New refiners for permutation group search

Contents lists available atScienceDirect

Journal

of

Symbolic

Computation

www.elsevier.com/locate/jsc

New

refiners

for

permutation

group

search

Christopher Jefferson

a

,

Markus Pfeiffer

a

,

Rebecca Waldecker

b

a_{UniversityofStAndrews,SchoolofComputerScience,NorthHaugh,StAndrews,KY169SX,Scotland,} United Kingdom

b_{Martin-Luther-UniversitätHalle-Wittenberg,InstitutfürMathematik,06099Halle,Germany}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received26May2017 Accepted11December2017 Availableonline11January2018

Keywords: Backtracksearch Refiners

Permutationgroups Algorithmicgrouptheory Computationalalgebra Partitionbacktrack

Partition backtrack is the current generic state of the art algorithm to search for subgroups of a given permutation group. We describe an improvement of partition backtrack for set stabilizers and in-tersections of subgroups by using orbital graphs. With extensive experiments we demonstrate that our methods improve perfor-mance of partition backtrack – in some cases by several orders of magnitude.

©2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Permutationgroupsare oneofthe mostnaturalandconvenientrepresentations offinitegroups. Theyhaveprovedparticularlyusefulforcomputationalpurposes,andsystemssuchasGAP (2016)and Magma(Bosmaetal.,1997)provideefficientimplementationsofmanyalgorithmstosolvearangeof problemsfrommembershiptestingtoidentificationoftheisomorphismtypeofagroup.

Given a permutation group acting on a finite set

, some problems – for example checking

whether or not a group contains a particular permutation, or computing the size of the group – canbesolvedintimepolynomialinthesizeof

.Thereareproblemsforwhichnopolynomialtime algorithmisknown,forexamplecomputingintersections,centralizersandnormalizersofsubgroups, andset andpartition stabilizers.Fortheseproblemsthe bestknownalgorithms performa very so-phisticatedexhaustivesearchthroughthegroupinquestion.Itisknown(seeforexampleChapter3

E-mailaddresses:[email protected](C. Jefferson),[email protected](M. Pfeiffer), [email protected](R. Waldecker).

URLs:http://caj.host.cs.st-andrews.ac.uk/(C. Jefferson),https://www.morphism.de/~markusp/(M. Pfeiffer), http://conway1.mathematik.uni-halle.de/~waldecker/index-english.html(R. Waldecker).

https://doi.org/10.1016/j.jsc.2017.12.003

inSeress,2003) thattheseproblemsare atleastasdifficultasgraphisomorphism,soit isunlikely thatapolynomialtimealgorithmcanbefound.

Thecurrentstateoftheartalgorithmisdescribed inLeon (1991)andiscommonlycalled parti-tion backtrack.It extendsideas introduced in McKay (1980) tosolve graph isomorphismproblems. Implementationsofpartitionbacktrackare availableinGAP (2016) andMagma(Bosmaetal., 1997), andtheysolvetheproblemsmentionedaboveveryefficientlyforafairrangeofexamples.

This doesnot mean that all permutation group problems can be solved easily orquickly using partitionbacktrack–andthisiswhereourorbitalgraphmethodscomeintoplay.Theideaspresented inthispaperimproveperformancebyseveralordersofmagnitudeforarangeofexamples,aswillbe demonstratedinSection 6.Partitionbacktrackshouldnot beviewedasamonolithic algorithm,but ratherasacombinationofalgorithmssolvingparticularsub-problems.Theconceptofarefinerisone ofitscrucialcomponents.Theyare usedtodetectandskippartsofthecomputation thatwouldbe superfluous:Betterrefinersleadtomoresuperfluouscomputationalstepstobeskipped.

Wedescribeanewclassofrefinersusingorbitalgraphs.Itcanbeusedtoimproveperformanceof partitionbacktrackimplementations,andwedemonstratethespeed-upthatcanbe achievedinour implementationofpartitionbacktrack.Thisarticleisorganizedasfollows:

Section2includesnecessarynotationandexamplesofpermutationgroupsandorderedpartitions. InSection3we givea briefdescriptionofbacktracksearch andtherole ofrefiners,andwediscuss somestandard refiners.Section4introduces orbitalgraphsandgivesacharacterizationofthecases where orbital graphs can benefit refinement, we briefly discuss the limitsof refiners using orbital graphs:Forexample,for2-transitivegroups,orbitalgraphsdonotprovideanyimprovement.Section5

then combines ideas introduced in Sections 3 and 4 to define new refiners using orbital graphs. Finally,we explainandpresentexperimentsandtheiroutcomesinSection6.Inparticular,thereare casesinwhichthecostsofcalculatingorbital graphsoutweightheadvantages inthesearch,butfor somesearchproblemswherethepreviouslybesttechniquesperformpoorly,ourmethodsprovetobe veryeffective.

Attheendofthepaperwecommentonrelatedquestionsandfurtherresearch.

Acknowledgements

AllauthorsthanktheDFG(Wa3089/6-1)andtheEPSRCCCPCoDiMa(EP/M022641/1)for support-ingthiswork.ThefirstauthorwouldliketothanktheRoyalSociety,andtheEPSRC(EP/M003728/1).

The second author would like to acknowledge support from the OpenDreamKit Horizon 2020

Eu-ropean ResearchInfrastructures Project (#676541). The third authorwishes to thankthe Computer Science Departmentofthe University ofSt Andrews forits hospitalityduring numerous visits, and KarinHelbichforthepicturesinthisarticle.

2. Notation,basicresultsandexamples

Wemostlyusestandardnotationforpermutationgroupsandrelatedobjectsandreferthereader toreferencessuchasDixonandMortimer (1996).

Throughoutwe let

be a finite setandn

∈

N

.We use Sym

()

asnotation forthe symmetric

groupon

and

Sn

forthesymmetricgroupon

{

1

,

...,

n

}

.

