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Graph Automorphisms

Graph Automorphisms

Bernard Knueven

CS 594 - Graph Theory

March 12, 2014

Bernard Knueven (CS 594 - Graph Theory) March 12, 2014 1 / 31

Graph Automorphisms

Agenda

1 Definitions

2 Group Theory

3 Examples

4 History

5 Applications

6 Open Problems

7 References

8 Homework


Graph Automorphisms Definitions

Isomorphism and Automorphism

Let X and Y be simple graphs.We say X and Y are isomorphic and write X – Y if thereexists a bijection φ : V pX q Ñ V pY q such that uv P EpX q ifand only if φpuqφpvq P EpY q.An isomorphism from X to itself is called anautomorphism of X .We denote the set of all automorphisms on X as AutpX q.We can see that AutpX q forms a group under thecomposition operator.


Graph Automorphisms Group Theory

Definition of Group

A nonempty set G together with an operation ˚ is called agroup if the following hold:

Closure: For all a,b P G, a ˚ b P G.Associativity: For all a,b, c P G, pa ˚ bq ˚ c “ a ˚ pb ˚ cq.

Identity Element: There exists e P G, such that for all a P G,e ˚ a “ a “ a ˚ e.

Inverse Element: For all a P G, there exists b P G such thata ˚ b “ b ˚ a “ e, where e is the identityelement.

We denote the group as pG, ˚q.



Some Results from Group Theory

The group axioms imply that the identity and inverses areunique.If pG, ˚q is a group, H Ď G is nonempty, and H with theoperation of G is a group, then H is called a subgroup ofG. We write H ď G.

Lagrange’s Theorem

Suppose G is a finite group. If H ď G, then |H| � |G|.



Examples of Groups

pZ,`q ď pQ,`q ď pR,`q ď pC,`qpQzt0u, ¨qThe set of all permutations of a set V is a group undercomposition and is denoted SympV q or Sympnq when|V | “ n. Notice that |Sympnq| “ n!For a graph X , we have then that AutpX q ď SympV pX qq.Question: For which graphs of degree n do we haveAutpX q – Sympnq?

When X “ Kn and the empty graph on n vertices, that is,X “ Kn.



A bit more notation

For v P V and g P SympV q, vg will denote the image ofvertex v under the permutation g.In this way, for G ď SympV q, we say that ‘G acts on V ’.Define

orbpvq “ tu P V | Dg P G such that vg “ uu.

In our context, when V is the vertex set of a graph X andG “ AutpX q, we will call orbpvq the orbit of vertex v .It can be shown that the orbits partition the vertex set [13].



Groups of Permutations

Let’s look at Symp3q. There are 3! “ 6 permutations

1 2 3Ó Ó Ó

1 2 3,

1 2 3Ó Ó Ó

2 1 3,

1 2 3Ó Ó Ó

3 2 1,

1 2 3Ó Ó Ó

1 3 2,

1 2 3Ó Ó Ó

2 3 1,

1 2 3Ó Ó Ó

3 1 2

which we will denote in cycle notation, respectively, astpq, p12q, p13q, p23q, p123q, p132qu.



Symp3q continued

Notice we can create the following ‘multiplication’ table (knownas a Cayley table), where we apply the row element first andthen the column element, i.e. row ˚ column:

˚ () (12) (13) (23) (123) (132)() () (12) (13) (23) (123) (132)

(12) (12) () (123) (132) (13) (23)(13) (13) (132) () (123) (23) (12)(23) (23) (123) (132) () (12) (13)(123) (123) (23) (12) (13) (132) ()(132) (132) (13) (23) (12) () (123)

This demonstrates that the group Symp3q is non-commutative,or non-abelian.


Graph Automorphisms Examples

Example: K3

1 2

3

We saw beforeAutpK3q “ Symp3q “ tpq, p12q, p13q, p23q, p123q, p132qu.What are the vertex orbits of K3?

t1,2,3u

When V pX q “ orbpvq for any v P V pX q, we say the graphX is vertex-transitive.



Example: P3

1 2 3

Using the labeling above, what is AutpP3q? What are thevertex orbits of P3?

