# Standard graph module

This chapter covers the graph facilities available in the standard graph module. The basic operations are related to edge, vertices and graph manipulations.

## Graph concepts

The `afnix-gfx` module provides the support for manipulating graphs. Formally a graph is a collection of edges and vertices. In a normal graph, an edge connects two vertices. On the other hand, a vertex can have several edges. When an edge connects several vertices, it is called an hyperedge and the resulting structure is called an hypergraph.

### Edge class

The `Edge` class is a class used for a graph construction in association with the `Vertex` class. An edge is used to connect vertices. Normally, an edge connects two vertices. The number of vertices attached to an edge is called the cardinality of that edge. When the edge cardinality is one, the edge is called a self-loop. This mean that the edge connects the vertex to itself. This last point is merely a definition but present the advantage of defining an hyperedge as a set of vertices.

### Vertex class

The `Vertex` is the other class used for the graph construction. and operates with the `edge` class. A vertex is used to reference edges. the number of edges referenced by a vertex is called the degree of that vertex.

### Graph

The `Graph` class is class that represent either a graph or a hypergraph. By definition, a graph is collection of edges and vertices. There are numerous property attached to graph. Formally, a graph consists of a set of edges, a set of vertices and the associated endpoints. However, the implementation is designed in a way so that each edge and vertex carry its associated objects. This method ensures that the graph is fully defined by only its two sets.

## Graph construction

The graph construction is quite simple and proceed by adding edges and vertices. The base system does not enforce rules on the graph structure. it is possible to add con connected vertices as well as unreferenced edges.

### Edge construction

An edge is constructed by simply invoking the default constructor. Optionally, a client object can be attached to the edge.

# create a default edge const edge (afnix:gfx:Edge)

# create an edge with a client object const hello (afnix:gfx:Edge "hello")

The `edge-p` predicate can be used to check for the object type. When an edge is created with client object, the `get-client` method can be used to access that object.

### Vertex construction

A vertex is constructed a way similar to the `Edge>` object. The vertex is constructed by simply invoking the default constructor. Optionally, a client object can be attached to the edge.

# create a default vertex const vrtx (afnix:gfx:Vertex)

# create an vertex with a client object const world (afnix:gfx:Vertex "world")

The `vertex-p` predicate can be used to check for the object type. When a vertex is created with a client object, the `get-client` method can be used to access that object.

### Graph construction

A graph is constructed by simply adding edges and vertices to it. The `graph-p` predicate can be use to assert the graph type. the graph class also supports the concept of client object which can be attached at construction or with the `set-client` method.

const graph (afnix:gfx:Graph)

The `add` method can be used to add edges or vertices to the graph. The important point is that during the construction process, the graph structure is updated with the proper number of edge and vertices.

# create a graph const g (afnix:gfx:Graph) assert true (afnix:gfx:graph-p g)

# create an edge and add vertices const edge (afnix:gfx:Edge) edge:add (afnix:gfx:Vertex "hello") edge:add (afnix:gfx:Vertex "world") assert 2 (edge:degree)

# add the edge to the graph and check g:add edge assert 1 (g:number-of-edges) assert 2 (g:number-of-vertices)

# check for nodes and edges assert true (afnix:gfx:edge-p (g:get-edge 0)) assert true (afnix:gfx:vertex-p (g:get-vertex 0)) assert true (afnix:gfx:vertex-p (g:get-vertex 1))