Graph isomorphism is the problem of determining whether two graphs are topologically identical. It is in NP but not known to be NP-complete. The refinement heuristic iteratively partitions vertices into equivalence classes to reduce possible mappings between graphs. For trees, there is a linear-time algorithm that labels vertices based on subtree structure to test rooted and unrooted tree isomorphism.

1. 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 2

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 V:
– {v,w} E {f(v),f(w)} F
Graph Isomorphism 3

4. 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 V: {v,w} E {f(v),f(w)} F
– For all v V: l(v) = l’(f(v)).
• Application: organic chemistry:
– determining if two molecules are identical.
Graph Isomorphism 4

5. Complexity of graph
isomorphism
• Problem is in NP, but
– No NP-completeness proof is known
– No polynomial time algorithm is known
If P NP
NP-complete ? isomorphism
Graph
NP P
Graph Isomorphism 5

6. 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, i.e., a bijective function f:
V V, not the identity, with for all
v,w V:
– {v,w} E, if and only if {f(v),f(w)} E.
Graph Isomorphism 6

7. 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 7

8. Special cases are easier
• Polynomial time algorithms for
– Graphs of bounded degree
– Planar graphs
This course
– Trees
• Expected polynomial time for random
graphs
Graph Isomorphism 8

9. 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 9

10. 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 10

11. Iterative vertex partition heuristic
• If |V| |W|, or |E| |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 11

12. 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 12

13. 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| |Bi|, then stop: no isomorphism.
Graph Isomorphism 13

14. Other refinements
• Partition upon other characteristics of
vertices
– Label
– Number of vertices at distance d (in a set Ai).
–…
Graph Isomorphism 14

15. 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 15

16. Isomorphism on trees
• Linear time algorithm to test if two
(labeled) trees are isomorphic.
• Algorithm to test if two rooted trees are
isomorphic.
• Used as a subroutine for unrooted trees.
Graph Isomorphism 16

17. 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 17

18. 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 18

19. 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 19

20. 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 20

21. 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 21

22. 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 22

23. Time
• One layer with n1 nodes with n2 nodes in
next layer costs O(n1 + n2) time.
• Total time: O(n).
Graph Isomorphism 23

24. Unrooted trees
• Center of a tree
– 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 24

25. 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 25

26. 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,…)
Graph Isomorphism 26