1. Algorithms and networks
Graph Isomorphism
Algorithms and networks

2. Today Graph isomorphism: definition
Complexity: isomorphism completeness
The refinement heuristic
Isomorphism for trees
Rooted trees
Unrooted trees
Graph Isomorphism

3. Graph Isomorphism
Two graphs G=(V,E) and H=(W,F) are isomorphic if there is a bijective function f: V ® W such that for all v, w Î V:
{v,w} Î E Û {f(v),f(w)} Î F
Graph Isomorphism

4. Applications Chemistry: databases of molecules (etc.)
Actually needed: canonical form of molecule structure / graph
Design verification
Software plagiarism detection
Speeding up algorithms for highly symmetric graphs
Graph Isomorphism

5. Variant for labeled graphs
Let G = (V,E), H=(W,F) be graphs with vertex labelings l: V ® L, l’ ® L.
G and H are isomorphic labeled graphs, if there is a bijective function f: V ® W such that
For all v, w Î V: {v,w} Î E Û {f(v),f(w)} Î F
For all v Î V: l(v) = l’(f(v)).
Application: organic chemistry:
determining if two molecules are identical.
Graph Isomorphism

6. Complexity of graph isomorphism
Problem is in NP, but
No NP-completeness proof is known
No polynomial time algorithm is known
If GI is NP-complete, then “strange things happen”
“Polynomial time hierarchy collapses to a finite level”
If P ¹ NP ?
NP-complete
Graph
isomorphism P NP
Graph Isomorphism

7. Isomorphism-complete
Problems are isomorphism-complete, if they are `equally hard’ as graph isomorphism
Isomorphism of bipartite graphs
Isomorphism of labeled graphs
Automorphism of graphs
Given: a graph G=(V,E)
Question: is there a non-trivial automorphism
Note: the identity is a (trivial) automorphism
Graph Isomorphism

8. Automorphism
An automorphism is a bijective function f: V ® V with for all v,wÎV:
{v,w} Î E, if and only if {f(v),f(w)} Î E.
A non-trivial automorphism is an automorphism that is not the identity
G1 has 6 automorphisms, and 5 non-trivial automorphisms
G2 has 2 automorphisms, and 1 non-trivial automorphism G1 w v G2
Graph Isomorphism

9. More isomorphism complete problems
Finding a graph isomorphism f
Isomorphism of semi-groups
Isomorphism of finite automata
Isomorphism of finite algebra’s
Isomorphism of
Connected graphs
Directed graphs
Regular graphs
Perfect graphs
Chordal graphs
Graphs that are isomorphic with their complement
Graph Isomorphism

10. Special cases are easier
Polynomial time algorithms for
Graphs of bounded degree
Planar graphs
Trees
Bounded treewidth
Expected polynomial time for random graphs
This course
Graph Isomorphism

11. An equivalence relation on vertices
Say v ~ w, if and only if there is an automorphism mapping v to w.
~ is an equivalence relation
Partitions the vertices in automorphism classes
Tells on structure of graph
Graph Isomorphism

12. Iterative vertex partition heuristic: the idea
Partition the vertices of G and H in classes
Each class for G has a corresponding class for H.
Property: vertices in class must be mapped to vertices in corresponding class
Refine classes as long as possible
When no refinement possible, check all possible ways that `remain’.
Graph Isomorphism

13. Iterative vertex partition heuristic skeleton
Partition the vertices of G and H in classes
If v and w are in different classes, there is no isomorphism or automorphism mapping v to w
Repeat
Refine the classes
Until … we do not find refinements
Solve
Graph Isomorphism

14. Iterative vertex partition heuristic
If |V| ¹ |W|, or |E| ¹ |F|, output: no. Done.
Otherwise, we partition the vertices of G and H into classes, such that
Each class for G has a corresponding class for H
If f is an isomorphism from G to H, then f(v) belongs to the class, corresponding to the class of v.
First try: vertices belong to the same class, when they have the same degree.
If f is an isomorphism, then the degree of f(v) equals the degree of v for each vertex v.
Graph Isomorphism

15. Scheme
Start with sequence SG = (A1) of subsets of G with A1=V, and sequence SH = (B1) of subsets of H with B1=W.
Repeat until …
Replace Ai in SG by Ai1,…,Air and replace Bi in SH by Bi1,…,Bir.
Ai1,…,Air is partition of Ai
Bi1,…,Bir is partition of Bi
For each isormorphism f: v in Aij if and only if f(v) in Bij
Graph Isomorphism