AutpP3q “ tpq, p13qu ď Symp3qvertex orbits: t1,3u, t2u



Example: P3

1 2 3


AutpP3q “ tpq, p13qu ď Symp3q

vertex orbits: t1,3u, t2u



Example: P3

1 2 3


AutpP3q “ tpq, p13qu ď Symp3qvertex orbits: t1,3u, t2u



Example: C4

1 2

34

What are the valid permutations? What are the vertexorbits?

AutpC4q “

tpq, p13q, p24q, p14qp23q, p12qp34q, p1234q, p13qp24q, p1432quvertex orbits: t1,2,3,4u

In general, AutpCnq – Dn, where Dn is the group of allsymmetries of a regular n-gon [11].



Example: C4

1 2

34


AutpC4q “

tpq, p13q, p24q, p14qp23q, p12qp34q, p1234q, p13qp24q, p1432qu

vertex orbits: t1,2,3,4u




Example: C4

1 2

34


AutpC4q “

tpq, p13q, p24q, p14qp23q, p12qp34q, p1234q, p13qp24q, p1432quvertex orbits: t1,2,3,4u




Vertex Invariants

When computing automorphism groups of a graph, it isuseful to distinguish vertices that cannot be symmetric.

For example, two vertices cannot be symmetric if they havedifferent degrees.Such properties are called vertex invariants.



Example: Kite Graph

1

2

3

4


AutpX q “ tpq, p13q, p24q, p13qp24quvertex orbits: t1,3u, t2,4u



Example: Kite Graph

1

2

3

4


AutpX q “ tpq, p13q, p24q, p13qp24qu

vertex orbits: t1,3u, t2,4u



Example: Kite Graph

1

2

3

4


AutpX q “ tpq, p13q, p24q, p13qp24quvertex orbits: t1,3u, t2,4u



Frucht’s TheoremAs an aside for the mathematicians

Theorem (Frucht, 1939 [10])Given any finite group G there exist infinitely manynon-isomorphic connected graphs X whose automorphismgroup is isomorphic to G.

Later, de Groot [8] and Sabidussi [26] independently made thefollowing improvement to Frucht’s Theorem.

Theorem

Every group is isomorphic to the automorphism group of somegraph.


Graph Automorphisms History

History I

The graph isomorphism problem (determining whetherthere is an isomorphism between two given graphs)became of practical interest to chemist in the 1960s as away of comparing two chemical structures [27].Of particular interest was finding a canonical form for agiven class of graphs closed under isomorphism.

Let X denote a class of graphs closed under isomorphisms.The function CF : X Ñ X is a canonical form for X if:

(i) For X P X , CF pX q – X(ii) For X ,Y P X , X – Y if and only if CF pX q “ CF pY q.

If we have the canonical form for a class of graphs, we canuse standard data structures to store and query adatabase of graphs.



History II

In the 1970s, several algorithms were developed forcomputing canonical forms [5, 3].However, the first algorithm of practical use was that ofMcKay [20], and was implemented by McKay in a programthat became known as nauty.

The main innovation of nauty was its use ofautomorphisms to prune the search space, and as a resultnauty computed the generators for the automorphismgroup of the input graph.nauty would be the program of choice for computingcanonical forms and automorphisms for the next severaldecades [21].



History III

As symmetry detection became to be recognized as agood tool for helping to solve ‘hard’ problems, the need fora specialized algorithm for computing the automorphismgroups of a graph became apparent.In 2004, Darga et al. introduced saucy, whichoutperformed nauty by several orders of magnitude formany graphs of practical interest [6].Two other programs made subsequent improvements,namely bliss [16] and Traces [24], both of whichcompute automorphisms and canonical labels.Since their release all the programs above have undergonefurther improvements [7, 17, 15, 18, 21].


Graph Automorphisms Applications

Applications

Graph Drawing [1]Graph Coloring [25]Boolean Satisfiability [2]Constraint Programming [12]Model Checking [9]Markov Models [22]Integer Programming [23]



Integer Programming Example

Integer programs are usually solved using abranch-and-bound technique.However, this can perform badly even for some problemsthat are easy to solve:

minxPt0,1u5

tx5 | 2x1 ` 2x2 ` 2x3 ` 2x4 ` x5 “ 3u.

x5 “ 1.

