Tree isomorphism

[kosii]

For #1602 . I'd just like to have an early feedback, I know it still misses handling eventual corner cases, tests and docs.

[jfinkels]

Might be more readable as if not T1 and not T2: return True.

[dschult]

But the suggestion is not the same meaning....
Original PR returns False but suggestion only returns True.

[jfinkels]

I thought the intention (not the implementation) was that when both are empty, immediately return True, otherwise run the algorithm.

[kosii]

It was to handle the case when T2 is empty (and so there is no center).
Should two empty graphs be considered as isomorph? Because nx.is_tree(nx.Graph()) raises NetworkXPointlessConcept, but nx.is_isomorphic(nx.Graph(), nx.Graph())

[jfinkels]

Hm, I believe that two null graphs should be considered isomorphic. The definition of isomorphic for graphs with node sets U and V is "there exists a function f from U to V such that f is a bijection and f preserves edges". Since both U and V are empty, there is just one function from U to V, the empty function, and it is a bijection. Furthermore, the edge preservation requirement is vacuously true.

However, the behavior of the tree isomorphism function should be "undefined" for non-tree inputs (that is, anything can happen). I think that, as with the bipartite package, the responsibility is on the calling code to decide whether the given graphs are actually trees before calling this is_isomorphic function. In other words, a pre-condition for calling this function will be that each graph is a tree. Since an empty graph is not a tree, you can assume that you will always get non-empty graphs as input.