edge/4`

, but requires you to specify the labelled edge to update

Returns the label for the given vertex. If no label was assigned, it returns []

Returns a list of all the vertices in the graph

# Link to this section Types

# Link to this section Functions

Gets the shortest path between `a`

and `b`

.

The A* algorithm is very much like Dijkstra’s algorithm, except in addition to edge weights, A*
also considers a heuristic function for determining the lower bound of the cost to go from vertex
`v`

to `b`

. The lower bound *must* be less than the cost of the shortest path from `v`

to `b`

, otherwise
it will do more harm than good. Dijkstra’s algorithm can be reframed as A* where `lower_bound(v)`

is always 0.

This function puts the heuristics in your hands, so you must provide the heuristic function, which should take
a single parameter, `v`

, which is the vertex being currently examined. Your heuristic should then determine what the
lower bound for the cost to reach `b`

from `v`

is, and return that value.

## Example

```
iex> g = Graph.new |> Graph.add_edges([{:a, :b}, {:b, :c}, {:c, :d}, {:b, :d}])
...> Graph.a_star(g, :a, :d, fn _ -> 0 end)
[:a, :b, :d]
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c, :d])
...> g = Graph.add_edges(g, [{:a, :c}, {:b, :c}, {:b, :d}])
...> Graph.a_star(g, :a, :d, fn _ -> 0 end)
nil
```

Like `add_edge/3`

or `add_edge/4`

, but takes a `Graph.Edge`

struct created with
`Graph.Edge.new/2`

or `Graph.Edge.new/3`

.

## Example

```
iex> g = Graph.new |> Graph.add_edge(Graph.Edge.new(:a, :b))
...> [:a, :b] = Graph.vertices(g)
...> Graph.edges(g)
[%Graph.Edge{v1: :a, v2: :b}]
```

Adds an edge connecting `v1`

to `v2`

. If either `v1`

or `v2`

do not exist in the graph,
they are automatically added. Adding the same edge more than once does not create multiple edges,
each edge is only ever stored once.

Edges have a default weight of 1, and an empty (nil) label. You can change this by passing options to this function, as shown below.

## Example

```
iex> g = Graph.new |> Graph.add_edge(:a, :b)
...> [:a, :b] = Graph.vertices(g)
...> Graph.edges(g)
[%Graph.Edge{v1: :a, v2: :b, label: nil, weight: 1}]
iex> g = Graph.new |> Graph.add_edge(:a, :b, label: :foo, weight: 2)
...> [:a, :b] = Graph.vertices(g)
...> Graph.edges(g)
[%Graph.Edge{v1: :a, v2: :b, label: :foo, weight: 2}]
```

add_edges(t(), [Graph.Edge.t()] | Enumerable.t()) :: t() | no_return()

This function is like `add_edge/3`

, but for multiple edges at once, it also accepts edge specifications
in a few different ways to make it easy to generate graphs succinctly.

Edges must be provided as a list of `Edge`

structs, `{vertex, vertex}`

pairs, or
`{vertex, vertex, edge_opts :: [label: term, weight: integer]}`

.

See the docs for `Graph.Edge.new/2`

or `Graph.Edge.new/3`

for more info on creating Edge structs, and
`add_edge/3`

for information on edge options.

If an invalid edge specification is provided, raises `Graph.EdgeSpecificationError`

.

## Examples

```
iex> alias Graph.Edge
...> edges = [Edge.new(:a, :b), Edge.new(:b, :c, weight: 2)]
...> g = Graph.new |> Graph.add_vertices([:a, :b, :c]) |> Graph.add_edges(edges)
...> Graph.edges(g)
[%Graph.Edge{v1: :a, v2: :b}, %Graph.Edge{v1: :b, v2: :c, weight: 2}]
iex> g = Graph.new |> Graph.add_edges([{:a, :b}, {:a, :b, label: :foo}, {:a, :b, label: :foo, weight: 2}])
...> Graph.edges(g)
[%Graph.Edge{v1: :a, v2: :b, label: :foo, weight: 2}, %Graph.Edge{v1: :a, v2: :b}]
iex> Graph.new |> Graph.add_vertices([:a, :b, :c]) |> Graph.add_edges([:a, :b])
** (Graph.EdgeSpecificationError) Expected a valid edge specification, but got: :a
```

Adds a new vertex to the graph. If the vertex is already present in the graph, the add is a no-op.

You can provide optional labels for the vertex, aside from the variety of uses this has for working with graphs, labels will also be used when exporting a graph in DOT format.