This problem comes from [23].Bernard Knueven (CS 594 - Graph Theory) March 12, 2014 20 / 31


minxPt0,1u5

tx5 | 2x1 ` 2x2 ` 2x3 ` 2x4 ` x5 “ 3u

The symmetries force the branch-and-bound algorithm tosearch the symmetric parts of the branch-and-bound tree:

A

B

D

G

L

p

x5 = 1

x4 = 0

x3 = 0

x2 = 0

x1 = 1

C

E

H

M

Q

x5 = 1

x4 = 0

x3 = 0

x2 = 1

F

I

N

R

x5 = 1

x4 = 0

x3 = 1

J

O

S

x5 = 1

x4 = 1

x3 = 0

x2 = 0

x1 = 0

Graphic is courtesy of Jim Ostrowski.Bernard Knueven (CS 594 - Graph Theory) March 12, 2014 21 / 31


minxPt0,1u5

tx5 | 2x1 ` 2x2 ` 2x3 ` 2x4 ` x5 “ 3u

However, we can capture at least some the structure in theproblem as a graph.We can then use nauty (or one of its cousins) to computethe generators of the automorphism group, which will alsobe automorphisms of our problem!This allows us to compute in the above problem thattx1, x2, x3, x4u are in the same orbit, and can thus betreated equivalently for branching purposes.



minxPt0,1u5

tx5 | 2x1 ` 2x2 ` 2x3 ` 2x4 ` x5 “ 3u

This is called orbital branching, and for highly symmetricproblems like this one significantly reduces the size of thebranch-and-bound tree:

A

B

L

P

x5 = 1

x2 + x3 + x4 = 0

x1 = 1

C

x1 + x2 + x3 + x4 = 0

Graphic is courtesy of Jim Ostrowski.Bernard Knueven (CS 594 - Graph Theory) March 12, 2014 23 / 31

Graph Automorphisms Open Problems

Computational Complexity

In 1979, Mathon showed that the following problems arepolynomial-time equivalent [19]:

isomorphism recognition for two graphs X and Y (GI),isomorphism map from X onto Y , if it exists,number of isomorphism from X onto Y ,order of the automorphism group of X ,generators of the automorphism group of X ,automorphism partition of X .

These problems are said to be isomorphism-complete orGI-complete.


Graph Automorphisms Open Problems

Open Problems

What is the computational complexity of GI?Is GI P P?Is GI NP-complete?

Considered unlikely, in 1991 Goldreich showed it wouldimply the collapse of the polynomial-time hierarchy [14].

Or is GI NP-intermediate?

Is graph canonization GI-complete?Fastest running time algorithm for GI was provided byBabai et al. in 1983 [4] and has stood for the last three

decades at eO´?

n log n¯

[21].


Graph Automorphisms References

References I

[1] David Abelson, Seok-Hee Hong, and Donald E Taylor. A group-theoretic method for drawing graphssymmetrically. In Graph Drawing, pages 86–97. Springer, 2002.

[2] Fadi A Aloul, Arathi Ramani, Igor L Markov, and Karem A Sakallah. Solving difficult SAT instances in thepresence of symmetry. In Proceedings of the 39th annual Design Automation Conference, pages 731–736.ACM, 2002.

[3] V.L. Arlazarov, I.I. Zuev, A.V. Uskov, and I.A. Faradzhev. An algorithm for the reduction of finite non-orientedgraphs to canonical form. {USSR} Computational Mathematics and Mathematical Physics, 14(3):195 – 201,1974.

[4] László Babai, William M Kantor, and Eugene M Luks. Computational complexity and the classification of finitesimple groups. In Foundations of Computer Science, 1983., 24th Annual Symposium on, pages 162–171.IEEE, 1983.

[5] Derek Gordon Corneil and Calvin C Gotlieb. An efficient algorithm for graph isomorphism. Journal of the ACM(JACM), 17(1):51–64, 1970.

[6] Paul T Darga, Mark H Liffiton, Karem A Sakallah, and Igor L Markov. Exploiting structure in symmetrydetection for CNF. In Proceedings of the 41st annual Design Automation Conference, pages 530–534. ACM,2004.