Definition1(Orderedpartitions).

•

An orderedpartition of

isan ordered listof non-empty disjointsubsets (called cells) of

whoseunionisallof

.Forall i

∈

we write

P

(

i

)

forthecellof P thatcontains thepoint i,

and we let OPart

()

denote the set of allorderedpartitionsof

. The notation for ordered partitionswillbe explainedinExample 2.If P

∈

OPart

()

andi

,

j

∈

,then wewrite i

∼P

j if andonlyifi

∈

P

(

j

)

.

•

Wesaythat Q isfinerthanP andwriteQ

P ifandonlyif,foralli

,

j

∈

,itistruethati

∼Q

j

Therelation definesapre-orderonOPart

()

andwepointoutthat,forall P

∈

OPart

()

,itis truethat P

P.Wecallan orderedpartitiondiscrete ifandonlyifeveryelement of

isina cellbyitself,andwecallanorderedpartitiontrivialifandonlyif

itselfisitsonlycell.

•

If P isan ordered partition of

and g

∈

Sym

()

,then we write Pg for theordered partition thatweobtainbyapplying gtotheelementsinthecellsofP.

•

ImportantorderedpartitionscomefromtheactionofsubgroupsofSym

()

on

.Wecallthem

orderedorbitpartitions.IfH

≤

Sym

()

andP

∈

OPart

()

,thenwesaythat P isanorderedorbit partitionforHifandonlyifthecellsofP areexactlytheorbitsofHon

.Wepointoutthat,if

H hasmorethanoneorbit,thenthereareseveraldistinctorderedorbitpartitions,differingonly bytheorderingofthecells.

•

Suppose that P and Q areordered orbit partitionsof

{

1

,

...,

n

}

,that k

∈

N

andthat

1

,

...,

k

are exactlythecells ofP.Then wewriteSym

(

P

)

forthesubgroupSym

(

1

)

× · · · ×

Sym

(

k

)

of

Sym

()

,i.e.thestabilizeroftheorderedpartitionPinSym

()

.WeuseCo

(

P

,

Q

)

forthesetof permutationsin Sym

()

that map P to Q.Notethat Co

(

P

,

Q

)

willeitherempty ora coset of Sym

(

P

)

.MoreoverCo

(

P

,

P

)

=

Sym

(

P

)

.

Example2.Given the algorithmic background of our work, we view partitions aslists. For exam-ple P

:= [

1

,

2

,

3

,

4

|

5

,

6

,

7

]

is an ordered partition of

:= {

1

,

2

,

3

,

4

,

5

,

6

,

7

}

with cells

{

1

,

2

,

3

,

4

}

and

{

5

,

6

,

7

}

. Then P

= [

2

,

1

,

3

,

4

|

5

,

6

,

7

]

, because the ordering of elements within a cell is irrel-evant. But P

= [

5

,

6

,

7

|

1

,

2

,

3

,

4

]

, because the ordering of cells is relevant. However, we see that

P

[

5

,

6

,

7

|

1

,

2

,

3

,

4

]

.LetQ

:= [

1

,

2

,

3

|

4

|

5

,

6

,

7

]

.Then Q

P and Q

[

4

,

5

,

6

,

7

|

3

,

2

,

1

]

.

Nextweconsiderthestabilizerin

S

7ofP.IfH1

≤

S

7isthesubgroupstabilizingtheset

{

1

,

2

,

3

,

4

}

andH2

≤

S

7isthesubgroupstabilizingtheset

{

5

,

6

,

7

}

,thenSym

(

P

)

isexactly H1

×

H2.

Finally,welookatthesubgroup H

:=

(

123

),

(

57

)

of

S

7 andwe writedowntwoofitsordered

orbitpartitions:

[

1

,

2

,

3

|

4

|

5

,

7

|

6

]

and

[

4

|

6

|

5

,

7

|

1

,

2

,

3

]

.

Definition3(Meet).Let P

,

Q

∈

OPart

()

. Then wedefine the meetofP andQ,denoted by P

∧

Q, asfollows:i

,

j

∈

areinthesamecell of P

∧

Q ifandonlyifi

∼P

j andi

∼

Q j.Foreverycell of P

∧

Q thereisauniquepairofcellsof P and Q thatitselementsare from,andthecellsof P

∧

Q

areorderedlexicographicallywithrespecttothesepairs.

Remark4.Thereareseveralreasonswhyorderedpartitionsareusedinpartitionbacktrack.One rea-son isthatwecanrepresentallelementsofthegroupascoset representativesofsomecosetofthe stabilizer ofan ordered partition. This approach doesnot work withunorderedpartitions. Another reasonisthat,withorderedpartitions,therelation“meet”iscompatiblewithtakingstabilizersof or-deredpartitions.Thispropertynolongerholdsifunorderedpartitionsareused.Onefinalcomment: Therelation“meet”isnotsymmetric,asisillustratedinthefollowingexample.

Example5.Let P

:= [

1

,

2

,

3

,

4

|

5

,

6

,

7

]

and Q

:= [

1

,

2

|

5

,

3

|

7

,

4

,

6

]

be ordered partitions of

:=

{

1

,

2

,

3

,

4

,

5

,

6

,

7

}

.Wecalculate P

∧

Q:Foreachelementi

∈

,wefindtheindicesofthecells

P

(

i

)

and

Q

(

i

)

,inthisorder.Thisgivesthefollowingpairs:

1 —

(

1

,

1

)

,2 —

(

1

,

1

)

,3 —

(

1

,

2

)

,4 —

(

1

,

3

)

,5 —

(

2

,

2

)

,6 —

(

2

,

3

)

,7 —

(

2

,

3

)

.

Therefore P

∧

Q

= [

1

,

2

|

3

|

4

|

5

|

6

,

7

]

. Calculating Q

∧

P instead gives the ordered partition

[

1

,

2

|

3

|

5

|

4

|

6

,

7

]

;ithasthesamecellsasP

∧

Q,butinadifferentorder.