## Example

```
iex> g = Graph.new |> Graph.add_vertex(:a, :mylabel) |> Graph.add_vertex(:a)
...> [:a] = Graph.vertices(g)
...> Graph.vertex_labels(g, :a)
[:mylabel]
iex> g = Graph.new |> Graph.add_vertex(:a, [:mylabel, :other])
...> Graph.vertex_labels(g, :a)
[:mylabel, :other]
```

Like `add_vertex/2`

, but takes a list of vertices to add to the graph.

## Example

```
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :a])
...> Graph.vertices(g)
[:a, :b]
```

Returns the root vertex of the arborescence, if one exists, otherwise nil.

Detects all maximal cliques in the provided graph.

Returns a list of cliques, where each clique is a list of vertices in the clique.

A clique is a subset `vs`

of the vertices in the given graph, which together form a complete graph;
or put another way, every vertex in `vs`

is connected to all other vertices in `vs`

.

Returns a list of connected components, where each component is a list of vertices.

A *connected component* is a maximal subgraph such that there is a path between each pair of vertices,
considering all edges undirected.

A *subgraph* is a graph whose vertices and edges are a subset of the vertices and edges of the source graph.

A *maximal subgraph* is a subgraph with property `P`

where all other subgraphs which contain the same vertices
do not have that same property `P`

.

## Example

```
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c, :d])
...> g = Graph.add_edges(g, [{:a, :b}, {:a, :c}, {:b, :c}, {:c, :d}, {:c, :a}])
...> Graph.components(g)
[[:d, :b, :c, :a]]
```

Calculates the k-coreness of vertex `v`

in graph `g`

.

The k-coreness of a vertex is defined as the maximum value of `k`

for which `v`

is found in the corresponding k-core of graph `g`

.

NOTE: This function decomposes all k-core components to determine the coreness
of a vertex - if you will be trying to determine the coreness of many vertices,
it is recommended to use `k_core_components/1`

and then lookup the coreness of a vertex
by querying the resulting map.

Determines the k-degeneracy of the given graph.

The degeneracy of graph `g`

is the maximum value of `k`

for which a k-core
exists in graph `g`

.

Calculates the degeneracy core of a given graph.

The degeneracy core of a graph is the k-core of the graph where the
value of `k`

is the degeneracy of the graph. The degeneracy of a graph
is the highest value of `k`

which has a non-empty k-core in the graph.

Returns the degree of vertex `v`

of graph `g`

.

The degree of a vertex is the total number of edges containing that vertex.

For directed graphs this is the same as the sum of the in-degree and out-degree of the given vertex. For undirected graphs, the in-degree and out-degree are always the same.

## Example

```
iex> g = Graph.new(type: :undirected) |> Graph.add_vertices([:a, :b, :c]) |> Graph.add_edge(:a, :b)
...> Graph.degree(g, :b)
1
iex> g = Graph.new() |> Graph.add_vertices([:a, :b, :c]) |> Graph.add_edge(:a, :b)
...> Graph.degree(g, :b)
1
```

Removes all edges connecting `v1`

to `v2`

, regardless of label.

If no such edge exists, the graph is returned unmodified.

## Example

iex> g = Graph.new |> Graph.add_edges([{:a, :b}, {:a, :b, label: :foo}]) …> g = Graph.delete_edge(g, :a, :b) …> [:a, :b] = Graph.vertices(g) …> Graph.edges(g) []

iex> g = Graph.new(type: :undirected) |> Graph.add_edges([{:a, :b}, {:a, :b, label: :foo}]) …> g = Graph.delete_edge(g, :a, :b) …> [:a, :b] = Graph.vertices(g) …> Graph.edges(g) []

Removes an edge connecting `v1`

to `v2`

. A label can be specified to disambiguate the
specific edge you wish to delete, if not provided, the unlabelled edge, if one exists,
will be removed.

If no such edge exists, the graph is returned unmodified.

## Example

```
iex> g = Graph.new |> Graph.add_edges([{:a, :b}, {:a, :b, label: :foo}])
...> g = Graph.delete_edge(g, :a, :b, nil)
...> [:a, :b] = Graph.vertices(g)
...> Graph.edges(g)
[%Graph.Edge{v1: :a, v2: :b, label: :foo}]
iex> g = Graph.new |> Graph.add_edges([{:a, :b}, {:a, :b, label: :foo}])
...> g = Graph.delete_edge(g, :a, :b, :foo)
...> [:a, :b] = Graph.vertices(g)
...> Graph.edges(g)
[%Graph.Edge{v1: :a, v2: :b, label: nil}]
iex> g = Graph.new(type: :undirected) |> Graph.add_edges([{:a, :b}, {:a, :b, label: :foo}])
...> g = Graph.delete_edge(g, :a, :b, :foo)
...> [:a, :b] = Graph.vertices(g)
...> Graph.edges(g)
[%Graph.Edge{v1: :a, v2: :b, label: nil}]
```

Like `delete_edge/3`

, but takes a list of edge specifications, and deletes the corresponding
edges from the graph, if they exist.

