CHAPTER
IV
ISOMORPHISM OF TR1V~.ENT GRAPHS AND OF CONE GRAPHS OF ]DEGI~E TWO
Considering
the length ol time the graph isomorphism
open despite extensive work, it is natural to seek more
problem.
problem
has remained
tractable restrictions of the
We now consider such a restriction and develop polynomial time isomor-
phism tests for graphs of fixed valence,
We define the v~der~ce of a vertex as the n u m b e r of edges incident to it and the
v a l e n c e of a graph X as t h e m a x i m u m v a l e n c e of its v e r t i c e s .
A s s u m e we have a n efficient i s o m o r p h i s m t e s t for g r a p h s of a c o n s t a n t v a l e n c e k,
a n d we now wish to t e s t i s o m o r p h i s m of g r a p h s of v a l e n c e k + l .
We could t r y to
r e d u c e t h i s p r o b l e m to t e s t i n g i s o m o r p h i s m of g r a p h s of v a l e n c e k by s u b s t i t u t i n g for
e a c h v e r t e x of v a l e n c e k+ 1 a s u i t a b l e s u b g r a p h F which t r a n s f o r m s t h e g r a p h into a
g r a p h of v a l e n c e k while p r e s e r v i n g i s o m o r p h i s m .
T h e r e is a g e n e r a l r e s u l t which
s t a t e s t h a t no s u c h s u b g r a p h F c a n e x i s t e x c e p t for r e d u c i n g g r a p h s of v a l e n c e 5 to
g r a p h s of v a l e n c e 4. F o r this r e a s o n , we believe t h a t t e s t i n g i s o m o r p h i s m of g r a p h s of
fixed v a l e n c e is n o t an i s o m o r p h i s m c o m p l e t e p r o b l e m ( s e e also C h a p t e r II, S e c tion 2).
In t h i s c h a p t e r , we give t h r e e p o l y n o m i a l t i m e i s o m o r p h i s m t e s t s for trivalent
graphs, i.e., g r a p h s of v a l e n c e t h r e e . T h e r e a r e s i m p l e p o l y n o m i a l t i m e i s o m o r p h i s m
t e s t s for g r a p h s of v a l e n c e one a n d two, so t h e t r i v a l e n t g r a p h s p r e s e n t t h e first nont r i v i a l case. We first outline t h e b a s i c a l g o r i t h m , o m i t t i n g t h e c r u c i a l c e n t r a l s t e p of
t a k i n g t h e s e t w i s e s t a b i l i z e r in c e r t a i n p e r m u t a t i o n g r o u p s .
We t h e n show how to
i m p l e m e n t this c e n t r a l s t e p in two e n t i r e l y d i f f e r e n t ways. One m e t h o d a p p l i e s t h e
r e s u l t s of C h a p t e r III, t h e o t h e r u s e s a new r e c u r s i v e p r o c e d u r e for c o m p u t i n g setwise
s t a b i l i z e r s in p - g r o u p s .
Our b e s t a l g o r i t h m is an O(n 4) i s o m o r p h i s m t e s t for t r i v a l e n t g r a p h s .
In it we
d e p a r t s l i g h t l y f r o m t h e b a s i c a p p r o a c h t a k e n by t h e first two m e t h o d s . In p a r t i c u l a r ,
we i m p r o v e t h e c o m p u t a t i o n a l t e c h n i q u e s for h a n d l i n g p - g r o u p s , do n o t t a k e setwise
s t a b i l i z e r s , a n d i n t r o d u c e a new t y p e of v a l e n c e r e d u c t i o n which is p o s s i b l e only
b e c a u s e of c e r t a i n p r o p e r t i e s of t r i v a l e n t g r a p h s .
1t5
The basic a p p r o a c h leading to t h e first two a l g o r i t h m g e n e r a l i z e s to g r a p h s of
h i g h e r valence. What m a k e s t r i v a l e n t g r a p h s a special case is t h a t the p e r m u t a t i o n
g r o u p s in which t h e setwise s t a b i l i z e r has to be d e t e r m i n e d a r e always T-groups. For
higher valence, m o r e g e n e r a l t y p e s of p e r m u t a t i o n g r o u p s m u s t be c o n s i d e r e d .
Most of the t e c h n i q u e s u s e d in the i s o m o r p h i s m t e s t for g r a p h s of valence k c a n
also be u s e d for a wider class of graphs.
In p a r t i c u l a r , the i s o m o r p h i s m t e s t for
g r a p h s of valence k m a y be used to t e s t ~somorphism of cone g r a p h s of degree k - 1 .
In this c h a p t e r , we will also c o n s i d e r cone g r a p h s of d e g r e e two, called
binary cone
graphs. Note, however, t h a t the 0 ( n 4) i s o m o r p h i s m t e s t for t r i v a l e n t g r a p h s is a n
e x c e p t i o n and c a n n o t be u s e d for all b i n a r y cone graphs.
1.
Basic Approach
1.1. Properties of the Automorphism Group
Recall t h a t t h e d i s t a n c e of a v e r t e x u f r o m a v e r t e x v in t h e g r a p h X is t h e l e n g t h
of a s h o r t e s t p a t h b e t w e e n u a n d v. Similarly, i f e = (vl,v2) is a n edge of X, we define
the d i s t a n c e of a v e r t e x u f r o m the edge e as the s m a l l e r of t h e d i s t a n c e s of u f r o m vl
a n d of u f r o m v 2. Note t h a t v 1 a n d v~ have d i s t a n c e 0 f r o m e.
The i s o m o r p h i s m t e s t s of this c h a p t e r are b a s e d on the o b s e r v a t i o n t h a t the a u t o m o r p h i s m group of a t r i v a l e n t g r a p h (and of a b i n a r y cone graph) is a l m o s t a T-group
(of. C h a p t e r ItI, Definition 7).
THEOREM 1 (Tutte)
Let X = (V,E) be a c o n n e c t e d t r i v a l e n t graph, e = (vl,v2) a n edge of X. T h e n Aute(X),
the g r o u p of all t h o s e a u t o m o r p h i s m s of X which stabilize t h e edge e, is a T-group.
Proof
Let Vk be t h e v e r t i c e s of X a t d i s t a n c e k f r o m e. Note t h a t Aute(X) is t h e
setwise s t a b i l i z e r of Vo i n t h e full a u t o m o r p h i s m g r o u p of X, a n d t h a t Aute(X) m u s t
also setwise stabilize the sets Vi. Generalizing Definition 4 of C h a p t e r III, we define
A(k)(X) as the pointwise s t a b i l i z e r in AuL(X) of all v e r t i c e s i n Vj, j -< k. Note t h a t A(°)
has i n d e x 1 or 2 in Aute(X). We will prove t h a t t h e i n d e x of A(k+~) in A(k), k ~ 0, is a
power of 2, f r o m which the t h e o r e m follows.
Let u be a n a r b i t r a r y v e r t e x in ¥~+1. There is a t l e a s t one v e r t e x u' e Vk s u c h t h a t
(u,u') is a n edge of X. Now if k = 0, t h e n t h e r e is a n o t h e r v e r t e x u" e Vk s u c h t h a t
116
( u ' , u ' ) is in E; o t h e ~ . s e ,
t h e r e is a v e r t e x u" E Vk_ x s u c h t h a t (u',u") is in E. We c o n -
s i d e r t h e o r b i t of u in AC4, Since u' is fixed in A(k), e v e r y v e r t e x w in t h e o r b i t of u
m u s t s a t i s f y w E V~+t a n d (w,u') E E. S i n c e t h e v a l e n c e of u ' is a t m o s t 8 a n d u" CVk+l,
t h e o r b i t of u i n A(k) h a s ! e n g t h e i t h e r 1 o r 2. S i n c e A{k+l) is o b t a i n e d f r o m A(~) b y s u c c e s s i v e l y s t a b i l i z i n g v e r t i c e s in Vk+ 1, t h e i n d e x (A(k):A(k+l)) m u s t be a p o w e r of 2 (see
C h a p t e r II, T h e o r e m 3).
By t h e s a m e c o n s i d e r a t i o n s , we r e a d i l y o b t a i n
COROLLARY 1
Let X = (V,E) be a b i n a r y cone g r a p h with r o o t v. Then Autv(X) is a 2-group.