It will be important later that taking meets of ordered partitions of

is compatible withthe actionofelementsofSym

()

on

.

Proof. Firstwenote:Ifg

∈

Sym

()

,theni

∼P

g j ifandonlyifig−

1

∼P

jg−1.LetT

:=

P

∧

Q.Thenby definitioni

∼T

jifandonlyifi

∼P

jandi

∼Q

j.Now,forallh

∈

H:

i

∼T

h j

⇔

ih

−1

∼T

jh−1

⇔

ih−1

∼P

jh−1 andih−1

∼Q

jh−1

⇔

i

∼

_Ph j andi

∼

_Qh j

⇔

i

∼P

h_∧Qh j

.

Sofarweprovedthat

(

P

∧

Q

)

h

Ph

_∧

_Qh _andPh

_∧

_Qh

₍

_P

_∧

_Q

₎

h_{,butthisdoesnotimplyequality.}

Wecandescribethecell

P∧Q

(

i

)

bythepair

[

P

(

i

),

Q

(

i

)

]

.Soweargueasfollows:

_(P∧Q)h

(

i

)

=

(P∧Q)

(

ih −1

)

= [

P

(

ih

−1

),

Q

(

ih

−1

)

] = [

_Ph

(

i

),

_Qh

(

i

)

] =

_Ph_∧Qh

(

i

).

Thisimpliesthat

(

P

∧

Q

)

h

=

Ph

_∧

_Qh_.

₂

3. Refiningorderedpartitions

3.1. Partitionbacktrack

Partitionbacktrackisatechnique forsolvingsearchproblemsonpermutationgroups. Itis imple-mentedinGAP (2016)andMagma(Bosmaetal.,1997)asthemaintechniqueforsolvingarangeof problems,includingcomputinggroup andcoset intersections,setandpartition stabilizers, centraliz-ers,andnormalizers.Thispaperwillnot provideafulldescriptionofpartitionbacktrack,insteadwe referreaderstoLeon (1991).

In brief, partition backtrack searches for elements of Sym

()

that satisfy a list of properties. These properties will typically be of the form “element lies in a subgroup of Sym

()

” or “ele-ment lies in a coset of a subgroup of Sym

()

”. For example, finding the stabilizer of a set S in a subgroup H of Sym

()

can be expressed as finding all elements that satisfy the list of prop-erties “element stabilizes S in Sym

()

” and “element is contained in the subgroup H”. We will not specify “properties” further because we have many different applications in mind, but each property should be easy to verify with an algorithm. The groups that appear in the search can be expressed in a variety of ways, for example by a set of generators, as the normalizer of a group, asthe automorphism group of a graph,or asthe stabilizer ofa set orordered partition in Sym

()

.

Togivea moreprecisedescription: Partitionbacktracktakesalistofpropertiesasinput,thenit startsfromthepair

(

P0

,

Q0

)

oftrivialordered partitionsandproceedstoperformsearchby

repeat-edlyalternatingbetweenthefollowingtwophases,startingati

=

0:

1. Refinementphase: Start withthe first property ofthe list andrefine the pair ofordered par-titions

(

Pi

,

Qi

)

to a newpair

(

P

,

Q

)

such that the following holds: Every element ofSym

()

inCo

(

Pi

,

Qi

)

isalsoinCo

(

P

,

Q

)

.Hence,wehavenotremoved anypermutationsthatsatisfyall

propertiesonthelist.

Repeatthisprocess withthesecondproperty fromthelistandcontinue through allproperties. Startagainatthebeginningofthelistuntiltheorderedpartitiondoesnotchangeanymore.The resultingpairoforderedpartitionswillbecalled

(

Pi+1

,

Qi+1

)

.

2. Branchingphase:Take Pi+1 andQi+1.Atthispoint,threecasesarepossible:

(a) ThereisnopermutationinSym

()

thatmapsPi+1to Qi+1.Thismeansthatthispartofthe

searchdoesnotproduceanypermutationthatsatisfiesallpropertiesonthelist.

(b) EverycellinPi+1 andQi+1 isofsizeone.Thenthereisexactlyonepermutationthatmaps Pi+1to Qi+1.Performafinalcheckthatthissatisfiesallpropertiesonthelist,andifitdoes,

thenrecorditasasolution.

(c) Splitthesearchbyproducingalistofpairsoforderedpartitions

(

P₁

,

Q₁

),

. . . ,

(

P_k

,

Q_k

)

where theCo

(

P_j

,

Q_j

)

for j

∈ {

1

,

. . . ,

k

}

formadisjointunionofCo

(

Pi+1

,

Qi+1

)

.

Thepracticalmethodthatwechooseforsplittingisthefollowing:

Wechoosel

∈

N

such thatthel-thcellc ofPi+1 hassizeatleast2,andwechooseasingle

candplacedinanewcellattheend,byitself.Foreachelementb ofthel-thcelldof Qi+1,

wecreateQ_j,whichisidenticalto Qi+1 exceptthatb isremovedfromcelldandplacedin

anewcellattheend,byitself.

Thisgivesnewpairsoforderedpartitionsthatarefinerthan

(

Pi+1

,

Qi+1

)

.Foreachofthese

pairs,thesearchcontinuesbygoingtotherefinementphase.

Thisprocesswillstopeventuallybecausethereisafinitenumberofbranches,theyallhavefinite length,andtherefinementstopsifthepartitiondoesnotchangeanymore.InLeon (1991)theauthor explainshowthisalgorithmcanbeimplementedefficiently.

Thispaperwillfocusonrefinement,becausethequalityofrefiners isone ofthemaininfluences

on performance. Refiners are only definedon a single ordered partition – Lemma 8 explainshow

refinersareusedoncosets.