Edge specifications can be `Edge`

structs, `{vertex, vertex}`

pairs, or `{vertex, vertex, label: label}`

triplets. An invalid specification will cause `Graph.EdgeSpecificationError`

to be raised.

## Examples

```
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c]) |> Graph.add_edge(:a, :b)
...> g = Graph.delete_edges(g, [{:a, :b}])
...> Graph.edges(g)
[]
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c]) |> Graph.add_edge(:a, :b, label: :foo)
...> g = Graph.delete_edges(g, [{:a, :b}])
...> Graph.edges(g)
[]
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c]) |> Graph.add_edge(:a, :b, label: :foo)
...> g = Graph.delete_edges(g, [{:a, :b, label: :bar}])
...> Graph.edges(g)
[%Graph.Edge{v1: :a, v2: :b, label: :foo}]
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c]) |> Graph.add_edge(:a, :b, label: :foo)
...> g = Graph.delete_edges(g, [{:a, :b, label: :foo}])
...> Graph.edges(g)
[]
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c]) |> Graph.add_edge(:a, :b)
...> Graph.delete_edges(g, [:a])
** (Graph.EdgeSpecificationError) Expected a valid edge specification, but got: :a
```

This function can be used to remove all edges between `v1`

and `v2`

. This is useful if
you are defining multiple edges between vertices to represent different relationships, but
want to remove them all as if they are a single unit.

## Examples

```
iex> g = Graph.new |> Graph.add_edges([{:a, :b}, {:a, :b, label: :foo}, {:b, :a}])
...> g = Graph.delete_edges(g, :a, :b)
...> Graph.edges(g)
[%Graph.Edge{v1: :b, v2: :a}]
iex> g = Graph.new(type: :undirected) |> Graph.add_edges([{:a, :b}, {:a, :b, label: :foo}, {:b, :a}])
...> g = Graph.delete_edges(g, :a, :b)
...> Graph.edges(g)
[]
```

Removes a vertex from the graph, as well as any edges which refer to that vertex. If the vertex does not exist in the graph, it is a no-op.

## Example

```
iex> g = Graph.new |> Graph.add_vertex(:a) |> Graph.add_vertex(:b) |> Graph.add_edge(:a, :b)
...> [:a, :b] = Graph.vertices(g)
...> [%Graph.Edge{v1: :a, v2: :b}] = Graph.edges(g)
...> g = Graph.delete_vertex(g, :b)
...> [:a] = Graph.vertices(g)
...> Graph.edges(g)
[]
```

Like `delete_vertex/2`

, but takes a list of vertices to delete from the graph.

## Example

```
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c]) |> Graph.delete_vertices([:a, :b])
...> Graph.vertices(g)
[:c]
```

Gets the shortest path between `a`

and `b`

.

As indicated by the name, this uses Dijkstra’s algorithm for locating the shortest path, which means that edge weights are taken into account when determining which vertices to search next. By default, all edges have a weight of 1, so vertices are inspected at random; which causes this algorithm to perform a naive depth-first search of the graph until a path is found. If your edges are weighted however, this will allow the algorithm to more intelligently navigate the graph.

## Example

```
iex> g = Graph.new |> Graph.add_edges([{:a, :b}, {:b, :c}, {:c, :d}, {:b, :d}])
...> Graph.dijkstra(g, :a, :d)
[:a, :b, :d]
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c, :d])
...> g = Graph.add_edges(g, [{:a, :c}, {:b, :c}, {:b, :d}])
...> Graph.dijkstra(g, :a, :d)
nil
```

Get an Edge struct for a specific vertex pair, or vertex pair + label.