In C h a p t e r V, we show how t o g e n e r a l i z e t h e s e r e s u l t s to g r a p h s of fixed v a l e n c e
and to cone g r a p h s of fixed d e g r e e . In t h i s c h a p t e r , we will c o n s i d e r t h e following two
problems:
PRo~
1
Given a c o n n e c t e d t r i v a l e n t g r a p h X a n d a n e d g e e of X, d e t e r m i n e g e n e r a t o r s for
Aute(X).
PROBf.mw 2
Given a b i n a r y c o n e g r a p h X with r o o t v, d e t e r m i n e g e n e r a t o r s for Autv(X).
Clearly, a polynomial time solution for Problem 2 gives us a polynomial time isom o r p h i s m test for binary cone graphs. Furthermore, a polynomial time solution for
Problem I will give us a polynomial time isomorphism test for connected trivalent
g r a p h s . To see this, l e t X a n d X' b e two c o n n e c t e d t r i v a l e n t g r a p h s to be t e s t e d for
i s o m o r p h i s m . C o n s i d e r t h e following p r o c e d u r e .
P i c k a n e d g e e of X. F o r e v e r y e d g e e' of X' do t h e following: Divide e b y i n s e r t i n g
as m i d p o i n t a new v e r t e x z, a n d likewise d i v i d e e' i n s e r t i n g a new v e r t e x z'. Add t h e
e d g e (z,z'), as shown in F i g u r e t below.
F o r t h e r e s u l t i n g g r a p h Z we d e t e r m i n e
Aut(z,z,)(Z). Note t h a t t h e r e is an i s o m o r p h i s m m a p p i n g t h e e d g e e to t h e edge e' iff
e v e r y g e n e r a t i n g s e t of Aut(z,z,)(Z) c o n t a i n s a p e r m u t a t i o n
e x c h a n g i n g z with z'.
Observe also t h a t t h e g r a p h Z d e p e n d s on t h e c h o i c e of t h e e d g e s e and e'.
Now it is also c l e a r t h a t a p o l y n o m i a l t i m e s o l u t i o n t o P r o b l e m 1 gives a p o l y n o m i a l t i m e i s o m o r p h i s m t e s t for t r i v a l e n t g r a p h s which a r e n o t c o n n e c t e d : S p l i t t h e
g r a p h s X a n d X' to b e t e s t e d for i s o m o r p h i s m i n t o c o n n e c t e d c o m p o n e n t s ; e a c h
117
X
X'
The g r a p h Z
Figure t
c o m p o n e n t is t h e n a c o n n e c t e d g r a p h of valence at m o s t three. Classify t h e s e c o m p o n e n t s into i s o m o r p h i s m classes. Then X' and X are isomorphic iff exactly half of the
c o m p o n e n t s in e a c h i s o m o r p h i s m class belong t o the g r a p h X.
1.8. Overall S t r u c t u r e of t h e Algorithm
We outline the basic a p p r o a c h to solving P r o b l e m t. Let X = (V,E) be a c o n n e c t e d
trivatent graph, e = (vvvz) an edge o f X, and a s s u m e we wish to d e t e r m i n e Aute(X).
Let Vk be the set of v e r t i c e s of distance k f r o m e, and let h be the height of X, i.e., let
V = VoUV1U • - • UVh. We define s u b s e t s E k of edges of X, 0 "~ k "~ h, by
Ek = ~ (u,w) I u E Vk, w e VkuVk.I
a s s u m i n g t h a t Vh+1 = ~b. We now define the g r a p h s X~, 0_< j _< h + l :
x o = (Vo, ¢)
x l = ( V o u V . Eo)
X~ = (VoUVIuVz, EoUEI)
x~ = (v, EoU - "
UE~_~)
xh+~ = (v, E) = x
We wish to d e t e r m i n e Aute(Xk) for e a c h g r a p h XkEXAm'I~ l
L e t X = (V,E), where V = tl ..... i2], E = t(I:2), (1,3), (1,4), (2,5), (8,6), (3,4), (3,7), (4,8),
(5,9), (5,10), (B,9), (6,I0), (7,11), (7,18), (8,11), (8,12), (9,11), (10,12)I. Xis a c o n n e c t e d
trivalent graph.
Let e = (1,8).
Then Vo = It,21, vl = 13,4,5,61, V~ = 17,8,9,10], and
Vs = ~t1,12{. Also, Eo = I(1,8), (1,3), (1,4), (2,5), (8,B)I, El = I(&4), (3,7), (4,8), (5,9),
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(5, i0), (6,9), (6,10)1, E~ = t(7,11), (7,18), (8,1t), (8,18), (9, i1), (lO.18)l, and Ea is
empty. The graphs X1 and X~ are shown in Figures 2 and 3 below. [3
3
4
5
\/ \/
8
!--2
The graph X1
Figure 8
7
8
9
a
4
;/
\/
l . - -
i0
\/
a
The graph X~
Figure 3
Note t h a t Aute(Xo) is always <(vl,v~)>. We will determine Aute(X) = Aute(Xh+l) by
determining generators for Aute(Xk.1) from generators for Aute(Xk), O ~ k ~ h. Let
A = Aute(Xk). The method for determining Aute(Xk+l) divides into two parts.
First,
determine the subgroup B of A consisting of all permutations in A which m a y be
extended to an automorphism of Aute(Xk+1), and extend B to the larger permutation
domain. Second, we determine generators for the pointwise stabilizer of all vertices
in V 0 U . • • u V k in the group Aute(Xk+l). We will see that the first step can be reduced
to finding the setwise stabilizer in 2-groups, whereas the second step, due to the
definition of the edge set Ek+ I, can be accomplished by inspection.
Specifically, we proceed as follows: Consider the group A obtained by restricting
A = Aute(Xk) to the vertex set V k, Generators for A are readily obtained from the generators for A since A stabilizes Vk setwise. We now determine the subgroup B consist-
ing of those permutations in A w ~ c h may be extended to automorphisms of the graph
(VkUVk+~, Ek). Since B is a subgroup of A, every element of B may be extended to an
automorphism in Aute(Xk÷~). Note that B is the restiction of the group Aute(Xk+1) to
Yk. We remark for the present that the extension of an element of B to an element of
119
Aute(Xk+l) is a simple m a t t e r .
Now a s s u m e t h a t we have e x t e n d e d e v e r y g e n e r a t o r of B to an e l e m e n t of
Aute(Xk+l). We have to add g e n e r a t o r s for the n o r m a l s u b g r o u p of Aute(Xk.l) fixing
pointwise the vertices in Vk. These additional g e n e r a t o r s c o m e f r o m two sources:
automorphisms
which
fix
VkuV~÷ 1 pointwise,
and
automorphisms
which
fix
V0U - - . UV k pointwise. The f o r m e r m u s t be a g e n e r a t i n g set for the pointwise stabilizer of Vk in Aute(Xk). It is easily obtained using the techniques of Chapter II. Using
the n o t a t i o n of Section 2 of C h a p t e r III, we note t h a t the l a t t e r set g e n e r a t e s the
group A(k)(Xk+l), and is obtained by inspection of the edge s e t E k, as we will now
explain.
Let us call an edge (u,w) of X a cross edge if u and w have the same distance f r o m
e. We observe t h a t Xk+l has no cross edges c o n n e c t i n g v e r t i c e s in Yk+l. So the gene r a t o r s for this group are transpositions (u,w), u, w ~ Vk+ 1, where for e a c h edge (u,z)
in Ek t h e r e is an edge (w,z) in Ek, and vice versa. (Note t h a t z is a v e r t e x in VIe.)
The union of t h e s e t h r e e sets clearly g e n e r a t e s Aute(Xk+l).
EXAi,~I~ 2
Consider the g r a p h X of Example t. We s k e t c h the d e t e r m i n a t i o n of Aute(X2) f r o m
Aute(Xl), where
e is the edge (1,2).
We find h e r e t h a t the g r o u p Aute(Xz) is
A = <(1,2)(3,5)(4,6), (3,4)>. The g r o u p A, therefore, contains the following eight p e r m u t a t i o n s : 0, (3,4), (5,6), (3,4)(5,6), (3,5)(4,6), (3,6)(4,5), (3,6,4,5), (8,5,4,6).
