# Module Pack.Graph

    module Graph: Sig_pack.S

Undirected imperative graphs with edges and vertices labeled with integer.

## Graph structure

    type t

abstract type of graphs

    module V: sig .. end

Vertices

    type vertex = V.t

    module E: sig .. end

Edges

    type edge = E.t

    val is_directed : bool

is this an implementation of directed graphs?

## Graph constructors and destructors

    val create : ?size:int -> unit -> t

Return an empty graph. Optionally, a size can be given, which should be on the order of the expected number of vertices that will be in the graph (for hash tables-based implementations). The graph grows as needed, so `size` is just an initial guess.

    val clear : t -> unit

Remove all vertices and edges from the given graph.
Since ocamlgraph 1.4

    val copy : t -> t

`copy g` returns a copy of `g`. Vertices and edges (and eventually marks, see module `Mark`) are duplicated.

    val add_vertex : t -> V.t -> unit

`add_vertex g v` adds the vertex `v` from the graph `g`. Do nothing if `v` is already in `g`.

    val remove_vertex : t -> V.t -> unit

`remove g v` removes the vertex `v` from the graph `g` (and all the edges going from `v` in `g`). Do nothing if `v` is not in `g`.

    val add_edge : t -> V.t -> V.t -> unit

`add_edge g v1 v2` adds an edge from the vertex `v1` to the vertex `v2` in the graph `g`. Add also `v1` (resp. `v2`) in `g` if `v1` (resp. `v2`) is not in `g`. Do nothing if this edge is already in `g`.

    val add_edge_e : t -> E.t -> unit

`add_edge_e g e` adds the edge `e` in the graph `g`. Add also `E.src e` (resp. `E.dst e`) in `g` if `E.src e` (resp. `E.dst e`) is not in `g`. Do nothing if `e` is already in `g`.

    val remove_edge : t -> V.t -> V.t -> unit

`remove_edge g v1 v2` removes the edge going from `v1` to `v2` from the graph `g`. Do nothing if this edge is not in `g`.
Raises `Invalid_argument` if `v1` or `v2` are not in `g`.

    val remove_edge_e : t -> E.t -> unit

`remove_edge_e g e` removes the edge `e` from the graph `g`. Do nothing if `e` is not in `g`.
Raises `Invalid_argument` if `E.src e` or `E.dst e` are not in `g`.

    module Mark: sig .. end

Vertices contains integers marks, which can be set or used by some algorithms (see for instance module `Marking` below)

## Size functions

    val is_empty : t -> bool

    val nb_vertex : t -> int

    val nb_edges : t -> int

Degree of a vertex

    val out_degree : t -> V.t -> int

`out_degree g v` returns the out-degree of `v` in `g`.
Raises `Invalid_argument` if `v` is not in `g`.

    val in_degree : t -> V.t -> int

`in_degree g v` returns the in-degree of `v` in `g`.
Raises `Invalid_argument` if `v` is not in `g`.

## Membership functions

    val mem_vertex : t -> V.t -> bool

    val mem_edge : t -> V.t -> V.t -> bool

    val mem_edge_e : t -> E.t -> bool

    val find_edge : t -> V.t -> V.t -> E.t

    val find_all_edges : t -> V.t -> V.t -> E.t list

## Successors and predecessors of a vertex

    val succ : t -> V.t -> V.t list

`succ g v` returns the successors of `v` in `g`.
Raises `Invalid_argument` if `v` is not in `g`.

    val pred : t -> V.t -> V.t list

`pred g v` returns the predecessors of `v` in `g`.
Raises `Invalid_argument` if `v` is not in `g`.
Labeled edges going from/to a vertex

    val succ_e : t -> V.t -> E.t list

`succ_e g v` returns the edges going from `v` in `g`.
Raises `Invalid_argument` if `v` is not in `g`.

    val pred_e : t -> V.t -> E.t list

`pred_e g v` returns the edges going to `v` in `g`.
Raises `Invalid_argument` if `v` is not in `g`.