## Example

```
iex> g = Graph.new |> Graph.add_edges([{:a, :b}, {:a, :b, label: :contains}, {:a, :b, label: :uses}])
...> Graph.edge(g, :b, :a)
nil
iex> g = Graph.new |> Graph.add_edges([{:a, :b}, {:a, :b, label: :contains}, {:a, :b, label: :uses}])
...> Graph.edge(g, :a, :b)
%Graph.Edge{v1: :a, v2: :b}
iex> g = Graph.new |> Graph.add_edges([{:a, :b}, {:a, :b, label: :contains}, {:a, :b, label: :uses}])
...> Graph.edge(g, :a, :b, :contains)
%Graph.Edge{v1: :a, v2: :b, label: :contains}
iex> g = Graph.new(type: :undirected) |> Graph.add_edges([{:a, :b}, {:a, :b, label: :contains}, {:a, :b, label: :uses}])
...> Graph.edge(g, :a, :b, :contains)
%Graph.Edge{v1: :a, v2: :b, label: :contains}
```

edge(t(), vertex(), vertex(), label()) :: Graph.Edge.t() | nil

Return a list of all the edges, where each edge is expressed as a tuple
of `{A, B}`

, where the elements are the vertices involved, and implying the
direction of the edge to be from `A`

to `B`

.

NOTE: You should be careful when using this on dense graphs, as it produces
lists with whatever you’ve provided as vertices, with likely many copies of
each. I’m not sure if those copies are shared in-memory as they are unchanged,
so it *should* be fairly compact in memory, but I have not verified that to be sure.

## Example

```
iex> g = Graph.new |> Graph.add_vertex(:a) |> Graph.add_vertex(:b) |> Graph.add_vertex(:c)
...> g = g |> Graph.add_edge(:a, :c) |> Graph.add_edge(:b, :c)
...> Graph.edges(g)
[%Graph.Edge{v1: :a, v2: :c}, %Graph.Edge{v1: :b, v2: :c}]
```

Returns a list of all edges inbound or outbound from vertex `v`

.

## Example

```
iex> g = Graph.new |> Graph.add_edges([{:a, :b}, {:b, :c}])
...> Graph.edges(g, :b)
[%Graph.Edge{v1: :a, v2: :b}, %Graph.Edge{v1: :b, v2: :c}]
iex> g = Graph.new |> Graph.add_edges([{:a, :b}, {:b, :c}])
...> Graph.edges(g, :d)
[]
```

Returns a list of all edges between `v1`

and `v2`

.

## Example

```
iex> g = Graph.new |> Graph.add_edge(:a, :b, label: :uses)
...> g = Graph.add_edge(g, :a, :b, label: :contains)
...> Graph.edges(g, :a, :b)
[%Graph.Edge{v1: :a, v2: :b, label: :contains}, %Graph.Edge{v1: :a, v2: :b, label: :uses}]
iex> g = Graph.new(type: :undirected) |> Graph.add_edge(:a, :b, label: :uses)
...> g = Graph.add_edge(g, :a, :b, label: :contains)
...> Graph.edges(g, :a, :b)
[%Graph.Edge{v1: :a, v2: :b, label: :contains}, %Graph.Edge{v1: :a, v2: :b, label: :uses}]
```

Builds a list of paths between vertex `a`

and vertex `b`

.

The algorithm used here is a depth-first search, which evaluates the whole graph until all paths are found. Order is guaranteed to be deterministic, but not guaranteed to be in any meaningful order (i.e. shortest to longest).

## Example

```
iex> g = Graph.new |> Graph.add_edges([{:a, :b}, {:b, :c}, {:c, :d}, {:b, :d}, {:c, :a}])
...> Graph.get_paths(g, :a, :d)
[[:a, :b, :c, :d], [:a, :b, :d]]
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c, :d])
...> g = Graph.add_edges(g, [{:a, :c}, {:b, :c}, {:b, :d}])
...> Graph.get_paths(g, :a, :d)
[]
```

See `dijkstra/1`

.

Returns true if the given vertex exists in the graph. Otherwise false.

## Example

```
iex> g = Graph.new |> Graph.add_vertices([:a, :b])
...> Graph.has_vertex?(g, :a)
true
iex> g = Graph.new |> Graph.add_vertices([:a, :b])
...> Graph.has_vertex?(g, :c)
false
```

Returns the in-degree of vertex `v`

of graph `g`

.

The *in-degree* of a vertex is the number of edges directed inbound towards that vertex.

For undirected graphs, the in-degree and out-degree are always the same - the sum total of all edges inbound or outbound from the vertex.

## Example

```
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c]) |> Graph.add_edge(:a, :b)
...> Graph.in_degree(g, :b)
1
```

Returns a list of `Graph.Edge`

structs representing the in edges to vertex `v`

.