Definition7(M-refiner).LetM

⊆

Sym

()

beasetofelementswithagivenproperty.AnM-refineris a map fM

:

OPart

()

→

OPart

()

that satisfiesthefollowingconditionsforanyorderedpartition P

of

:

(a) f

(

P

)

P.

(b) Forallg

∈

M,itistruethat f

(

Pg

₎

₌

_f

₍

_P

₎

g_.

Part(b)ofthedefinitionimpliesthateveryelementinSym

(

P

)

thatsatisfiestheproperty(referring to M) and stabilizes P also stabilizes f

(

P

)

. This turns out to be a very natural condition – our experienceisthatrefinerstendtosatisfy(b).Itisnotonlyanaturalrequirementforarefiner,butit isalsooneofthemainreasonswhypartition backtrackissoefficient.Formoredetails seeSections 6and7inLeon (1991).

Lemma 8 shows how, using an M-refiner (which refers to only a single ordered partition), we can performfilteringon cosets,asrequiredby oursearch definedabove.Thisgreatly simplifiesour implementation.

Lemma8.LetP

,

Q

∈

OPart

()

andlet fMbeanM-refinerforM

⊆

S ym

()

.Thenforallg

∈

M suchthat g

∈

Co

(

P

,

Q

)

,itfollowsthatg

∈

Co

(

fM

(

P

),

fM

(

Q

))

.

Proof. Ifg

∈

Co

(

P

,

Q

)

,then Q

=

Pg_{,andbydefinitionofanM-refineritholdsthatf}

M

(

Q

)

=

fM

(

Pg

)

=

fM

(

P

)

g,andthereforeCo

(

fM

(

P

),

fM

(

Q

))

=

Co

(

fM

(

P

),

fM

(

P

)

g

)

,andso g

∈

Co

(

fM

(

P

),

fM

(

Q

))

.

2

3.2. Refinersforpermutationgroups

We will now give a definitionof the standardrefiner forpermutation groups givenby a list of generatorsfromLeon (1991).WewillintroducerefinersthatuseorbitalgraphsinSection4.

Definition9(FixedM).LetM beasubgroupofSym

()

.Then themapFixedM

:

OPart

()

→

OPart

()

isdefinedasfollows:

•

Given P

∈

OPart

()

,letk

∈

N

and

α

1

,

. . . ,

α

k

∈

betheelementsinsingletoncellsofP.LetM0

denotethepoint-wisestabilizerof

α

1

,

. . . ,

α

kin M.

•

ReturnthemeetofP andanorderedorbitpartitionofM0.

Lemma10.LetM beasubgroupofSym

()

.ThenthemapFixedMisarefiner.

Proof. Let P

∈

OPart

()

. Then FixedM

(

P

)

P, because FixedM

(

P

)

is the meet of P with another

orderedpartitionanditisthereforefinerthan P.

Letk

∈

N

and

α

1

,

. . . ,

α

k

∈

besuchthatthesearepreciselytheelementsinsingletoncellsof P.

Now we let M0 denote the point-wise stabilizer of

α

1

,

. . . ,

α

k in M and we note that F

:=

M0

stabilizes

α

1

,

. . . ,

α

k.Let g

∈

G.Weneedtoshow thatFixedM

(

Pg

)

=

FixedM

(

P

)

g.Firstwe notethat Fg fixes

α

₁g

,

. . . ,

α

_kg, which are exactly the singleton cells of Pg. Now we fix some ordered orbit partition Q of F asdescribedbeforethelemma.Then Qg _{isanorderedorbitpartitionof F}g_.Using

Lemma 6wededuce:

FixedM

(

P

)

g

=

(

P

∧

Q

)

g

=

Pg

∧

Qg

=

FixedM

(

Pg

).

2

The major limitation of Fixed is that it ignores non-singleton cells. More concretely, given a transitivegroup G andan ordered partition P that contains no singletoncells, FixedG

(

P

)

=

P. We

cannoteasilyusethesamestrategyasFixed fornon-singletoncells(findingthestabilizerofthe non-singletoncellsin G),becausethiswouldrequiresolving thesetstabilizerproblem,whichis exactly oneoftheproblemsthatissolvedviabacktrack!

Instead,welookatotherpropertiesofgroupswecanuse,whichallowustorefinenon-singleton cellsefficiently.Wewillnowshowhoworbitalgraphscanbeusedinrefinersthatcomplement exist-ingrefinersforgroupsexpressedbyalistofgenerators.Theserefinerswillprovideusefulrefinement evenfortransitivegroupsandnon-singletoncells.Wearenotthefirstonestousethisidea:Theißen (1995)usesorbitalgraphsasaningredientforrefinersfornormalizersearch.However,ourworkdoes notbuildonhis–partlybecauseourhypothesisismoregeneral,andpartlybecausehisresultshave not beenpublished except forinhis PhDthesis. We show that the concepthas not yetbeen fully exploited.

4. Orbitalgraphs

Herewe introducethe graphsthat weuseinour newrefiners–orbital graphs–andprove the properties that are necessary in order to decide whether ornot a refinement by orbital graphs is computationallybeneficial.

Definition11(Digraphs).Forthe purposes ofthispaper,a digraph

is a pair

=

(

V

,

A

)

where V

denotesthesetofvertices(orpoints)and A denotesthesetofarcs,i.e.directededges. Ifx

,

y

∈

V, thenan arcfromx to y in

willbe denotedby

(

x

,

y

)

.An isolatedvertexofa digraphis avertex withnoarcs goingintoitorcomingoutofit.Adigraphiscompleteifandonlyifitssetofarcsis exactly

{

(

x

,

y

)

|

x

,

y

∈

V

,

x

=

y

}

.

isacompletebipartitedigraphifandonlyifthereexistdisjointsubsets S

,

E ofverticessuch that V is the unionof S (the “starting”vertices) and E (the “end” vertices) andthe set ofarcs is exactly A

= {

(

x

,

y

)

|

x

∈

S

,

y

∈

E

}

.