Since t h e r e is the edge (3,4) b u t not an edge (5,6) in E 2, we find t h a t only the following four p e r m u t a t i o n s
in A are also in B: 0 ,
(3,4), (5,6), (3,4)(5,6).
Thus,
= <(3,4), (5,6)>, an e l e m e n t a r y Abelian g r o u p of o r d e r 4. The g e n e r a t o r s of B m a y
be e x t e n d e d to the p e r m u t a t i o n s (3,4)(7,8), and (5,6)(& 10) in Aute(X2). Next, i n s p e c t ing the g r a p h X2, we find t h a t the transposition (9,10) is the only g e n e r a t o r for
A(1)(Xa). F u r t h e r m o r e , the pointwise stabilizer of f&4,5,6~ in Aute(Xz) is t h e trivial
group. We t h e r e f o r e conclude t h a t Aute(X2) = <(3,4)(7,8), (5,6)(9,10), (9,10)>. []
1.3. R e d u c t i o n t o t h e S e t w i s e S t a b i l i z e r i n a 2 ~ r o u p
Having outlined the global s t r u c t u r e of the a l g o r i t h m for P r o b l e m l, we now go
into the details of its design. We begin with the two parts of Step one: determining
120
g e n e r a t o r s f o r B, t h e r e s t r i c t i o n of Aute(Xk+l) t o Yk, a n d e x t e n d i n g t h e s e g e n e r a t o r s
to e l e m e n t s in Aute(Xk+1).
Throughout, we assume that X = (V,E) is a connected triva!ent graph, e = (vl,vs)
an edge of X. As before, V k is the set of vertices at distance k from e and h is the
h e i g h t of X. The e d g e s e t s El, ..., Eh a n d t h e s u b g r a p h s X0
.....
Xh÷ 1 a r e d e f i n e d as in
S u b s e c t i o n 1.2.
L e t A = Aute(Xk). S i n c e A s t a b i l i z e s Vk setwise, we c a n o b t a i n A, t h e r e s t r i c t i o n of
A to t h e s e t Vk, s i m p l y by r e s t r i c t i n g e a c h g e n e r a t o r of A to Vk. By T h e o r e m 1, b o t h A
and A a r e 2 - g r o u p s . Given t h e s u b s e t Y of X, t h e r e is a n a t u r a l e m b e d d i n g of Sym(Y)
as a s u b g r o u p i n t o Sym(X). With t h i s e m b e d d i n g in mind, we define t h e s u b g r o u p B o~
Aby
B = I ~ e A ! (~
C Sym(Vk+1))(w~ E Aute(Xk+1) )
and let B be the restriction of B to V k. Clearly B < A.
Our first task will be to show h o w to determine B from a setwise stabilizer in some
2-group.
Then we will address the problem of how to extend B to a subgroup of
Aute(Xk+1).
Recall Theorem 7 of Chapter If. The theorem showed how to obtain the automorphism group of an arbitrary graph from the intersection of a specific permutation
group with a direct product of synunetric groups, i.e.,from the setwise stabilizer in a
particular permutation group. The theorem was obtained by considering the induced
action of a vertex permutation on the set of all vertex pairs, thereby translating
edges into a suitable labelling of points in the n e w permutation domain.
We will now
use a similar trick to obtain B as the setwise stabilizer of an isomorphic representation of A. A minor difficulty arises from the fact that there are edges in E k connecting
vertices in V k with vertices in Vk+ I. These edges do not translate into pairs of vertices
of Vk, but certain collections of these edges can be represented as small subsets of V k
as we now explain:
Let u e Vk+ i. Define the =~%cesfry of u as the set of vertices w e Vk such that (u,w)
is an edge of X (and thus in Ek). Since X is connected, every vertex in Vk+1 has a
nonempty ancestry. Furthermore, since X is trivaient, every ancestry has cardinality
at most 3.
We will classify the edges in Ek into hypes and will group t h e m into Isrn///es. Let
(w,v) E Ek. If (w,v) is a cross edge, i.e.,if both w and v are in V k, then the edge has the
121
type to.z. Otherwise, let w E Vk+1, v E V:k, Then the type of the edge (w,v) is tLj, where
is the cardinality of the ancestry of w, and i-I is the n u m b e r of vertices in Vk+ I with
the s a m e ancestry as w. In the trivalent ease, note that j ~ 3 and i ~ 2. Next, if (w,v)
h a s t y p e t0,g, t h e n its f a m i l y is I(w,v)l and t h u s c o n s i s t s of a single c r o s s e d g e only.
Otherwise, ff (w,v) h a s t y p e ti,j, i > 0, t h e n t h e f a m i l y of t h i s e d g e is t h e s e t
I(w,v), (w,vs) ..... (w,vj), (wz,v) ..... (wi,wj) l, c o n s i s t i n ~ of t h e i.j e d g e s in E k c o n n e c t i n g
t h e v e r t i c e s w, w 2. . . . . w i in Vk+ I with t h e i r c o m m o n a n c e s t r y , t h e v e r t i c e s v, v2 . . . . . vj in
Vk. F i g u r e 4 below shows t h e f a m i l i e s a n d t y p e s w h i c h o c c u r in t h e t r i v a l e n t case.
Note that a family F of edges of type ti,j, i > 0, spans the bipartite graph Ki,j.
VI
V2
V3
VI
ts,~
/T\
Vl
V2
V2
tal
t2,2
V3
/\
v1
tI,3
V1
v2
t1,~
V1
T
vl
tl,1
Vs
to.~
Types a n d F a m i l i e s of T r i v a l e n t G r a p h s
Figure 4
If F is a f a m i l y of e d g e s of t y p e ti3, t h e n (F)k d e n o t e s t h e s e t of j v e r t i c e s of Vk
i n c i d e n t t o t h e family, a n d (F)k. I d e n o t e s t h e s e t of i v e r t i c e s in Vk. 1 i n c i d e n t t o t h e
family.
L~L~ 1
Let ~ E Aute(Vk÷1). F o r e v e r y f a m i l y F in E~ of t y p e tLj. e i t h e r a s t a b i l i z e s F o r i t m a p s
F onto a f a m i l y F' of t h e s a m e t y p e .
122
Proof
Let F be a n y family of t y p e t W. If ~ stabilizes (F)k, t h e n ~ m u s t also s t a b i l -
ize (F)k+l. T h e r e f o r e a stabilizes t h e family F of edges. If a m a p s (F)k into a set (F')k,
then, by the d e f i n i t i o n of family, ~ m u s t m a p (F)k+l into (F')k+l, a n d so t h e family F is
m a p p e d i n t o a family F' of edges of t y p e tr, s, r -> i, s -> j. By t h e p i g e o n h o l e p r i n c i p l e ,
r = i a n d s = j, f r o m which the l e m m a follows. -
tf for every family F i n Ek t h e a u t o m o r p h i s m ~ c Aute(Xt) e i t h e r stabilizes (F)k or
m a p s this s e t i n t o t h e s e t (F')k of a family F' of type equal to F, t h e n a c a n be
e x t e n d e d to a n a u t o m o r p h i s m in Au%(Xk.1).
Proof
If a satisfies the h y p o t h e s i s of t h e l e m m a , t h e n it c l e a r l y m a p s cross
edges to cross edges. So, !et F a n d F' be f a m i h e s of edges which a r e n o t cross edges.
Then, by the d e f i n i t i o n of family, (F)k÷l (3 (F')k+l = ¢, f r o m w h i c h t h e l e m m a follows. =
COROLL~Y 2 ( F u r s t , Hopcroft,
Luks)
Let a c Aute(Xk+l) be a n a u t o m o r p h i s m which fixes e v e r y v e r t e x in Vk. Then ~ s t a b i l izes e v e r y s e t (F)k+~, F a family in Ek.