16. Possible refinement
Count for each vertex in Ai and Bi how many neighbors they have in Aj and Bj for some i, j.
Set Ais = {v in Ai | v has s neighbors in Aj}.
Set Bis = {v in Bi | v has s neighbors in Bj}.
Invariant: for all v in the ith set in SG, f(v) in the ith set in SH.
If some |Ai| ¹ |Bi|, then stop: no isomorphism.
Graph Isomorphism

17. Other refinements Partition upon other characteristics of vertices
Label
Number of vertices at distance d (in a set Ai). …
Graph Isomorphism

18. After refining
If each Ai contains one vertex: check the only possible isomorphism.
Otherwise, cleverly enumerate all functions that are still possible, and check these.
Works well in practice!
Graph Isomorphism

19. Isomorphism on trees
Linear time algorithm to test if two (labeled) trees are isomorphic. (Aho, Hopcroft, Ullman, 1974)
Algorithm to test if two rooted trees are isomorphic.
Used as a subroutine for unrooted trees.
Graph Isomorphism

20. Rooted tree isomorphism
For a vertex v in T, let T(v) be the subtree of T with v as root.
Level of vertex: distance to root.
If T1 and T2 have different number of levels: not isomorphic, and we stop. Otherwise, we continue:
Graph Isomorphism

21. Structure of algorithm
Tree is processed level by level, from bottom to root
Processing a level:
A long label for each vertex is computed
This is transformed to a short label
Vertices in the same layer whose subtrees are isomorphic get the same labels:
If v and w on the same level, then
L(v)=L(w), if and only if T(v) and T(w) are isomorphic with an isomorphism that maps v to w.
Graph Isomorphism

22. Labeling procedure
For each vertex, get the set of labels assigned to its children.
Sort these sets.
Bucketsort the pairs (L(w), v) for T1, w child of v
Bucketsort the pairs (L(w), v) for T2, w child of v
For each v, we now have a long label LL(v) which is the sorted set of labels of the children.
Use bucketsort to sort the vertices in T1 and T2 such that vertices with same long label are consecutive in ordering.
Graph Isomorphism

23. On sorting w.r.t. the long lists (1)
Preliminary work:
Sort the nodes is the layer on the number of children they have.
Linear time. (Counting sort / Radix sort.)
Make a set of pairs (j,i) with (j,i) in the set when the jth number in a long list is i.
Radix sort this set of pairs.
Graph Isomorphism

24. On sorting w.r.t. the long lists (2)
Let q be the maximum length of a long list
Repeat
Distribute among buckets the nodes with at least q children, with respect to the qth label in their long list
Nodes distributed in buckets in earlier round are taken here in the order as they appear in these buckets.
The sorted list of pairs (j,i) is used to skip empty buckets in this step.
q --;
Until q=0.
Graph Isomorphism

25. After vertices are sorted with respect to long label
Give vertices with same long label same short label (start counting at 0), and repeat at next level.
If we see that the set of labels for a level of T1 is not equal to the set for the same level of T2, stop: not isomorphic.
Graph Isomorphism

26. Time
One layer with n1 nodes with n2 nodes in next layer costs O(n1 + n2) time.
Total time: O(n).
Graph Isomorphism

27. Unrooted trees Center of a tree Finding the center:
A vertex v with the property that the maximum distance to any other vertex in T is as small as possible.
Each tree has a center of one or two vertices.
Finding the center:
Repeat until we have a vertex or a single edge:
Remove all leaves from T.
O(n) time: each vertex maintains current degree in variable. Variables are updated when vertices are removed, and vertices put in set of leaves when their degree becomes 1.
Graph Isomorphism

28. Isomorphism of unrooted trees
Note: the center must be mapped to the center
If T1 and T2 both have a center of size 1:
Use those vertices as root.
If T1 and T2 both have a center of size 2:
Try the two different ways of mapping the centers
Or: subdivide the edge between the two centers and take the new vertices as root
Otherwise: not isomorphic.
1 or 2 calls to isomorphism of rooted trees: O(n).
Graph Isomorphism

29. Conclusions Similar methods work for finding automorphisms
We saw: heuristic for arbitrary graphs, algorithm for trees
There are algorithms for several special graph classes (e.g., planar graphs, graphs of bounded degree,…)
The related Subgraph Isomorphism problem is NP-complete
Graph Isomorphism