WereferthereadertoBang-JensenandGutin (2008)forstandardnotationandforthedefinitions ofconnectedcomponents,graphisomorphismsetc.Wepointoutthatbyaproperdigraphwemean adigraphthathasatleastonearcsuchthatitsreversearcisnotinthegraph.Alldigraphsconsidered herehavenomultiplearcsandnoloops.Wealsopointout thatwheneverwereferto thesizeofa connectedcomponentwemeanthenumberofverticesinthecomponent.

Definition12(Orbitalgraphs).Let

be a finiteset, G

:=

Sym

()

and H

≤

G.Forall vertices

γ

∈

andallh

∈

H wewrite

γ

h _{fortheimageof}

_γ

_{underhintheoriginalpermutationaction.}

FollowingDixonandMortimer (1996)wesaythatanorbitalgraphisself-pairedifandonlyif,for all

γ

,

δ

∈

,itistruethat

(

γ

,

δ)

isanarcifandonlyif

(δ,

γ

)

isanarc.

Example13.If we build an orbital graph for

S

3, then for each base-pair we obtain the complete

digraph on

{

1

,

2

,

3

}

. The reason is that this group is 2-transitive: Given

α

,

β

∈ {

1

,

2

,

3

}

such that

(

α

,

β)

is a base-pair, and given any distinct

γ

,

δ

∈ {

1

,

2

,

3

}

, there exists some g

∈

S

3 such that

α

g

₌

_γ

_and

_β

g

₌

_δ

_.Therefore

₍

_γ

_,

_δ)

_{isalso an arc, andthismeansthat all possible arcsexist.For}

groups that are not 2-transitive, the choice of the base-pair becomes much more important. Let

H

:=

(

123

),

(

45

),

(

46

)

≤

S

6.

Then,startingwiththebase-pair

(

1

,

2

)

,weobtainthefollowingdigraph:

But,startingwiththebase-pair

(

5

,

6

)

,wefind:

Foraconnecteddigraphwestartwiththepair

(

2

,

4

)

:

SomepropertiesoforbitalgraphscanbefoundinCameron (1999)andDixonandMortimer (1996), butwe decidedtoincludeshortproofsforthe statementsinthenext lemmainordertomake this articlemoreself-contained.

Hypothesis14.Let

be a finite set, let H

≤

G

:=

Sym

()

andlet

α

,

β

∈

be distinct. Let

:=

(

H

,

,

(

α

,

β))

andlet A denotethesetofarcsof

.

Lemma15.SupposethatHypothesis 14holds.Thenwehavethefollowing:

(i)

=

(

H

,

,

(

γ

,

δ))

ifandonlyif

(

γ

,

δ)

∈

A.

(ii)

isself-pairedifandonlyifsomeh

∈

H interchanges

α

and

β

.

(iii)

α

H_{ispreciselythesetofverticesof}

_{thatarethestartingpointofsomearc.}

(iv)

β

H ispreciselythesetofverticesof

thataretheendpointofsomearc.

(v) Thenumberofarcsstartingat

α

is

|

β

Hα

|

_{andthenumberofarcsgoinginto}

_β

_is

|

_α

Hβ

|

_.

Proof. (i)If

=

(

H

,

,

(

γ

,

δ))

,thenbydefinition

(

γ

,

δ)

isanarcin

.

Conversely, supposethat

(

γ

,

δ)

isanarcin

.Then thereexistssome h

∈

Hsuchthat

(

α

h

_,

_β

h

₎

₌

(

γ

,

δ)

.Hence theorbitalgraphwithbase-pair

(

α

,

β)

isthesameastheorbitalgraphwithbase-pair

(

γ

,

δ)

.

(iii)Let

γ

∈

α

H _andh

_∈

_{H be such that}

_γ

₌

_α

h_{. Then}

₍

_γ

_,

_β

h

₎

_{is an arc withstarting point}

_γ

_.

Conversely, if

δ

∈

is such that

(

γ

,

δ)

is an arc in

, then there exists some h

∈

H such that

(

α

h

_,

_β

h

₎

₌

₍

_γ

_,

_δ)

_andhence

_γ

₌

_α

h

_∈

_α

H_{.Similarargumentsshow(iv).}

(v)Thenumberofarcsstartingat

α

is

|{

(

α

,

γ

)

|

γ

∈

}|

= |{

(

α

h

_,

_β

h

₎

_|

_h

_∈

_H

_,

_α

h

₌

_α

_}|

_{= |}

_β

Hα

|

_{andthenumberofarcsgoinginto}

_β

_is

|{

(δ,

β)

|

δ

∈

}|

= |{

(

α

h

_,

_β

h

₎

_|

_h

_∈

_H

_,

_β

h

₌

_β

_}|

_{= |}

_α

Hβ

|

_.

2

Remark16.Somecomments:

(a)Parts (iii)and(v)ofthelemma, together,givethe totalnumberofarcsin

.The numberof arcsstartingat

α

isexactly

|

β

Hα

|

_,soweobtain

|

_A

|

= |

_α

H

|

· |

_β

Hα

|

_.

(b)In(ii)itisnottruethat Hmustcontainthetransposition

(

α

,

β)

.Acounterexampleisprovided by H

:=

(

12

)(

34

)

≤

S

4 acting naturally on

:= {

1

,

2

,

3

,

4

}

andits orbital graph with base-pair

(

1

,

2

)

.

(c)Parts(iii)and(iv)ofthelemmaimplythat,ifH actstransitivelyon

,then

hasnoisolated vertices.

Lemma17.SupposethatHypothesis 14holds.ThenH actson

asagroupofgraphautomorphisms.