Let W be t h e c o l l e c t i o n of all s u b s e t s of Vk of size 1, 2, or 3. Note t h a t G, defined
by t h e i n d u c e d a c t i o n of A on W, is a g a i n a 2-group. We label the p o i n t s in W with t h e
labels ~,t, 0 ~ i -~ 2, 1 -- j ~ 3, w h e r e t h e p o i n t z is l a b e l l e d t W if z = (F) k for s o m e f a m ily F of edges in Ek of type rid. The r e m a i n i n g , u n l a b e l l e d p o i n t s of W are now labelled
t0,0. Observe t h a t a n y p o i n t in W c a n have a t m o s t two labels; t h a t is, the only m u l t i ple labelling possible is t~, 2 a n d t0,2. S u c h p o i n t s m a y be l a b e l l e d t0,L2. Note t h a t we
have u s e d up to n i n e labels. Let H be t h e s u b g r o u p of G c o n s i s t i n g of all e l e m e n t s in G
which r e s p e c t the labelling of W, a n d n o t e t h a t H is the setwise s t a b i l i z e r of the subsets of p o i n t s of W with the s a m e label. Let ~ be t h e s u b g r o u p of A c o r r e s p o n d i n g to
the s u b g r o u p H of G. By L e m r n a t a 1 a n d 2, it is now c l e a r t h a t B is the s u b g r o u p of
c o n s i s t i n g of all p e r m u t a t i o n s of Vk which m a y be e x t e n d e d to a n a u t o m o r p h i s m in
Aut~(Xk+l).
Consider the g r a p h X of E x a m p l e 1. The edge s e t Ee c o n t a i n s the following four families:
f(3,4)t,
a family of type to, ~,
f(3,7)~,
a family of type tL1,
t(4,a)t,
also of type tl, l,
a n d /(5,9), (5,10), (6,9), (6,10)t, a family of type t2, 2. The s e t W c o n s i s t s of the following 14 points: x I = Ial, x~ = i41, xs = [5t, x4 = ~6t, x5 = I3,4t, x~ = t3,5~, x7 = t3,61,
x~ = t4,5~, x9 = i4,8{, Xlo = 15,6f, xll =
f3,4,5t,
xlz
TM
[3,4,6/, xi3 = [3,5,6l, Xl4 = [4,5,6~.
123
The g r o u p G defined by t h e i n d u c e d a c t i o n of A = <(3,4), (3,5)(4,6)> on W is g e n erated
by
the
two
permutations
(xl,xs)(xe,x4)(x~,xl0)(xT,xs)(xn,xls)(xl~,xl4)
and
(x.x2)(x~,xs)(xT,x~)(x13,x.).
The edge s e t E s i n d u c e s t h e following four blocks on W via the labelling d e s c r i b e d
above: J1 = Ixs], labelled to, e, Je = ~xt,x21, labelled tL~, Js = ~xlo], labelled t~,2. The
r e m a i n i n g p o i n t s i n W f o r m i n t h e block J4 a n d are labelled t0,0. [:3
Because of the presence of singletons in W, it is easy to obtain generators for
from generators of %he subgroup H of G, the setwise stabilizer of the induced partition
blocks of W.
We will now show how to extend these generators to elements of
Aute(Xk+a).
Let 7r e B be a p e r m u t a t i o n , of Vk~ Since g e n e r a t o r s for Aute(Xk) a r e a l r e a d y
known, we m a y a s s u m e t h a t a r e p r e s e n t a t i o n m a t r i x for Aute(Xk) is also k n o w n where
we a s s u m e t h a t the p o i n t s in Vk are s t a b i l i z e d first. Thus, in o r d e r to e x t e n d 7r to an
a u t o m o r p h i s m ~ in Aute(Xk), we p r o c e e d as follows: We t e s t m e m b e r s h i p of ~r i n
Aute(Xk). In g e n e r a l , we will discover t h a t 7r is n o t i n this g r o u p , b u t i n t h e c o u r s e of
the m e m b e r s h i p t e s t we will have c o m p u t e d 0 ' = ~r~i" 1 . . . ~r-1 where 7r' fixes Vk
pointwise (cf. C h a p t e r II, S e c t i o n 3). So ~ = ~r " " " @1 is a n e l e m e n t of Aute(Xk) a n d
m u s t agree with ~r on the s e t Vk, since 7r e A. Next, n o t e t h a t it is easy to e x t e n d ~ to
a p e r m u t a t i o n X of "COL) • . • UVk+~ which p r e s e r v e s t h e edges in EL, s i n c e t h e families
F i n Ek i n d u c e a p a r t i t i o n of Vk+I with t h e sets (F)k+1 as its blocks. Therefore, i t is
clear how to find t h e d e s i r e d e x t e n s i o n of ~r to a n a u t o m o r p h i s m i n Aute(Xk+~).
I,~T~La 3
Let *~1 be a g e n e r a t i n g s e t for B, K1 its e x t e n s i o n in Aute(Xk). Let Ke be a g e n e r a t i n g
s e t for C, the pointwise s t a b i l i z e r of V~ in Aute(Xk). Then K1L)Ks g e n e r a t e s B, t h e s u b group of all a u t o m o r p h i s m s in Aute(Xk) which m a y be e x t e n d e d to a u t o m o r p h i s m s i n
Aute(X~÷l).
Proof
Clear f r o m t h e definition of B. -
As a c o n s e q u e n c e of L e m m a 3 a n d the p r e c e d i n g discussion, we now have a way of
finding g e n e r a t o r s for t h e g r o u p B and, f u r t h e r m o r e , a way of e x t e n d i n g t h e s e g e n e r a tors to e l e m e n t s in AUte(Xk+I). Thus, we have a n a l g o r i t h m for Step one of t h e overall
c o n s t r u c t i o n o u t l i n e d previously.
t24
We now t u r n to S t e p two, t h e d e t e r m i n a t i o n of t h o s e a u t o m o r p h i s m s in Aute(Xk÷l)
which p o i n t w i s e s t a b i l i z e t h e v e r t i c e s of V0U • " - UVk.
By C o r o l l a r y 2, e v e r y a u t o m o r p h i s m in A(k)(Xk+l) m u s t s e t w i s e s t a b i l i z e t h e v e r t e x
s e t s (F)k+l, F t h e farnilies of E k. Clearly, if F is a n y family, t h e n t h e v e r t i c e s in (F)k+l
m a y b e p e r m u t e d a r b i t r a r i l y , t h u s A(k)(Xk+l) m u s t be t h e d i r e c t p r o d u c t of t h e s y m m e t r i c g r o u p s of t h e s e sets. Since we a s s u m e t h a t X is a t r i v a l e n t g r a p h , no s e t (F)k+l
has c a r d i n a l i t y e x c e e d i n g 2, t h u s A(k)(Xk+i) is g e n e r a t e d b y d i s j o i n t t r a n s p o s i t i o n s a n d
is t h e r e f o r e a n e l e m e n t a r y Abelian 8-group.
EXA/m'LE 4
Let X be t h e g r a p h of E x a m p l e 1, a n d c o n s i d e r t h e v e r t e x s e t V s = 17,8,9,10t. The f a m ilies of E2 p a r t i t i o n Y2 into t h e b l o c k s B~ = ~7], B~ = Is1, a n d Bs = ~9,I0t. Thus, t h e
g r o u p A(2)(Xs), c o n s i s t i n g of all a u t o m o r p h i s m s in Aute(Xs) which fix t h e v e r t i c e s i
t h r o u g h 6, is p r e c i s e l y <(9,10)>. []
1.4. B i n a r y C o n e G r a p h s
Having o u t l i n e d t h e d e s i g n of a n a l g o r i t h m for P r o b l e m 1, we now d i s c u s s b r i e f l y
how t h e s e t e c h n i q u e s a p p l y to P r o b l e m 2, d e t e r m i n i n g t h e a u t o m o r p h i s m s of b i n a r y
cone g r a p h s .
Let X b e a b i n a r y c o n e g r a p h of h e i g h t h w i t h r o o t v, Vk t h e s e t of v e r t i c e s of dist a n c e k f r o m t h e r o o t . Let Ek be t h e s u b s e t of e d g e s c o n s i s t i n g of t h e B F S - t r e e e d g e s
c o n n e c t i n g t h e v e r t i c e s in Vk with t h e v e r t i c e s in Vk+ 1, a n d all c r o s s e d g e s i n c i d e n t to
v e r t i c e s in Vk. Define t h e g r a p h s Xk, t -< k ~ h, b y
xk =
(vocJ
~.
uvk, EoU - . .
uE,-D
As in the trivalent case, we determine Autv(Xk) for each graph X k by determining generators for Au~(Xk÷1) from generators for Aut~(Xk) and the edge set E k. Note that
Autv(Xl) has order either I or 2 and is determined easily from the graph X I.
