Onauty

An OCaml library for determining whether provided graphs are isomorphic or not.
In case they are, a map between vertices can also be calculated. It uses nauty library as its backbone.

Credits

This software uses nauty and Traces library (http://pallini.di.uniroma1.it/) licensed under the Apache License 2.0. A boilerplate declaration is available in LICENSE-Nauty&Traces.txt file. The full license is available at: https://www.apache.org/licenses/LICENSE-2.0

Usage

Creating a graph

To create a graph one should specify its vertices and adjacency matrix¹.

open Onauty

let g = Graph.empty ()
    |> Graph.add_vertices 4 
    |> Graph.add_conns [(0,1);(1,2);(3,2)]

It is possible to create a graph all at once (as shown above) or partially.

open Onauty

let g = Graph.empty () ;;
Graph.add_vertex g |> Graph.add_vertex ;;
.
. (* doing some stuff*)
.
Graph.add_conn (0,1) g;;

¹ The order of connections is only important if a graph will be later considered as a digraph (directed graph).

Checking for isomorphism

Having two graphs, we can check whether they are isomorphic to each other.

let g1 = Onauty.Graph.empty ()
    |> Onauty.Graph.add_vertices 4 
    |> Onauty.Graph.add_conns [(0,1);(1,2);(3,2)]
let g2 = Onauty.Graph.empty () 
    |> Onauty.Graph.add_vertices 4 
    |> Onauty.Graph.add_conns [(0,1);(2,1);(3,2)]
(* g1 and g2 will be considered as undirected graphs *)
let are_iso1 = Onauty.Iso.are_graphs_iso g1 g2    
(* g1 and g2 will be considered as directed graphs*)
let are_iso2 = Onauty.Iso.are_digraphs_iso g1 g2    

Coloring vertices

It is also possible to partition a set of vertices into groups of colors.

(*this creates a graph with vertex 0 colored with first color, vertices 2 and 3 are colored with second color and vertex 1 is colored with third color *)
let g = Onauty.Graph.empty () 
    |> Onauty.Graph.add_vertices 4 
    |> Onauty.Graph.add_conns [(0,1);(2,1);(3,2)]
    |> Onauty.Graph.set_colors 
      (Common.StringMap.empty 
        |> Common.StringMap.add "C1" [0]
        |> Common.StringMap.add "C2" [2;3]
        |> Common.StringMap.add "C3" [1]

Please note that order of colors is important for determing the ismorphism. For example:

open Onauty
let g1 = Graph.empty () 
    |> Graph.add_vertices 4 
    |> Graph.add_conns [(0,1);(2,1);(3,2)]
    |> Graph.set_colors 
      (Common.StringMap.empty 
        |> Common.StringMap.add "A" [0]
        |> Common.StringMap.add "B" [2;3]
        |> Common.StringMap.add "C" [1] 
      )
      
let g2 = Graph.empty () 
    |> Graph.add_vertices 4 
    |> Graph.add_conns [(0,1);(2,1);(3,2)]
    |> Graph.set_colors 
      (Common.StringMap.empty 
        |> Common.StringMap.add "C" [0]
        |> Common.StringMap.add "A" [2;3]
        |> Common.StringMap.add "B" [1]
      )
(*this results in false because colors (represented as strings "A", "B" and "C") are sorted alphbetically so the first graph has vertex 0 assigned to first color while the second graph has vertex 0 assigned to third color. Structurally graphs are identical but colors are different.  *)
let are_iso = Iso.are_graphs_iso ~check_colors:true g1 g2 ;;

Mappig between graphs

Similarly as before, we can generate a mapping between isomorphic graphs (or digraphs). For example:

let g1 = Graph.empty () 
    |> Graph.add_vertices 4 
    |> Graph.add_conns [(0,1);(2,1);(3,2)]
    |> Graph.set_colors 
      (Common.StringMap.empty 
        |> Common.StringMap.add "A" [0]
        |> Common.StringMap.add "B" [1]
        |> Common.StringMap.add "C" [2;3] 
      )   
	
let g2 = Graph.empty () 
	|> Graph.add_vertices 4 
	|> Graph.add_conns [(0,1);(2,1);(3,2)]
	|> Graph.set_colors 
	  (Common.StringMap.empty 
		|> Common.StringMap.add "A" [3]
		|> Common.StringMap.add "B" [2]
		|> Common.StringMap.add "C" [0;1]
	  )
let are_graphs_iso,mapping = Iso.graphs_iso_map ~check_colors:true g1 g2 (* this results in a tuple of the form: (true,[|(0,3);(1,2);(2,1);(3,0)|]) *)
let are_digraphs_iso,mapping = Iso.digraphs_iso_map ~check_colors:true g1 g2 (* this results in a tupel of the form: (false,[||]) *)