Proof. FirstwenotethatH actsfaithfullyontheset

.Nowwelet

γ

,

δ

∈

.

If

(

γ

,

δ)

isan arc, then there exists some h

∈

H such that

(

γ

,

δ)

=

(

α

h

_,

_β

h

₎

_{by definitionof}

_.

Hence

(

γ

g

_,

_δ

g

₎

₌

₍

_α

hg

_,

_β

hg

₎

_{isan arc.Conversely, if}

₍

_γ

h

_,

_δ

h

₎

_{isanarc, thenthereexists somea}

_∈

_H

suchthat

(

γ

h

_,

_δ

h

₎

₌

₍

_α

a

_,

_β

a

₎

_andhence

₍

_γ

_,

_δ)

₌

₍

_α

ah−1

,

β

ah−1

)

isanarc.As Hisagroup,theinduced mapsarebijectiveandhenceeveryh

∈

H inducesagraphautomorphismon

.

2

Lemma18.SupposethatHypothesis 14holdsandlet

denotetheconnectedcomponentthatcontains

(

α

,

β)

. Theneveryconnectedcomponentof

thathassizeatleast2isisomorphicto

.

Proof. Let

denoteanarbitraryconnectedcomponentof

ofsizeatleast2 andlet

(

γ

,

δ)

bean arcin

.

Fromthedefinitionoforbitalgraphs leth

∈

H be suchthat

(

α

h

_,

_β

h

₎

₌

₍

_γ

_,

_δ)

_{.Thenh inducesan}

automorphismon

byLemma 17anditmovesallarcsfrom

toarcsin

.Conversely,h−1_induces

anautomorphismon

thatmovesallarcsof

into

.Thusitfollowsthat

and

areisomorphic

asgraphs.

2

Lemma 19 shows how to choose a set of base-pairs that determines all orbital graphs for a group H. Parts (i) and(ii) showto take a representativefrom eachorbit of H asthe firstelement ofthebase-pair,andthenpart(iii)showsthatwemuststabilizethisrepresentativeinH,andtakea representativefromeachorbitinthisstabilizerforthesecond elementofourbase-pair.These base-pairs willallow usto analyzethe setoforbital graphs ofa group, before weconstruct anyorbital graphsexplicitly.

Lemma19.Let

beafinitesetandletH

≤

G

:=

Sym

()

.

(i) Supposethat

α

,

β

∈

and

α

∈

β

H.ThenthesetoforbitalgraphsofH withbase-pairsstartingwith

α

is equaltothesetoforbitalgraphsofH withbase-pairsstartingwith

β

.

(ii) Supposethat

α

,

γ

∈

and

α

∈

/

γ

H_{.ThenthesetoforbitalgraphsofH withbase-pairsstartingwith}

_α

_is disjointfromthesetoforbitalgraphsofH withbase-pairsstartingwith

γ

.

(iii)Supposethat

α

,

β,

γ

∈

andthat

α

=

β

,

α

=

γ

.Let

1

:=

(

H

,

,

(

α

,

β))

and

2

:=

(

H

,

,

(

α

,

γ

))

. Then

1

=

2ifandonlyif

γ

∈

β

Hα.

Proof. (i) Let h

∈

H be such that

α

h

₌

_β

_{. Then for all}

_γ

_∈

_{, it follows that}

₍

_H

_,

_,

₍

_α

_,

_γ

₎₎

₌

(

H

,

,

(

α

h

_,

_γ

h

₎₎

₌

₍

_H

_,

_,

_(β,

_γ

h

₎₎

_{. Conversely}

₍

_H

_,

_,

_(β,

_γ

₎₎

₌

₍

_H

_,

_,

_(β

(h−1₎

,

γ

(h−1₎

))

=

(

H

,

,

(

α

,

γ

(h−1₎

(ii) Suppose that the pairs

(

α

,

β)

and

(

γ

,

δ)

generate the same orbital graph of H. Then by

Lemma 15(i)thereish

∈

Hsuchthat

(

α

h

_,

_β

h

₎

₌

₍

_γ

_,

_δ)

_{,whichimpliesthat}

_α

_∈

_γ

H_.Thisproves

thestatement.

(iii) If

1

=

2,then

(

α

,

β)

and

(

α

,

γ

)

generatethesameorbitalgraph.SobyLemma 15(i) thereis h

∈

H suchthat

(

α

h

_,

_β

h

₎

₌

₍

_α

_,

_γ

₎

_{.Thismeansthat}

_α

h

₌

_α

_{andthereforeh}

_∈

_H

α,whichimplies

that

γ

∈

β

Hα_{.Conversely, if}

_γ

∈

_β

Hα _{thenthere exists h}

∈

_H

α such that

(

α

h

,

β

h

)

=

(

α

,

γ

)

and

hence

1

=

2.

2

4.1. Futileorbitalgraphs

Afterthepreparatory resultsabove,we nowcharacterizethe situationswhereorbital graphsare beneficialinpartitionbacktrack.

Definition20(Futileorbitalgraph).Suppose thatHypothesis 14holdsandthat P is anordered orbit partition of H. We denote the stabilizerof P in G by Sym

(

P

)

, aswe did in Definition 1, andwe emphasize that Sym

(

P

)

stabilizesevery H-orbit(i.e.every cell ofthe orderedpartition P) asaset andthatitactsasthefullsymmetricgrouponeveryorbit.

Wesaythattheorbitalgraph

isfutileifandonlyifSym

(

P

)

,initsnaturalactionon

,induces

graphautomorphismson

.

Example21.We refertothegraphsinExample 13 forthegroup H

:=

(

123

),

(

45

),

(

46

)

≤

S

6,and

weusetheorderedorbitpartition P

:= [

1

,

2

,

3

|

4

,

5

,

6

]

.Thefirstgraph

1,withbase-pair

(

1

,

2

)

,isnot

futile.Weobservethatthetransposition

(

12

)

∈

S

6stabilizesthepartition P,butitdoesnotinducea

graphautomorphismon

1 because

(

1

,

2

)

isanarcin

1 and

(

2

,

1

)

isnot.