Just as before, we classify the edges in Ek into types and group t h e m into families. Since X is a binary cone graph, the only possible types are t0,a, t~,i, and tz,i, thus
we c a n r e p r e s e n t A on t h e s e t W c o n s i s t i n g of all s u b s e t s of Vk of size t or 2. Again,
t h e f a m i l i e s F in E k i n d u c e a p a r t i t i o n of W, in t h i s e a s e i n t o a t m o s t f o u r b l o c k s . The
setwise stabilizer of these blocks then gives us the subgroup B, the restriction of
125
Aut~Xk+l) to the vertex set V~.
From these observations, it should be clear that Problem 2 can be solved by the
same techniques as the ones developed for Problem 1. Because W is of size 0(n 2) for
binary cone graphs, instead of O(ns) in the trivalent case, the algorithm for Problem 2
has a faster running time. However, this difference in performance can be eliminated
by replacing certain edges in Ek+ 1 with specially labelled cross edges. We will discuss
this technique in Section 4 of this chapter.
In the remainder of the chapter we
merely state the results for binary cone graphs without explicit proofs. The reader
should have no difficulty in working out the necessary details.
2.
An AlEorithm for Determinin£ the Automorphisms of Trivalent Graphs
In Section 1 we have developed the basic design of a polynomial time algorithm
for Problem 1 (and thus also for Problem 2). Algorithm 1 below formally specifies the
method. The major work is done in the main loop which successively determines generating sets for the groups AUte(Xk) given the connected, trivalent graph X.
By far the largest part of the loop is devoted to Step one, constructing the subgroup B of A = AUte(Xk) consisting of those automorphisms in A which m a y be
extended to an automorphism of Xk÷l. This happens in Lines 6-31, and includes
finding generators for B, extending those generators, and finding generators for the
pointwise stabilizer of Vk in A. Step two, determining the group A(k)(Xk+I), the pointwise stabilizer of VoU . . . uVk in Aute(Xk+l), is done in Lines 22 and 23 and is, as
already remarked, very straightforward.
The algorithm has certain inefficiencies. We discuss some of the sources of
inefficiency at the end of this section, and improve the algorithm with the material of
the next section. Finally, in Section 4, we design an almost practical algorithm for
trivalent graph isomorphism.
126
AL~RrlR~ I (Automorphism of a Connected, Trivalent Graph)
Input
The c o n n e c t e d , t r i v a l e n t g r a p h X = (V,E) a n d t h e e d g e e = (vl,v~) of X.
Output
A g e n e r a t i n g s e t for Aute(X).
Method
L
2,
begin
Using b r e a d t h - f i r s t s e a r c h , d e t e r m i n e t h e v e r t e x s e t s "Co. . . . . Vh a n d t h e e d g e
s e t s E o ..... Eh;
3.
D e t e r m i n e Ko, a g e n e r a t i n g s e t for Aute(X0), b y i n s p e c t i o n ;
4.
f o r k := 0 t o h do b e g i n
5.
F r o m t h e s e t Kk, c o n s t r u c t t h e s e t K, g e n e r a t i n g t h e g r o u p A t h a t is t h e r e s t r i c t i o n of Aute(Xk) to Vk;
6.
C o n s t r u c t W, t h e s e t of all s u b s e t s of Vk of size 1, 2, o r 3;
7.
F r o m K, c o n s t r u c t D, a g e n e r a t i n g s e t for t h e g r o u p G d e f i n e d b y t h e i n d u c e d
a c t i o n of A on W;
8.
Classify t h e e d g e s in E~ b y t y p e a n d g r o u p t h e m into t h e f a m i l i e s F 1. . . . . Fr;
9.
P a r t i t i o n W i n t o t h e b l o c k s J1 . . . . . J9 a c c o r d i n g to t h e t y p e s of t h e f a m i l i e s F
a n d t h e s e t s (F)k;
10.
F i n d a g e n e r a t i n g s e t for H, t h e s e t w i s e s t a b i l i z e r of t h e b l o c k s Jl, -.-, J9 in t h e
2 - g r o u p G;
ti.
F r o m t h e g e n e r a t i n g s e t for H, c o n s t r u c t D, a g e n e r a t i n g s e t for B, t h e r e s t r i c t i o n of Aute(Xk+1) to Vk;
12.
C o n s t r u c t a r e p r e s e n t a t i o n m a t r i x for A = Aute(Xk), w h e r e t h e v e r t i c e s in Vk
a r e s t a b i l i z e d first;
13.
i n i t i a l i z e Kk+l, t h e g e n e r a t i n g s e t for Aute(Xk÷l), to c o n t a i n t h e g e n e r a t o r s of
the pointwise s t a b i l i z e r of Vk in Aute(Xk);
14.
f o r e a c h g e n e r a t o r ~ in D do b e g i n
15.
find ~ E A whose
restriction to V k is ~;
t6.
if k < h t h e n b e g i n
17.
find X, t h e e x t e n s i o n of ~ t o a n a u t o m o r p h i s m of Xk÷l;
18.
a d d X t o Kk+t;
t 9.
end
20.
e l s e a d d ~ to K~+I;
2 i.
end;
127
22.
for each set (F)k+1 do
23.
24.
25.
if (F)i+1 = lu,wl then add ~ = (u,w) to Kk+1;
end;
output(Kh÷1);
36. end~
We analyze Algorithm I firstwithout accounting for the exact bound on Step 10
which is in charge of finding the setwise stabilizer in a 2-group. W e assume that Step
I0 requires T(m) steps, where m
is the degree of G, assumin£ further that G is
presented by a generating set of size O ( m ~) and that the generating set determined
for H also is at most of size O(m2). Clearly these assumptions are realistic.
Assume that X has n vertices, and observe that X cannot have more than 3n.
2
edges. Let n~ be the cardinality of V~, and note that Ek is 0(nk+nk÷1) in size.
We begin by estimating the time required in each iteration of the for-loop extending through Lines 4-24. Assuming that K k has cardinality O(n~), Line 5 requires O(n 3)
steps. The resulting generating set is for a group of degree nk. Therefore, by
Chapter II, we m a y reduce the set K to size O(n~) at the cost of O(nZ.n~+n~) steps.
Line 6 requires 0(n~) steps, and results in a set W of size O(n~). Thus, we can construct D from K in 0(n~) steps (Line 7), processing O(n~) generators. Classificationof
the 0(nk+nk+1) edges in Ek requires at most O(nk+nk+l) steps since there are only 7
distinct types and the largest possible family consists of 6 edges, i.e.,since the families have constant size. Subsequently, the set W can be partitioned in O(n~+nk+~)
steps, so that Lines 5-9 require a total of O(n3+n&n~+nkS+nk+1) steps.
By assumption, Line 10 requires T(n~) steps and delivers a generating set of size
at most n~, since H has degree 0(n~).
Line t I constructs a generating set for B. Because of the possibly very large generating set for H, this line costs 0(n~) steps. The resulting generating set can be
reduced to size 0(n~), in the same time bound.
Line 12 now requires O(n~.nZ+n s)
steps and allows us to initialize Kk+1 in O(n 3) steps in Line 13.
The loop in Lines 14-21 is executed at m o s t 0(nk2) times. Step 15 is done by partially sifting 7r, t h u s requires O(n 2) steps. Step 16 can be done in O(nk+nk+1) steps,
thus Lines 14-21 take at m o s t 0(n2-n~+n~+n~÷l) steps.
128
The loop in Lines 22 a n d 23 tak.es n o m o r e t h a t O((nk+nk÷l).n) s t e p s , a s s u m i n g t h e
p e r m u t a t i o n s a r e s t o r e d as v e c t o r s of l e n g t h n. T h e r e f o r e , Lines 5-23 r e q u i r e a t o t a l
of a t m o s t O(n~+n~+T(n~)) s t e p s , o b s e r v i n g t h a t n ~ n~. Note t h a t t h e s e t K~÷I is of
size at m o s t 0(ha), and t h a t h is a t m o s t 0(n). Thus, t h e loop of Lines 4-24 r e q u i r e s a
h
total of O(n~+ ~ T(n~)) steps.
k=1
Under the assumption that T(m) increases monotonically with m and at least as
h
fast as linearly in m, we obtain ~ T(n~) -< T(nS). Clearly, the running time of the loop
k=l
d o m i n a t e s all o t h e r s t e p s . T h e r e f o r e , in s u m m a r y , we have j u s t shown
T~om~
2
Let X be a c o n n e c t e d , t r i v a l e n t g r a p h with n v e r t i c e s . A s s u m e we have a p r o c e d u r e
w h i c h d e t e r m i n e s , in T(m) s t e p s , a n O(m z) g e n e r a t i n g s e t for t h e setwise s t a b i l i z e r in
a 2 - g r o u p G of d e g r e e m, p r e s e n t e d b y a g e n e r a t i n g s e t also of size 0(m2).