In the case of undirected graphs, it delegates to `edges/2`

.

## Example

```
iex> g = Graph.new |> Graph.add_edges([{:a, :b}, {:a, :b, label: :foo}, {:b, :c}])
...> Graph.in_edges(g, :b)
[%Graph.Edge{v1: :a, v2: :b, label: :foo}, %Graph.Edge{v1: :a, v2: :b}]
```

Returns a list of vertices which all have edges coming in to the given vertex `v`

.

In the case of undirected graphs, it delegates to `neighbors/2`

.

## Example

```
iex> g = Graph.new |> Graph.add_edges([{:a, :b}, {:a, :b, label: :foo}, {:b, :c}])
...> Graph.in_neighbors(g, :b)
[:a]
```

info(t()) :: %{num_edges: non_neg_integer(), num_vertices: non_neg_integer()}

Returns a map of summary information about this graph.

NOTE: The `size_in_bytes`

value is an estimate, not a perfectly precise value, but
should be close enough to be useful.

## Example

```
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c, :d])
...> g = g |> Graph.add_edges([{:a, :b}, {:b, :c}])
...> match?(%{type: :directed, num_vertices: 4, num_edges: 2}, Graph.info(g))
true
```

Returns true if and only if the graph `g`

is acyclic.

Returns true if the graph is an aborescence, a directed acyclic graph,
where the *root*, a vertex, of the arborescence has a unique path from itself
to every other vertex in the graph.

Returns true if the graph `g`

is not acyclic.

Returns true if graph `g1`

is a subgraph of `g2`

.

A graph is a subgraph of another graph if it’s vertices and edges are a subset of that graph’s vertices and edges.

## Example

```
iex> g1 = Graph.new |> Graph.add_vertices([:a, :b, :c, :d]) |> Graph.add_edge(:a, :b) |> Graph.add_edge(:b, :c)
...> g2 = Graph.new |> Graph.add_vertices([:b, :c]) |> Graph.add_edge(:b, :c)
...> Graph.is_subgraph?(g2, g1)
true
iex> g1 = Graph.new |> Graph.add_vertices([:a, :b, :c, :d]) |> Graph.add_edges([{:a, :b}, {:b, :c}])
...> g2 = Graph.new |> Graph.add_vertices([:b, :c, :e]) |> Graph.add_edges([{:b, :c}, {:c, :e}])
...> Graph.is_subgraph?(g2, g1)
false
```

Returns true if and only if the graph `g`

is a tree.

This function always returns false for undirected graphs.

NOTE: Multiple edges between the same pair of vertices in the same direction are considered a single edge when determining if the provided graph is a tree.

Detects all maximal cliques of degree `k`

.

Returns a list of cliques, where each clique is a list of vertices in the clique.

Calculates the k-core for a given graph and value of `k`

.

A k-core of the graph is a maximal subgraph of `g`

which contains vertices of which all
have a degree of at least `k`

. This function returns a new `Graph`

which is a subgraph
of `g`

containing all vertices which have a coreness >= the desired value of `k`

.

If there is no k-core in the graph for the provided value of `k`

, an empty `Graph`

is returned.

If a negative integer is provided for `k`

, a RuntimeError will be raised.

NOTE: For performance reasons, k-core calculations make use of ETS. If you are sensitive to the number of concurrent ETS tables running in your system, you should be aware of it’s usage here. 2 tables are used, and they are automatically cleaned up when this function returns.

k_core_components(t()) :: %{optional(k :: non_neg_integer()) => [vertex()]}

Groups all vertices by their k-coreness into a single map.

More commonly you will want a specific k-core, in particular the degeneracy core, for which there are other functions in the API you can use. However if you have a need to determine which k-core each vertex belongs to, this function can be used to do just that.

As an example, you can construct the k-core for a given graph like so:

```
k_core_vertices =
g
|> Graph.k_core_components()
|> Stream.filter(fn {k, _} -> k >= desired_k end)
|> Enum.flat_map(fn {_, vs} -> vs end)
Graph.subgraph(g, k_core_vertices)
```

Updates the labels for the given vertex.

If no such vertex exists in the graph, `{:error, {:invalid_vertex, v}}`

is returned.

## Example

```
iex> g = Graph.new |> Graph.add_vertex(:a, :foo)
...> [:foo] = Graph.vertex_labels(g, :a)
...> g = Graph.label_vertex(g, :a, :bar)
...> Graph.vertex_labels(g, :a)
[:foo, :bar]
iex> g = Graph.new |> Graph.add_vertex(:a)
...> g = Graph.label_vertex(g, :a, [:foo, :bar])
...> Graph.vertex_labels(g, :a)
[:foo, :bar]
```

Returns a list of vertices from graph `g`

which are included in a loop, where a loop is a cycle of length 1.