Thesecondgraph

2,withbase-pair

(

5

,

6

)

,isfutile.

Forthiswenotethat Sym

(

P

)

=

(

12

),

(

23

),

(

45

),

(

46

)

.Nowif g

∈

Sym

(

P

)

,then g permutesthe three isolatedverticesinthegraphanditpermutesthesetofverticesintheconnectedcomponent containing 4

,

5 and 6. Giventhat thisconnected componentis a complete graph,it followsthat g

inducesagraphautomorphismonthiscomponentandhenceonallofG.

Finally welookatthethirdgraph

3 withbase-pair

(

2

,

4

)

.Thisisacompletebipartite graph,so

againweseethatSym

(

P

)

inducesgraphautomorphisms.

Theintuitionbehindthisdefinitionisthatwewouldliketobeabletocharacterizeorbitalgraphs wherethegraphstructuredoesnotgiveusanyadditionalinformationcomparedtotheorbitstructure on

that comesfrom theaction of H. In afutile orbital graph, all theinformation that could be gainedfromthegraphstructurecanalreadybeseeninanorderedorbitpartition of

withrespect to H.Buildingsuchagraphwouldbe uselessfromacomputational perspective.Thereforeourmain theoretical resulton this topicclassifies futile orbital graphs. We also discusshow to detect futile orbitalgraphswithoutevenbuildingthem.

Wenotethatthefollowingresultdoesnotplaceanyrestrictionsaboutthenumberoforbitsof H

on

.Inparticulartherecouldbearbitrarilymanyisolatedpointsin

.

Theorem22.SupposethatHypothesis 14holds.Then

isfutileifandonlyifithasauniqueconnected com-ponent

ofsizeatleast2andmoreoveroneofthefollowingholds:

(a)

isacompletebipartitedigraphor (b)

isacompletedigraph.

Proof. Let P be an orderedorbitpartitionof H.Then Sym

(

P

)

actsontheset oforbits of H andit actsfaithfullyonthesetofverticesof

.Hencetoanswerthequestionwhether

isfutileornot,we onlyhavetoconsiderarcsin

.

We splitour proof into two casesdepending on whetherornot

isa proper digraph. In both caseswebeginbyproving thatthefutility of

impliesthat

istheuniqueconnectedcomponent ofsizeatleast2 andthat(a)or(b)holds,andthenwediscusstheconverse.

Case1:

isaproperdigraph.

Then

isnotself-pairedandLemma 15(i)and(ii)implythat,forall

ω

1

,

ω

2

∈

,thereisatmost

onearcbetweenthem.InthefollowingargumentswewilloftenrefertoLemma 15(iii)and(iv)as well.

Wesupposethat

isfutileandweproveinaseriesoflittlestepsthat

istheuniqueconnected componentofsizeatleast2 andthat(a)istrue.

(1)Suppose that

γ

,

δ

∈

are distinct andin thesame H-orbit. Then theyare not onan arc. In particular

α

H

₌

_β

H_.

Proof. As

γ

and

δ

arein thesame H-orbit, they lie inthe samecell ofthe partition P.It follows from the futility of

that the transposition

(

γ

,

δ)

∈

Sym

()

, which stabilizes P, induces a graph automorphismon

.Thereforeneither

(

γ

,

δ)

nor

(δ,

γ

)

isanarc.Fromthisandthefactthat

(

α

,

β)

∈

A itfollowsthat

α

H

₌

_β

H_.

₂

(2)Suppose that

ω

∈

ison an arc. Then itis eithera starting pointor an endpoint,butnot both.

Proof. ThisfollowsfromLemma 15(iii)and(1).

2

LetS

:=

α

H _andE

_:=

_β

H_{,andlet I}

_⊆

_{denotethesetofisolatedverticesof}

_. (3)

=

S

˙∪

E

˙∪

I.Moreover S

∪

E spans

,and

isacompletebipartitedigraph.

Proof. The firststatementfollowsfrom(2).Moreoverthereareno arcsbetweenverticesin S orE, respectively,by(1).Weshowthatallelementsof E areonanarcwith

α

:

Forall

γ

∈

E,we findthetransposition g

:=

(β,

γ

)

∈

Sym

(

P

)

,andit fixes

α

H _{point-wiseby (1).}

Thefutilityof

impliesthat gmapsthearc

(

α

,

β)

tothearc

(

α

,

γ

)

.Now itfollowsthat A

=

S

×

E

andhencethedigraphspannedby S

∪

E isacompletebipartite digraph.Thenitmustcoincidewith

and

istheuniqueconnectedcomponentofsizeatleast2 of

.

2

Conversely,wesupposethat

istheuniqueconnectedcomponentofsizeatleast2 of

andthat (a)holds.Weprovethat

isfutile.

LetS andE denotethesubsetsofthevertexsetof

suchthat allarcsstartat S andendatE. LetIbethesetofisolatedverticesof

,sothat

=

S

˙∪

E

˙∪

I.

Now

α

H

_⊆

_{S andthe bipartite structure impliesthat even}

_α

H

₌

_S.Similarly

_β

H

₌

_{E. Therefore}

Sym

(

P

)

stabilizes the sets S, E and I. We already knowthat Sym

(

P

)

permutes the vertices of

faithfully,sonowwelookatarcs.

Letg

∈

Sym

(

P

)

andlet

(

ω

1

,

ω

2

)

∈

A.Then

ω

1

∈

S,

ω

2

∈

E andthereexistssomeh

∈

H suchthat

(

α

h

_,

_β

h

₎

₌

₍

_ω

1

,

ω

2

)

. SinceSym

(

P

)

stabilizes thesets S and E, weseethat

ω

1g

∈

S and

ω

g

2

∈

E.The

completenesspropertythenimpliesthat

(

ω

₁g

,

ω

₂g

)

∈

A.