Then a
g e n e r a t i n g s e t for Aute(X ) c a n b e f o u n d in O(ng+T(n~)) s t e p s .
CoROU.~%r 3
Let X be a b i n a r y c o n e g r a p h with n v e r t i c e s a n d r o o t v. A s s u m e we have a p r o c e d u r e
w h i c h d e t e r m i n e s , in T(m) s t e p s , a n 0 ( m z) g e n e r a t i n g s e t for t h e setwise s t a b i l i z e r in
a 2 - g r o u p G of d e g r e e m, p r e s e n t e d b y a g e n e r a t i n g s e t also of size O(mS). Then a
g e n e r a t i n g s e t for Autv(X) c a n b e f o u n d in O(n~+T(nS)) s t e p s .
We now a p p l y t h e r e s u l t s of C h a p t e r III, S e c t i o n 3. By T h e o r e m 18 of C h a p t e r III,
we o b t a i n i m m e d i a t e l y
CORO~
4 (Furst, Hoffmann, Hoperoft, Luks)
Let X b e a c o n n e c t e d , t r i v a l e n t g r a p h with n v e r t i c e s . Then g e n e r a t o r s for Aute(X) c a n
be f o u n d in 0 ( n 2~) s t e p s .
COROUm~I~ 5
Let X be a b i n a r y c o n e g r a p h w i t h n v e r t i c e s a n d r o o t v. Then g e n e r a t o r s for Autv(X)
c a n b e f o u n d in 0 ( n 14) s t e p s .
We have j u s t solved P r o b l e m s i a n d 2 in p o l y n o m i a l t i m e . As a c o n s e q u e n c e , we
now have a p o l y n o m i a l t i m e i s o m o r p h i s m t e s t for b i n a r y c o n e g r a p h s a n d for t r i v a l e n t
graphs.
The b o u n d s in C o r o l l a r i e s 4 a n d 5 c a n be s i g n i f i c a n t l y i m p r o v e d . The i m p r o v e m e n t s c o m e f r o m s e v e r a l s o u r c e s : F o r one, we should s e e k t o r e d u c e t h e d e g r e e of
129
the 2-group G in which we d e t e r m i n e the setwise stabilizer H, In the trivalent case, it
is n o t h a r d to r e d u c e this d e g r e e to 0 ( n z) by replacing families F of t y p e tj, 3 by t h r e e
specially labelled cross edges connecting the vertices in (F)k. Note t h a t the newly
a d d e d edges for families of type tl, 3 m u s t have labels different f r o m the labels for the
edges a d d e d for families of type tz, 3. The o t h e r s o u r c e for improving the running time
is to seek b e t t e r t e c h n i q u e s for Step 10, and we will explore this idea in the n e x t section. Finally, a c e r t a i n a m o u n t of duplication of work can be eliminated, resulting in a
f u r t h e r savings in the running time.
3.
Setwise Stabilizers in p-Groups (Method 2)
As we have seen above, the efficiency of Algorithm 1 depends crucially on the t i m ing of Step I0, the d e t e r m i n a t i o n of a setwise stabilizer in a 2-group. In C h a p t e r III,
we gave a polynomial time algorithm for finding setwise stabilizers in p-groups
(Chapter llI, T h e o r e m 18). We will now develop a different m e t h o d for this p r o b l e m
based on new techniques.
The t e c h n i q u e s t o be i n t r o d u c e d a r e only indirectly r e l a t e d to t h e t e c h n i q u e s of
the p r e c e d i n g chapters.
We t h e r e f o r e begin with an intuitive, p e r h a p s s o m e w h a t
vague, d e v e l o p m e n t of the underlying ideas.
Let G < Sym(X) be an intransitive p e r m u t a t i o n g r o u p with orbits A1..... As. Let Y
be a s u b s e t of X. Assume we wish to d e t e r m i n e Gy, t h e setwise stabilizer of Y in G.
Let Y~ = Y(~Ai, G (°) = G, and G(~) = (G(i-1))yi, 1 <- i m s .
Since G stabilizes e a c h of its
orbits, clearly Gy = G(s).
We have just dissected Y into the sets ¥i and have r e d u c e d the p r o b l e m of finding
Gy to t h e p r o b l e m of finding the stabilizer of the (pOssibly smaUer) sets Y~. A priori, it
is n o t obvious t h a t the stabilizer of a smaller set Yi can be c o m p u t a t i o n a l l y d e t e r m i n e d m o r e easily t h a n t h e stabilizer of the l a r g e r set. However, if we s u c c e e d in
breaking up the sets Yi in turn, and r e p e a t this recursively until Y is eventually
d e c o m p o s e d into singleton sets, t h e n t h e r e is h o p e for a polynomial time a l g o r i t h m
for finding Cry since, by Chapter II, we know how to d e t e r m i n e pointwise stabilizers
efficiently.
We consider how to b r e a k up the sets Yi. The m a j o r difficulty is t h a t G acts t r a n sitively on the orbit A~. However, if t h e r e is a s u b g r o u p H of G acting intransitively on
130
Ai, then, for this subgroup H, we m a y d i s s e c t Yi by i n t e r s e c t i n g it with the o r b i t s of H.
In fact, since I < G, such a subgroup H always exists.
Not every intransitive subgroup H is of use: Let G be a transitive group, H any
intransitive subgroup of G, and let H¥ be the setwise stabilizer of Y in H. To obtain Gy,
it is not sufficient to know the subgroup Hy. Instead, we also need to determine the
setwise stabilizer of Y in each right coset H ~ of H in G, i,e.,we need to find
Note that (Hzr)y could be empty. Since we have to find (HTr)y for each right coset HTr
in G, only subgroups H of sr~etL index in G are useful.
While Hy is a subgroup and thus is easily represented by a small generating set,
the representation of the sets (Hn)y is not immediately clear. Here, we will use the
following
L~:UlgA4
Let G, H < S n, Tr any permutation. If ( G w ) A H is not empty, then it is a right coset of
GNH,
Proof
Let
~, ~ c (G~)NH.
(G~)c~H c (GAH)#.
more, since G A H
Then
~3~-I ~- G
and
Conversely, let @ £ (Grr)NH, ~ e G A H .
< G, ~
~#-I e H,
and
so
Clearly ~# e H. Further-
e G~#. Thus ( G N H ) ~ c (G~)NH, from which the l e m m a fol-
lows. "
Observe that (H~r)y = (Hw)N(Sym(Y)xSym(Y)).
Thus, we m a y represent nonempty
sets (H~)y succinctly by a small generating set for Hy and a eoset representative
c (H~)~.
As a furLher consequence of L e m m a
4, the step for breaking up Y in the intransi-
tive case has a generalization to determining the stabilizer of Y in cosets of intransitive groups. The details of this will be outlined later.
Summarizing the ideas just sketched: We aim at finding a reeursive procedure for
determining the setwise stabilizer in groups and in right cosets of groups. The procedure is to split the set Y to be stabilized into smaller sets by considering a subgroup tower consisting of intransitive groups with small index in the containing subgroup. We do not expect all permutation groups to possess a suitable subgroup tower.
However, in the ease of p-groups such a tower always exists, and below we will develop
a suitable algorithm realizing these concepts.
131
3.1. T h e A l g o r i t h m
We now d e v e l o p t h e d e t a i l s of t h e a l g o r i t h m for finding t h e s e t . s o
stabilizer in p-
grouPs. We p o i n t e d o u t e a r l i e r t h a t t h e e s s e n t i a l p r e r e q u i s i t e for t h e a l g o r i t h m is t h e
e x i s t e n c e of a s u b g r o u p t o w e r c o n s i s t i n g of i n t r a n s i t i v e s u b g r o u p s of s m a l l index. We
show t h a t p - g r o u p s always have s u c h t o w e r s b e c a u s e of t h e i r i m p r i m i t i v i t y s t r u c t u r e
(cf. C h a p t e r III, S u b s e c t i o n &3).
