Question

What are the suitable data structures for graphs ?

I guess the answer varies on the type of graph?

in any case, any recommendations?

Answer 1

Seeing as you want an abstract data type, I won’t mention anything about implementation.

A graph G = <V, E> is a set of vertices V, and a set of edges E where each element in E is a tuple e = <v1, v2>, and v1 and v2 are in V.

The following may look like Java but I’m really thinking of any sufficiently expressive language:

interface Graph {
    Set getVertices();
    Set getEdges();
    void addVertex(Object v);
    void addEdge(Object v1, Object v2);
    void removeVrtex(Object v);
    void removeEdge(Object v1, Object v2);
}

I think this is the bare minimum. I haven’t specified what is returned by getEdges; if the language has a native tuple type it would a set of those. In practice you would want to add extra methods such as:

int getDegree(Object v);
void contractEdge(Object v1, Object v2);
boolean isIsthmus(Object v1, Object v2);  /* does removal of this edge increase the number of components? */
int numComponents();
boolean isConnected();  /* return numComponents() <= 1 */
boolean isCycle(Set path);  /* path is a set of edges. */

etc.

Extending this to digraphs is left as an exercise :-).