## Graph iterators

iter/fold on all vertices/edges of a graph

    val iter_vertex : (V.t -> unit) -> t -> unit

    val iter_edges : (V.t -> V.t -> unit) -> t -> unit

    val fold_vertex : (V.t -> 'a -> 'a) -> t -> 'a -> 'a

    val fold_edges : (V.t -> V.t -> 'a -> 'a) -> t -> 'a -> 'a

    val map_vertex : (V.t -> V.t) -> t -> t

map iterator on vertex
iter/fold on all labeled edges of a graph

    val iter_edges_e : (E.t -> unit) -> t -> unit

    val fold_edges_e : (E.t -> 'a -> 'a) -> t -> 'a -> 'a

## Vertex iterators

Each iterator `iterator f v g` iters `f` to the successors/predecessors of `v` in the graph `g` and raises `Invalid_argument` if `v` is not in `g`.
iter/fold on all successors/predecessors of a vertex.

    val iter_succ : (V.t -> unit) -> t -> V.t -> unit

    val iter_pred : (V.t -> unit) -> t -> V.t -> unit

    val fold_succ : (V.t -> 'a -> 'a) -> t -> V.t -> 'a -> 'a

    val fold_pred : (V.t -> 'a -> 'a) -> t -> V.t -> 'a -> 'a

iter/fold on all edges going from/to a vertex.

    val iter_succ_e : (E.t -> unit) -> t -> V.t -> unit

    val fold_succ_e : (E.t -> 'a -> 'a) -> t -> V.t -> 'a -> 'a

    val iter_pred_e : (E.t -> unit) -> t -> V.t -> unit

    val fold_pred_e : (E.t -> 'a -> 'a) -> t -> V.t -> 'a -> 'a

## Basic operations

    val find_vertex : t -> int -> V.t

`vertex g i` returns a vertex of label `i` in `g`. The behaviour is unspecified if `g` has several vertices with label `i`. Note: this function is inefficient (linear in the number of vertices); you should better keep the vertices as long as you create them.

    val transitive_closure : ?reflexive:bool -> t -> t

`transitive_closure ?reflexive g` returns the transitive closure of `g` (as a new graph). Loops (i.e. edges from a vertex to itself) are added only if `reflexive` is `true` (default is `false`).

    val add_transitive_closure : ?reflexive:bool -> t -> t

`add_transitive_closure ?reflexive g` replaces `g` by its transitive closure. Meaningless for persistent implementations (then acts as `transitive_closure`).

    val mirror : t -> t

`mirror g` returns a new graph which is the mirror image of `g`: each edge from `u` to `v` has been replaced by an edge from `v` to `u`. For undirected graphs, it simply returns a copy of `g`.

    val complement : t -> t

`complement g` builds a new graph which is the complement of `g`: each edge present in `g` is not present in the resulting graph and vice-versa. Edges of the returned graph are unlabeled.

    val intersect : t -> t -> t

`intersect g1 g2` returns a new graph which is the intersection of `g1` and `g2`: each vertex and edge present in `g1` *and* `g2` is present in the resulting graph.

    val union : t -> t -> t

`union g1 g2` returns a new graph which is the union of `g1` and `g2`: each vertex and edge present in `g1` *or* `g2` is present in the resulting graph.

## Traversal

    module Dfs: sig .. end

Depth-first search

    module Bfs: sig .. end

Breadth-first search

    module Marking: sig .. end

Graph traversal with marking

## Graph generators

    module Classic: sig .. end

Classic graphs

    module Rand: sig .. end

Random graphs

    module Components: sig .. end

Strongly connected components

## Classical algorithms

    val shortest_path : t -> V.t -> V.t -> E.t list * int

Dijkstra's shortest path algorithm. Weights are the labels.

    val ford_fulkerson : t ->
           V.t -> V.t -> (E.t -> int) * int

Ford Fulkerson maximum flow algorithm

    val goldberg : t ->
           V.t -> V.t -> (E.t -> int) * int

Goldberg maximum flow algorithm

    val bellman_ford : t -> V.t -> E.t list

`bellman_ford g v` finds a negative cycle from `v`, and returns it, or raises `Not_found` if there is no such cycle

    module PathCheck: sig .. end

Path checking

    module Topological: sig .. end

Topological order

    val spanningtree : t -> E.t list

Kruskal algorithm

## Input / Output

    val dot_output : t -> string -> unit

DOT output in a file

    val display_with_gv : t -> unit

Displays the given graph using the external tools "dot" and "gv" and returns when gv's window is closed

    val parse_gml_file : string -> t

    val parse_dot_file : string -> t

    val print_gml : Format.formatter -> t -> unit

    val print_gml_file : t -> string -> unit