R e c a l l t h e c o n s t r u c t i o n of a Sylow p - s u b g r o u p P of Sn c o n t a i n i n ~ a given p - g r o u p
as s u b g r o u p , a n d c o n s i d e r t h e a s s o c i a t e d c o n e g r a p h ( s ) . If X is t h e ( d i r e c t e d , r e g u l a r )
c o n e g r a p h of h e i g h t h a s s o c i a t e d with P, t h e n t h e l e a v e s of t h e p s u b t r e e s of h e i g h t
h - 1 m u s t f o r m a s y s t e m of i m p r i m i t i v i t y for P, a n d t h e s u b g r o u p of P w h i c h (setwise)
s t a b i l i z e s e a c h s e t of i m p r i r n i t i v i t y h a s i n d e x p in P.
Developing t h i s a r g u m e n t
slightly, we o b t a i n i m m e d i a t e l y
l.~g~l[A 5
Let G be a p - g r o u p with t h e n o n t r i v i a l o r b i t A. Then h m a y b e p a r t i t i o n e d i n t o p s e t s
F I ..... Fp of equal size which f o r m a s y s t e m of i m p r i m i t i v i t y for (the a c t i o n of) G on A,
and t h e s u b g r o u p H of G s t a b i l i z i n g e a c h s e t Fi setwise has i n d e x p in G.
L e m m a 5 p r o v i d e s t h e b a s i c t o o l for finding s u i t a b l e i n t r a n s i t i v e s u b g r o u p s . Note
t h a t t h e r e q u i r e d s y s t e m of i m p r i r n i t i v i t y m a y be f o u n d using t h e t e c h n i q u e s of Subs e c t i o n 3.3 in C h a p t e r III.
Let GO be a n i n t r a n s i t i v e p - g r o u p w i t h o r b i t s hi, ..., As. R e c a l l t h a t we p l a n t o s t a bilize t h e s e t Y in G O b y s u c c e s s i v e l y s t a b i h z i n g t h e s e t s Yi = YC~&~. Clearly, t h e s t a b i l i z a t i o n of Yi d e p e n d s only on t h e a c t i o n of GO on t h e s e t ~i. One m i g h t now c o n c l u d e
t h a t we should s t a b i l i z e Yi in t h e t r a n s i t i v e c o n s t i t u e n t of Go in Sym(~i). But t h i s is
n o t r i g h t , for t h e s t a b i l i z a t i o n of Y~ in Go m a y affect t h e way in which t h e g r o u p c a n
a c t on t h e r e m a i n i n g o r b i t s . F o r t h i s r e a s o n we o n l y d e a l with s u b g r o u p s G t h a t h a v e
d e g r e e e q u a l to t h e d e g r e e of G0. Along with t h e s u b g r o u p G of G O (or a r i g h t c o s e t of
G) we will k e e p t r a c k of a s e t Z. H e r e G is known t o s e t w i s e s t a b i l i z e Z a n d we a r e c o n s i d e r i n g t h e a c t i o n of G on Z for t h e p u r p o s e of s t a b i h z i n g Y(~Z. We a l t e r n a t e r e c u r sively b e t w e e n t h e following two s t e p s :
(a)
G a c t s i n t r a n s i t i v e l y on Z with o r b i t s &l ..... ~ :
Successively consider the sets
Yi = Y~AI, 1 - < i - < s .
F o r t h e g r o u p G, s t a b i l i z e t h e s e t s Yi b y calling S t e p (b) r e c u r s i v e t y .
~32
F o r t h e c o s e t Gx of G, find all Lhose p e r m u t a t i o n s in G~ which m a p Y1 into Y by
calling S t e p (b) r e c u r s i v e l y . Next, in t h e r e s u l t i n g set, find all t h o s e p e r m u t a tions m a p p i n g Yz i n t o Y, e t c e t e r a .
(b~,~ G ~s t r a n s i t i v e on Z ( b u t s t a b i l i z e s Z setwise): F i n d a s u b g r o u p H of i n d e x p in G
t h a t s t a b i l i z e s a m a x i m a l s y s t e m of i m p r i m i t i v i t y for t h e a c t i o n of G on Z. Let
G=H~:+
...
+H~p.
F o r t h e g r o u p G, d e t e r m i n e t h e s t a b i l i z e r of Yf~Z in e a c h H~: b y r e c u r s i v e l y c a l ling S t e p (a).
F o r t h e c o s e t G~ of G, s t a b i l i z e ZC~Y (in Y) in t h e c o s e t s H~i~ for e a c h i, a g a i n
using S t e p (a).
In e i t h e r case, " p a s t e t o g e t h e r " t h e n o n e m p t y c o s e t s (and s u b g r o u p s ) , o b t a i n i n g
GyN z (or (G~)yc~z), This p a r t is d i s c u s s e d in m o r e d e t a i l below.
F o r e a c h r e c u r s i v e a p p l i c a t i o n of S t e p (a), n o t e t h a t t h e size of Z is s m a l l e r b y a fact o r of 1__
P
The r e e u r s i o n e n d s w h e n Z = ~z~ is a s i n g l e t o n a n d G fixes z. In this case,
Gyc~z is c l e a r l y G. Moreover, b y T h e o r e m 3 of C h a p t e r II, (G~r)y(~z is n o n e m p t y iff
z ~ ~=Y w h e n e v e r z c Y. In t h a t c a s e , we have (G~)y(sz = G~.
We now s p e c i f y m o r e s u c c i n c t l y t h e r e c u r s i v e p r o c e d u r e j u s t s k e t c h e d .
Let
G < Sym(X) b e a g r o u p , Y a n d Z s u b s e t s of X. F o r ~ c Sym(X) we define
Sy(G~,Z) = ~ ~ c G~ I (Vz c Z)(z~ ~ ¥ iff z ~ V)t.
If Z is s e t w i s e s t a b l e in G, t h e n , by L e r n m a 4 above, S¥(G~,Z) is e i t h e r e m p t y , or i t is a
r i g h t c o s e t of Sy(G,Z) = GyN z. Rules (a) a n d (b) a b o v e now t r a n s l a t e as follows:
(a)
If Z is s t a b l e in G a n d G is i n t r a n s i t i v e on Z with t h e o r b i t s Al . . . . As, t h e n
Sy(G~,Z) = s y ( s ~ . . . S y ( G ~ , ~ D , . . - , 4 - ~ ) , 4 ) .
(b)
If G = H ~ I + H ~ +
' " " +H~p, t h e n
Sy(G~,Z) = S y ( H ~ l u , Z ) + S y ( H ~ e n , Z ) + ' ' " +Sy(H~pn,Z).
Finally, for t h e b a s e c a s e of t h e r e c u r s i o n ,
(c)
tf G fixes t h e p o i n t z, t h e n Sy(Gu,!z~) = G~ p r o v i d e d t h a t z n c Y w h e n e v e r z E ¥;
o t h e r w i s e , Sv(OU, fzl) = ¢.
133
We now t u r n to t h e p r o b l e m of pasting cosets t o g e t h e r . The basic r e s u l t used is
Lemma 4. Specifically, if
G~=H~ITr+H~27r+
--- +H~p~
then, by Rule (b) above,
Sy(G~,Z) = Sy(H91~,Z) + Sy(H~2~,Z) + '' • + Sy(H~p~,Z),
where "+" denotes the disjoint union. By Lemrna 4, a nonempty set Sy(Hgi~,Z) is a
eoset <Ki>!~i and
the nonempty
set Sy(H~j~,Z) is the
coset <Kj>'~j,
where
<I~> = <Kj> = Sy(H,Z). Thus,
S y ( H ~ I ~ , Z ) + S y ( H ~ , Z ) + . " • +Sy(H~p~,Z) = <Sy(H,Z),K>@,
where • is one of the @j and K = I ~/i#-I I Sy(H~i~,Z) ~ ¢I- This gives us a method for
collecting the nonempty cosets in Step (b) and representing their union succinctly.
It is important to note that the above rules are valid for all permutation groups
although their usefulness is limited since a permutation group G need not have a suitable subgroup tower of intransitive subgroups. In general, the problem is Rule (b)
which creates as m a n y reeursive calls as the index of H in G. With p-groups it is
always possible to find a subgroup H of index p in G, but typically there m a y not be an
intransitive subgroup H of acceptably small index.
