How to partially compare two graphs

Question

For example, these two graphs is considered to be a perfect partial match:

0 - 1

1 - 2

2 - 3

3 - 0

AND

0 - 1

1 - 2

Answer 1

This is the subgraph isomorphism problem: http://en.wikipedia.org/wiki/Subgraph_isomorphism_problem

There is one algorithm mentioned in the article due to Ullmann.

Ullmann's algorithm is an extension of a depth-first search. A depth-first search would work like this:

def search(graph,subgraph,assignments):
  i=len(assignments)

  # Make sure that every edge between assigned vertices in the subgraph is also an
  # edge in the graph.
  for edge in subgraph.edges:
    if edge.first

For the basic algorithm, possible_assignments[i] = range(0,graph.n_vertices). That is, all the vertices are a possibility.

Ullmann extends this basic algorithm by narrowing the possibilities:

def update_possible_assignments(graph,subgraph,possible_assignments):
  any_changes=True
  while any_changes:
    any_changes = False
    for i in range(0,len(subgraph.n_vertices)):
      for j in possible_assignments[i]:
        for x in subgraph.adjacencies(i):
          match=False
          for y in range(0,len(graph.n_vertices)):
            if y in possible_assignments[x] and graph.has_edge(j,y):
              match=True
          if not match:
            possible_assignments[i].remove(j)
            any_changes = True

The idea is that if node i of the subgraph could possibly match node j of the graph, then for every node x that is adjacent to node i in the subgraph, it has to be possible to find a node y that is adjacent to node j in the graph. This process helps more than might first be obvious, because each time we eliminate a possible assignment, this may cause other possible assignments to be eliminated, since they are interdependent.

The final algorithm is then:

def search(graph,subgraph,assignments,possible_assignments):
  update_possible_assignments(graph,subgraph,possible_assignments)

  i=len(assignments)

  # Make sure that every edge between assigned vertices in the subgraph is also an
  # edge in the graph.
  for edge in subgraph.edges:
    if edge.first