[7] Paul T. Darga, Karem A. Sakallah, and Igor L. Markov. Faster symmetry discovery using sparsity ofsymmetries. In Proceedings of the 45th Annual Design Automation Conference, DAC ’08, pages 149–154,New York, NY, USA, 2008. ACM.

[8] Johannes de Groot. Groups represented by homeomorphism groups I. Mathematische Annalen,138(1):80–102, 1959.

[9] Alastair F Donaldson and Alice Miller. Automatic symmetry detection for model checking using computationalgroup theory. In FM 2005: Formal Methods, pages 481–496. Springer, 2005.



References II

[10] Robert Frucht. Herstellung von Graphen mit vorgegebener abstrakter Gruppe. Compositio Mathematica,6:239–250, 1939.

[11] Ashwin Ganesan. Automorphism groups of graphs. arXiv preprint arXiv:1206.6279, 2012.

[12] Ian P Gent, Karen E Petrie, and Jean-François Puget. Symmetry in constraint programming. Handbook ofConstraint Programming, pages 329–376, 2006.

[13] Chirs Godsil and Gordon Royle. Algebraic Graph Theory. Springer-Verlag, 2001.

[14] Oded Goldreich, Silvio Micali, and Avi Wigderson. Proofs that yield nothing but their validity or all languages innp have zero-knowledge proof systems. Journal of the ACM (JACM), 38(3):690–728, 1991.

[15] Tommi Junttila and Petteri Kaski. Conflict propagation and component recursion for canonical labeling. InTheory and Practice of Algorithms in (Computer) Systems, pages 151–162. Springer, 2011.

[16] Tommi A Junttila and Petteri Kaski. Engineering an efficient canonical labeling tool for large and sparsegraphs. In ALENEX, volume 7, pages 135–149. SIAM, 2007.

[17] Hadi Katebi, Karem A Sakallah, and Igor L Markov. Symmetry and satisfiability: An update. In Theory andApplications of Satisfiability Testing–SAT 2010, pages 113–127. Springer, 2010.

[18] Hadi Katebi, Karem A Sakallah, and Igor L Markov. Conflict anticipation in the search for graphautomorphisms. In Logic for Programming, Artificial Intelligence, and Reasoning, pages 243–257. Springer,2012.

[19] Rudolf Mathon. A note on the graph isomorphism counting problem. Information Processing Letters,8(3):131–136, 1979.

[20] Brendan D McKay. Practical graph isomorphism. Department of Computer Science, Vanderbilt University,1981.



References III

[21] Brendan D McKay and Adolfo Piperno. Practical graph isomorphism, II. Journal of Symbolic Computation,60:94–112, 2014.

[22] W Douglas Obal, Michael G McQuinn, and William H Sanders. Detecting and exploiting symmetry indiscrete-state markov models. Reliability, IEEE Transactions on, 56(4):643–654, 2007.

[23] James Ostrowski. Symmetry in Integer Programming. PhD thesis, Lehigh University, 2009.

[24] Adolfo Piperno. Search space contraction in canonical labeling of graphs. arXiv preprint arXiv:0804.4881,2008.

[25] Arathi Ramani, Fadi A Aloul, Igor L Markov, and Karem A Sakallah. Breaking instance-independentsymmetries in exact graph coloring. In Design, Automation and Test in Europe Conference and Exhibition,2004. Proceedings, volume 1, pages 324–329. IEEE, 2004.

[26] Gert Sabidussi. Graphs with given infinite group. Monatshefte für Mathematik, 64(1):64–67, 1960.

[27] Edward H Sussenguth. A graph-theoretic algorithm for matching chemical structures. Journal of ChemicalDocumentation, 5(1):36–43, 1965.

[28] Douglas B. West. Introduction to Graph Theory. Prentice-Hall, 2001.


Graph Automorphisms Homework

Homework I

1. Show that for any graph X , Aut pX q “ Aut`

X˘

. [13]2. Let X be a graph. Show that if u, v P V pX q and g P AutpX q,

then dpu, vq “ dpug , vgq. [13]3. Find the automorphism group of the following graphs

(expressed in cycle notation) and the vertex orbits (graphsfrom [28]).

a. 4

2

3

1

b. 4

2

3

1


Graph Automorphisms Homework

Homework II

c.

1

42

53

d. 12

3

5

4


Graph Automorphisms

Questions?


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