## Example

```
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c]) |> Graph.add_edge(:a, :a)
...> Graph.loop_vertices(g)
[:a]
```

Return all neighboring vertices of the given vertex.

## Example

```
iex> g = Graph.new |> Graph.add_edges([{:a, :b}, {:b, :a}, {:b, :c}, {:c, :a}])
...> Graph.neighbors(g, :a)
[:b, :c]
iex> g = Graph.new |> Graph.add_edges([{:a, :b}, {:b, :a}, {:b, :c}, {:c, :a}])
...> Graph.neighbors(g, :d)
[]
```

Creates a new directed graph.

Creates a new graph using the provided options.

## Options

`type: :directed | :undirected`

, specifies what type of graph this is

## Example

```
iex> g = Graph.new(type: :undirected) |> Graph.add_edges([{:a, :b}, {:b, :a}])
...> Graph.edges(g)
[%Graph.Edge{v1: :a, v2: :b}]
iex> g = Graph.new(type: :directed) |> Graph.add_edges([{:a, :b}, {:b, :a}])
...> Graph.edges(g)
[%Graph.Edge{v1: :a, v2: :b}, %Graph.Edge{v1: :b, v2: :a}]
```

Returns the number of edges in the graph.

Pseudo-edges (label/weight pairs applied to an edge) are not counted, only distinct vertex pairs where an edge exists between them are counted.

## Example

```
iex> g = Graph.add_edges(Graph.new, [{:a, :b}, {:b, :c}, {:a, :a}])
...> Graph.num_edges(g)
3
```

Returns the number of vertices in the graph

## Example

```
iex> g = Graph.add_vertices(Graph.new, [:a, :b, :c])
...> Graph.num_vertices(g)
3
```

Returns the out-degree of vertex `v`

of graph `g`

.

The *out-degree* of a vertex is the number of edges directed outbound from that vertex.

For undirected graphs, the in-degree and out-degree are always the same - the sum total of all edges inbound or outbound from the vertex.

## Example

```
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c]) |> Graph.add_edge(:a, :b)
...> Graph.out_degree(g, :a)
1
```

Returns a list of `Graph.Edge`

structs representing the out edges from vertex `v`

.

In the case of undirected graphs, it delegates to `edges/2`

.

## Example

```
iex> g = Graph.new |> Graph.add_edges([{:a, :b}, {:a, :b, label: :foo}, {:b, :c}])
...> Graph.out_edges(g, :a)
[%Graph.Edge{v1: :a, v2: :b, label: :foo}, %Graph.Edge{v1: :a, v2: :b}]
```

Returns a list of vertices which the given vertex `v`

has edges going to.

In the case of undirected graphs, it delegates to `neighbors/2`

.

## Example

```
iex> g = Graph.new |> Graph.add_edges([{:a, :b}, {:a, :b, label: :foo}, {:b, :c}])
...> Graph.out_neighbors(g, :a)
[:b]
```

Returns all vertices of graph `g`

. The order is given by a depth-first traversal of the graph,
collecting visited vertices in postorder. More precisely, the vertices visited while searching from an
arbitrarily chosen vertex are collected in postorder, and all those collected vertices are placed before
the subsequently visited vertices.

## Example

Our example code constructs a graph which looks like so:

```
:a
:b
/ :c :d
/
:e
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c, :d, :e])
...> g = Graph.add_edges(g, [{:a, :b}, {:b, :c}, {:b, :d}, {:c, :e}])
...> Graph.postorder(g)
[:e, :c, :d, :b, :a]
```

Returns all vertices of graph `g`

. The order is given by a depth-first traversal of the graph,
collecting visited vertices in preorder.

## Example

Our example code constructs a graph which looks like so:

```
:a
:b
/ :c :d
/
:e
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c, :d, :e])
...> g = Graph.add_edges(g, [{:a, :b}, {:b, :c}, {:b, :d}, {:c, :e}])
...> Graph.preorder(g)
[:a, :b, :c, :e, :d]
```

Returns an unsorted list of vertices from the graph, such that for each vertex in the list (call it `v`

),
there is a path in the graph from some vertex of `vs`

to `v`

.

As paths of length zero are allowed, the vertices of `vs`

are also included in the returned list.

