Is it possible to develop an algorithm to solve a graph isomorphism?

Question

Or will I need to develop an algorithm for every unique graph? The user is given a type of graph, and they are then supposed to use the interface to add nodes and edges to an initial graph. Then they submit the graph and the algorithm is supposed to confirm whether the user's graph matches the given graph.

The algorithm needs to confirm not only the neighbours of each node, but also that each node and each edge has the correct value. The initial graphs will always have a root node, which is where the algorithm can start from.

I am wondering if I can develop the logic for such an algorithm in the general sense, or will I need to actually code a unique algorithm for each unique graph. It isn't a big deal if it's the latter case, since I only have about 20 unique graphs.

Thanks. I hope I was clear.

Answer 1

Graph isomorphism problem might not be hard. But it's very hard to prove this problem is not hard.

There are three possibilities for this problem.
1. Graph isomorphism problem is NP-hard .
2. Graph isomorphism problem has a polynomial time solution.
3. Graph isomorphism problem is neither NP-hard or P.

If two graphs are isomorphic, then there exist a permutation for this isomorphism. Take this permutation as a certificate, we could prove this two graphs are isomorphic to each other in polynomial time. Thus, graph isomorphism lies in the territory of NP set. However, it has been more than 30 years that no one could prove whether this problem is NP-hard or P. Thus, this problem is intrinsically hard despite its simple problem description.

Answer 2

If I understand the question properly, you can have ONE single algorithm, which will work by accepting one of several reference graphs as its input (in addition to the input of the unknown graph which isomorphism with the reference graph is to be asserted).

It appears that you seek to assert whether a given graph is exactly identical to another graph rather than asserting if the graphs are isomorph relative to a particular set of operations or characteristics. This implies that the algorithm be supplied some specific reference graph, rather than working off some set of "abstract" rules such as whether neither graphs have loops, or both graphs are fully connected etc. even though the graphs may differ in some other fashion.

Edit , following confirmation that:
Yeah, the algorithm would be supplied a reference graph (which is the answer), and will then check the user's graph to see if it is isomorphic (including the values of edges and nodes) to the reference

In that case, yes, it is quite possible to develop a relatively simple algorithm which would assert isomorphism of these two graphs. Note that the considerations mentioned in other remarks and answers and relative to the fact that the problem may be NP-Hard are merely indicative that a simple algorithm [or any algorithm for that matter] may not be sufficient to solve the problem in a reasonable amount of time for graphs which size and complexity are too big. However, assuming relatively small graphs and taking advantage (!) of the requirement that the weights of edges and nodes also need to match, the following algorithm should generally be applicable.

General idea :
For each sub-graph that is disconnected from the rest of the graph, identify one (or possibly several) node(s) in the user graph which must match a particular node of the reference graph. By following the paths from this node [in an orderly fashion, more on this below], assert the identity of other nodes and/or determine that there are some nodes which cannot be matched (and hence that the two structures are not isomorphic).

Rough pseudo code :
1. For both the reference and the user supplied graph, make the the list of their Connected Components ie the list of sub-graphs therein which are disconnected from the rest of the graph. Finding these connected components is done by following either a breadth-first or a depth-first path from starting at a given node and "marking" all nodes on that path with an arbitrary [typically incremental] element ID number. Once a given path has been fully visited, repeat the operation from any other non-marked node, and do so until there are no more non-marked nodes.

2. Build a "database" of the characteristics of each graph . This will be useful to identify matching candidates and also to determine, early on, instances of non-isomorphism. Each "database" would have two kinds of "records" : node and edge, with the following fields, respectively: - node_id, Connected_element_Id, node weight, number of outgoing edges, number of incoming edges, sum of outgoing edges weights, sum of incoming edges weight. node - edge_id, Connected_element_Id, edge weight, node_id_of_start, node_id_of_end, weight_of_start_node, weight_of_end_node

3. Build a database of the Connected elements of each graph
Each record should have the following fields: Connected_element_id, number of nodes, number of edges, sum of node weights, sum of edge weights.

4. [optionally] Dispatch the easy cases of non-isomorphism :
4.a mismatch of the number of connected elements
4.b mismatch of of number of connected elements, grouped-by all fields but the id (number of nodes, number of edges, sum of nodes weights, sum of edges weights)

5. For each connected element in the reference graph
5.1 Identify candidates for the matching connected element in the user-supplied graph. The candidates must have the same connected element characteristics (number of nodes, number of edges, sum of nodes weights, sum of edges weights) and contain the same list of nodes and edges, again, counted by grouping by all characteristics but the id.
5.2 For each candidate, finalize its confirmation as an isomorph graph relative to the corresponding connected element in the reference graph. This is done by starting at a candidate node-match, ie a node, hopefully unique which has the exact same characteristics on both graphs. In case there is not such a node, one needs to disqualify each possible candidate until isomorphism can be confirmed (or all candidates are exhausted). For the candidate node match, walk the graph, in, say, breadth first, and by finding matches for the other nodes, on the basis of the direction and weight of the edges and weight of the nodes.

The main tricks with this algorithm is are to keep proper accounting of the candidates (whether candidate connected element at higher level or candidate node, at lower level), and to also remember and mark other identified items as such (and un-mark them if somehow the hypothetical candidate eventually proves to not be feasible.)

I realize the above falls short of a formal algorithm description, but that should give you an idea of what is required and possibly a starting point, would you decide to implement it.

You can remark that the requirement of matching nodes and edges weights may appear to be an added difficulty for asserting isomorphism, effectively simplify the algorithm because the underlying node/edge characteristics render these more unique and hence make it more likely that the algorithm will a) find unique node candidates and b) either quickly find other candidates on the path and/or quickly assert non-isomorphism.