Conversely, if

(

ω

₁g

,

ω

₂g

)

∈

A, then there exists some h

∈

H such that

(

α

h

_,

_β

h

₎

₌

₍

_ω

g 1

,

ω

g 2

)

. Now

ω

1

=

α

hg− 1

∈

Sand

ω

2

=

β

hg− 1

∈

E whence

(

ω

1

,

ω

2

)

∈

Abycompleteness.

Hence

isfutile.

Case2:

isnotaproperdigraph,whichmeansthatitisself-paired.

Westillhaveourconnectedcomponent

andwebegin, oncemore,withthehypothesis that

isfutile.Let

γ

∈

beanarbitrary,non-isolatedvertex.

We knowthat

β

H

=

α

H _{by Lemma 15 (iii)and(iv), because}

_{is self-paired. As}

_γ

_{was chosen}

α

,

β,

γ

areallinthesame H-orbitandhenceinacommoncellofthepartition P.Inparticularthe transposition g

:=

(β,

γ

)

iscontainedinSym

(

P

)

and,becauseoffutility,itinducesagraph

automor-phismon

.

Then

(

α

,

β)

∈

A impliesthat

(

α

,

γ

)

=

(

α

g

_,

_β

g

₎

_∈

_{A.Thisargumentshowsthat}

_istheonly

con-nectedcomponentofsizeatleast2 in

andthat(b)istrue.

We conversely suppose that

is the unique connected componentof size atleast 2 of

and

that (b)holds.Together withthe definitionoforbitalgraphs (andthefact thatarcs alwaysgo both waysinthepresentcase)thisimpliesthat

α

H

₌

_β

H _spans

_{andthattheisolated vertices,viewed}

aselementsof

,arenotcontainedin

α

H_.

WeknowthatSym

(

P

)

actsfaithfullyonthevertexsetof

.Nowlet g

∈

Sym

(

P

)

andlet

ω

1

,

ω

2

∈

.Werecallthat

α

H

₌

_β

H _isSym

₍

_P

₎

_-invariant.

ThenitfollowsasinCase1,usingthecompleteness,that

(

ω

1

,

ω

2

)

∈

A ifandonlyif

(

ω

₁g

,

ω

g₂

)

∈

A.

ConsequentlySym

(

P

)

actsasagroupofgraphautomorphismson

,i.e.

isfutile.

2

Wegive anexampleinordertoillustratethatfutility ofan orbitalgraphisnot obviousandwhy furtherinvestigationsintothecomputationalusefulnessoforbitalgraphsshouldbepursued.

Example23.WeletG

:=

S

9 andwelookatthesubgroupH

:=

(

12

),

(

13

),

(

45

),

(

46

),

(

14

)(

25

)(

36

),

(

789

)

.Let

betheorbitalgraphforHwithbase-pair

(

1

,

2

)

.Then

hasthefollowingshape: Onthevertices1

,

2

,

3 and4

,

5

,

6 wehaveacompletedigraph,respectively,thereisnoarcbetween the sets

{

1

,

2

,

3

}

and

{

4

,

5

,

6

}

, andthe points 7

,

8 and9 are isolated. This might looklike a futile graph,butaccordingtothetheoremitisnot.Consideranorderedorbitpartition P

:= [

1

,

2

,

3

,

4

,

5

,

6

|

7

,

8

,

9

]

ofH.

The group Sym

(

P

)

contains thetransposition

(

24

)

∈

G.Thiselement interchanges thevertices2 and4 of

andfixes 1,sothiselement doesnotinduce anautomorphismon

.(Otherwisethearc

(

1

,

2

)

wouldbemapped tothearc

(

1

,

4

)

,whichdoesnot exist.) Thisgraphcan beusedto deduce, forexample,thatanyelementwhichswaps1 and4 mustalsoswap

{

2

,

3

}

with

{

5

,

6

}

.

Hence Sym

(

P

)

doesnotactasagroupofautomorphismson

andweseethat

isnotfutile.

It isimportant that we can detect futile graphs easily,without having to build them explicitly. We will nowgive a collectionoflemmasthat allow futileorbital graphsto be detected usingonly information aboutorbits andstabilizers of a group, without explicit construction of entire orbital graphs.

Lemma24.SupposethatHypothesis 14holdsandthat

=

α

H

_˙∪

_β

H

_˙∪

_{I,where I}

_⊆

_{isthesetofisolated} verticesof

.Then

isfutileifandonlyifHαactstransitivelyon

β

H.

Proof. Suppose that

is futile. Then Theorem 22 and Lemma 15 (iii) and (iv)imply that

is a complete bipartitedigraph. Inparticular,forall

δ

∈

β

H _{itfollowsthat}

₍

_α

_,

_δ)

_∈

_{A andsothereexists}

some h

∈

H such that

(

α

,

δ)

=

(

α

h

_,

_β

h

₎

_{. In particular H}

α is transitive on

β

H. Conversely we

sup-posethat Hα istransitiveon

β

H.Itfollowsthatforall

β

∈

β

H thereexistssome h

∈

Hα such that

β

h

=

β

.

We prove that Hβ acts transitively on

α

H, so we let

α

∈

α

H and we choose g

∈

H such that

α

=

α

g_{.Then, usingthetransitivity argumentabove,we leth}

_∈

_H

α be suchthat

β

h

=

β

g− 1

,which implies

β

hg

=

β

and

α

hg

₌

_α

_.ThereforeH

β actstransitivelyon

α

H.Now thedefinitionofanorbital

graphimpliesthat

isacompletebipartitedigraphandhencefutile,byTheorem 22.

2

We finish this section by giving some concrete bounds on the number of edges in futile and

non-futileorbitalgraphs.