## Example

```
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c, :d])
...> g = Graph.add_edges(g, [{:a, :b}, {:a, :c}, {:b, :c}, {:c, :d}])
...> Graph.reachable(g, [:a])
[:d, :c, :b, :a]
```

Returns an unsorted list of vertices from the graph, such that for each vertex in the list (call it `v`

),
there is a path in the graph of length one or more from some vertex of `vs`

to `v`

.

As a consequence, only those vertices of `vs`

that are included in some cycle are returned.

## Example

```
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c, :d])
...> g = Graph.add_edges(g, [{:a, :b}, {:a, :c}, {:b, :c}, {:c, :d}])
...> Graph.reachable_neighbors(g, [:a])
[:d, :c, :b]
```

Returns an unsorted list of vertices from the graph, such that for each vertex in the list (call it `v`

),
there is a path from `v`

to some vertex of `vs`

.

As paths of length zero are allowed, the vertices of `vs`

are also included in the returned list.

## Example

```
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c, :d])
...> g = Graph.add_edges(g, [{:a, :b}, {:a, :c}, {:b, :c}, {:c, :d}])
...> Graph.reaching(g, [:d])
[:b, :a, :c, :d]
```

Returns an unsorted list of vertices from the graph, such that for each vertex in the list (call it `v`

),
there is a path of length one or more from `v`

to some vertex of `vs`

.

As a consequence, only those vertices of `vs`

that are included in some cycle are returned.

## Example

```
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c, :d])
...> g = Graph.add_edges(g, [{:a, :b}, {:a, :c}, {:b, :c}, {:c, :a}, {:b, :d}])
...> Graph.reaching_neighbors(g, [:b])
[:b, :c, :a]
```

Replaces `vertex`

with `new_vertex`

in the graph.

## Example

```
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c, :d])
...> g = Graph.add_edges(g, [{:a, :b}, {:b, :c}, {:c, :a}, {:c, :d}])
...> [:a, :b, :c, :d] = Graph.vertices(g)
...> g = Graph.replace_vertex(g, :a, :e)
...> [:b, :c, :d, :e] = Graph.vertices(g)
...> Graph.edges(g)
[%Graph.Edge{v1: :b, v2: :c}, %Graph.Edge{v1: :c, v2: :d}, %Graph.Edge{v1: :c, v2: :e}, %Graph.Edge{v1: :e, v2: :b}]
```

Splits the edges between `v1`

and `v2`

by inserting a new vertex, `v3`

, deleting
the edges between `v1`

and `v2`

, and inserting new edges from `v1`

to `v3`

and from
`v3`

to `v2`

.

The resulting edges from the split will share the same weight and label as the old edges.

## Example

```
iex> g = Graph.new |> Graph.add_vertices([:a, :c]) |> Graph.add_edge(:a, :c, weight: 2)
...> g = Graph.split_edge(g, :a, :c, :b)
...> Graph.edges(g)
[%Graph.Edge{v1: :a, v2: :b, weight: 2}, %Graph.Edge{v1: :b, v2: :c, weight: 2}]
iex> g = Graph.new(type: :undirected) |> Graph.add_vertices([:a, :c]) |> Graph.add_edge(:a, :c, weight: 2)
...> g = Graph.split_edge(g, :a, :c, :b)
...> Graph.edges(g)
[%Graph.Edge{v1: :a, v2: :b, weight: 2}, %Graph.Edge{v1: :b, v2: :c, weight: 2}]
```

Returns a list of strongly connected components, where each component is a list of vertices.

A *strongly connected component* is a maximal subgraph such that there is a path between each pair of vertices.

See `components/1`

for the definitions of *subgraph* and *maximal subgraph* if you are unfamiliar with the
terminology.

## Example

```
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c, :d])
...> g = Graph.add_edges(g, [{:a, :b}, {:a, :c}, {:b, :c}, {:c, :d}, {:c, :a}])
...> Graph.strong_components(g)
[[:d], [:b, :c, :a]]
```

Builds a maximal subgraph of `g`

which includes all of the vertices in `vs`

and the edges which connect them.

See the test suite for example usage.

Converts the given Graph to DOT format, which can then be converted to
a number of other formats via Graphviz, e.g. `dot -Tpng out.dot > out.png`

.

If labels are set on a vertex, then those labels are used in the DOT output in place of the vertex itself. If no labels were set, then the vertex is stringified if it’s a primitive type and inspected if it’s not, in which case the inspect output will be quoted and used as the vertex label in the DOT file.

Edge labels and weights will be shown as attributes on the edge definitions, otherwise they use the same labelling scheme for the involved vertices as described above.

NOTE: Currently this function assumes graphs are directed graphs, but in the future it will support undirected graphs as well.

## Example

```
> g = Graph.new |> Graph.add_vertices([:a, :b, :c, :d])
> g = Graph.add_edges([{:a, :b}, {:b, :c}, {:b, :d}, {:c, :d}])
> g = Graph.label_vertex(g, :a, :start)
> g = Graph.label_vertex(g, :d, :finish)
> g = Graph.update_edge(g, :b, :d, weight: 3)
> IO.puts(Graph.to_dot(g))
strict digraph {
start
b
c
finish
start -> b [weight=1]
b -> c [weight=1]
b -> finish [weight=3]
c -> finish [weight=1]
}
```

Returns a topological ordering of the vertices of graph `g`

, if such an ordering exists, otherwise it returns false.
For each vertex in the returned list, no out-neighbors occur earlier in the list.

Multiple edges between two vertices are considered a single edge for purposes of this sort.

## Example

```
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c, :d])
...> g = Graph.add_edges(g, [{:a, :b}, {:a, :c}, {:b, :c}, {:c, :d}])
...> Graph.topsort(g)
[:a, :b, :c, :d]
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c, :d])
...> g = Graph.add_edges(g, [{:a, :b}, {:a, :c}, {:b, :c}, {:c, :d}, {:c, :a}])
...> Graph.topsort(g)
false
```

The transposition of a graph is another graph with the direction of all the edges reversed.

## Example

```
iex> g = Graph.new |> Graph.add_vertices([:a, :b, :c]) |> Graph.add_edge(:a, :b) |> Graph.add_edge(:b, :c)
...> g |> Graph.transpose |> Graph.edges
[%Graph.Edge{v1: :b, v2: :a}, %Graph.Edge{v1: :c, v2: :b}]
```

update_edge(t(), vertex(), vertex(), Graph.Edge.edge_opts()) :: t() | {:error, :no_such_edge}

Given two vertices, this function updates the metadata (weight/label) for the unlabelled edge between those two vertices. If no unlabelled edge exists between them, an error tuple is returned. If you set a label, the unlabelled edge will be replaced with a new labelled edge.

## Example

```
iex> g = Graph.new |> Graph.add_edge(:a, :b) |> Graph.add_edge(:a, :b, label: :bar)
...> %Graph{} = g = Graph.update_edge(g, :a, :b, weight: 2, label: :foo)
...> Graph.edges(g)
[%Graph.Edge{v1: :a, v2: :b, label: :bar}, %Graph.Edge{v1: :a, v2: :b, label: :foo, weight: 2}]
```

Like `update_edge/4`

, but requires you to specify the labelled edge to update.

Th implementation of `update_edge/4`

is actually `update_edge(g, v1, v2, nil, opts)`

.

## Example

```
iex> g = Graph.new |> Graph.add_edge(:a, :b) |> Graph.add_edge(:a, :b, label: :bar)
...> %Graph{} = g = Graph.update_labelled_edge(g, :a, :b, :bar, weight: 2, label: :foo)
...> Graph.edges(g)
[%Graph.Edge{v1: :a, v2: :b, label: :foo, weight: 2}, %Graph.Edge{v1: :a, v2: :b}]
iex> g = Graph.new(type: :undirected) |> Graph.add_edge(:a, :b) |> Graph.add_edge(:a, :b, label: :bar)
...> %Graph{} = g = Graph.update_labelled_edge(g, :a, :b, :bar, weight: 2, label: :foo)
...> Graph.edges(g)
[%Graph.Edge{v1: :a, v2: :b, label: :foo, weight: 2}, %Graph.Edge{v1: :a, v2: :b}]
```

Returns the label for the given vertex. If no label was assigned, it returns [].

## Example

```
iex> g = Graph.new |> Graph.add_vertex(:a) |> Graph.label_vertex(:a, :my_label)
...> Graph.vertex_labels(g, :a)
[:my_label]
```

Returns a list of all the vertices in the graph.

NOTE: You should be careful when using this on large graphs, as the list it produces contains every vertex on the graph. I have not yet verified whether Erlang ensures that they are a shared reference with the original, or copies, but if the latter it could result in running out of memory if the graph is too large.

## Example

```
iex> g = Graph.new |> Graph.add_vertex(:a) |> Graph.add_vertex(:b)
...> Graph.vertices(g)
[:a, :b]
```