Algorithm 2 below gives the details of the algorithm for p-groups. Rule (a) is handled in Lines 12-25 and Rule (b) in Lines 26-4t. The base case, Rule (c), is d e a l t with in
Lines 7-11.
The h e a r t of the algorithm is the r e c u r s i v e p r o c e d u r e STABILIZE. It
a c c e p t s as input p a r a m e t e r s a d e s c r i p t i o n of a (right c o s e t of a) p-group, <K>~, and a
subset Z of the p e r m u t a t i o n domain.
<K> is known to stabilize Z. The p r o c e d u r e
c o m p u t e s S-f(<K>Tr,Z) and so e i t h e r r e t u r n s t h e c o s e t <K'>@, where <K'> = <K>yoz,
or it r e t u r n s an indication t h a t Sy(<K>~,Z) is e m p t y in the boolean variable
isernpty.
Correctness of the a l g o r i t h m is s t r a i g h t f o r w a r d from the p r e c e d i n g discussion
and from L e m m a t a 4 and 5. The time r e q u i r e m e n t s will be d i s c u s s e d subsequently.
~34
AhC~OR1TH~I2 (Setw~se S t a b i l i z e r i n a p ~ r o u p )
Input
G e n e r a t i n g s e t K for a p - g r o u p G a c t i n g on t h e s e t X of size n, a n d a s u b ~et Y of X.
GeneratLng s e t •' of C~, t h e setwise s t a b i l i z e r of Y in G. In c a s e t h a t C~ is
Output
the trivial group, the set K' equals IOi.
The r e c u r s i v e p r o c e d u r e STABILIZE, Lines 5-44, w o r k s t h r o u g h t h e t o w e r
Comment
of intransitive s u b g r o u p s .
Method
t,
begin
2.
STABILIZE( O, K, X; "~, K', i s e m p t y ) ;
3.
output(K');
4.
end.
5.
p r o c e d u r e STABILIZE (~, K, Z; ~p, K', i s e m p t y ) ;
c o m m e n t The p r o c e d u r e s e a r c h e s t h e c o s e t <K>~ of t h e p - g r o u p <K>, w h e r e
<K> is k n o w n to s t a b i l i z e t h e s e t Z. It deterr~Anes t h e s e t of all p e r m u t a t i o n s
in t h e c o s e t which m a p Z N Y i n t o Y a n d m a p Z - Y into X-Y. This s e t is e i t h e r
empty (and then the variable
isempty is ~ u e ) , or i t is t h e e o s e t <K'>~, w h e r e
<K'> is t h e setwtse s t a b i l i z e r of Zf~Y in <K>, (and t h e n / s e r n p f y is f a l s e ) .
8.
7.
begin
if G fixes Z p o i n t w i s e t h e n b e g i n
8.
K' := K;
9.
~ := ~r;
iO.
i s e m p t y := ((ZNY)~cY) a n d ( ( Z - Y ) * c X - Y ) ;
i t,
end
12.
else begin
13,
d e t e r m i n e t h e o r b i t s h 1. . . . . h s of <K> in Z;
!4.
if s > I t h e n b e g i n
c o m m e n t <K> a c t s i n t r a n s i t i v e l y on Z;
15.
No := K;
16.
~0 := ~;
i7.
i := i;
t35
lB.
i s e m p t y := false;
19.
while i ~ s a n d n o t i s e m p t y do begin
20.
STABILIZE(~-i_I, Ki-1, Ai; hi, Ki, isempty)
21.
i := i+l;
22.
end;
23.
~ := Tr.;
24.
K' := K~;
25.
end
36.
else begin
c o m m e n t <K> a c t s transitively on Z, so t h e r e m u s t be a s y s t e m of imprirnitivity for <K> partitioning Z into exactly p subsets of equal size;
27.
find p sets of imprimitivity, F1 ..... Fp, for the action of <K> on Z;
28.
find a g e n e r a t i n g s e t K for the s u b g r o u p H of <K> which stabilizes e a c h of
the Fi setwise, and find a c o m p l e t e r i g h t t r a n s v e r s a l ~ ..... ~p] for H in
<K>;
29.
i s e m p t y := true;
c o m m e n t find t h e first n o n e m p t y coset;
30.
i := i;
31.
while i -< p a n d i s e m p t y do begin
32.
STABILIZE(~iTr, K, Z; 3k, K', isempty);
33.
i := i+l;
34.
end;
c o m m e n t find the remaining n o n e m p t y cosets;
w h i l e i ~ p do begin
35.
36.
STABILIZE(~in, K, Z; ~i,Ki, e e m p t y ) ;
37.
ff n o t c e m p t y t h e n
K' := K' U t ~i~ -I ];
38.
39.
i := i+l;
40.
end;
41.
4~.
43.
44.
end;
end;
r e t u r n ( ~ , K', isempty);
end-
136
&2. Analysis o~ Algorithm 2
We now analyze Algorithm 2's running tLule. The key issue here is the running
t i m e of t h e p r o c e d u r e STABILIZE. A s s u m e t h a t STABILIZE is c a l l e d with t h e i n p u t
parameters
~, K, a n d Z, <K> a p - g r o u p .
Clearly, t h e t i m e r e q u i r e d to d e t e r m i n e
Sy(<K>~,Z) d e p e n d s on:
n,
t h e d e g r e e of t h e p e r m u t a t i o n s ,
m, t h e c a r d i n a l i t y of Z,
p,
t h e p r i m e dividing t h e o r d e r of <K>, a n d
k,
t h e l a r g e s t c a r d i n a l i t y of a n y g e n e r a t i n g s e t c o n s i d e r e d .
Without loss of g e n e r a l i t y , we will a s s u m e t h a t k is O(n2).
Under the additional
a s s u m p t i o n t h a t Z is a n o r b i t of <K>, i.e., t h a t <K> a c t s t r a n s i t i v e l y on Z, we l e t
T(m,n,p) be t h e r u n n i n g t i m e of p r o c e d u r e STABILIZE a n d p r o c e e d t o give an e s t i m a t e
for t h e a s y m p t o t i c g r o w t h of t h i s f u n c t i o n .
R a t h e r t h a n e s t i m a t i n g t h e t i m e r e q u i r e d b y a c c o u n t i n g for e a c h line in t h e p r o c e d u r e , we will i d e n t i f y t h e w o r k d o n e b y STABILIZE a t e a c h level of r e c u r s i o n a n d t h e
n u m b e r of r e c u r s i v e calls m a d e . S i n c e <K> is a p - g r o u p , t h e c a r d i n a l i t y m of Z m u s t
be a p o w e r of p if <K> is t o a c t t r a n s i t i v e l y on t h i s set. Thus, ff m > t, STABILIZE
p r o c e e d s as follows:
t.
Z is p a r t i t i o n e d i n t o p s e t s of i m p r i m i t i v i t y , F 1. . . . . £p, of e q u a l size.
2.
A g e n e r a t i n g s e t K for t h e s u b g r o u p H of <K> is d e t e r m i n e d , w h e r e H s t a b i l i z e s
e a c h s e t £i. F u r t h e r m o r e , a c o m p l e t e r i g h t t r a n s v e r s a l I~1 . . . . . ~p] for H in <K> is
found. Note t h a t H is i n t r a n s i t i v e on Z.
3.
The s e t s S7(H~i~,Z) a r e d e t e r m i n e d .
B e c a u s e of t h e i n t r a n s i t i v i t y of H, t h i s
involves p c o n s e c u t i v e r e c u r s i v e calls with t h e new s e t s Z = Fi to b e c o n s i d e r e d .
m
Note t h a t H is t r a n s i t i v e on Fi, a n d t h a t t h e s e s e t s have c a r d i n a l i t y - - .
P
4.
The n o n e m p t y s e t s f o u n d in S t e p 3 a r e c o m b i n e d into t h e c o s e t Sy(<K>~T,Z).
C o n s e q u e n t l y , calling STABILIZE w i t h ~, K, a n d Z c a u s e s pZ r e e u r s i v e calls on t h e p r o e e d u r e with s e t s Z' of size m_m. f r o m w h i c h we o b t a i n t h e b a s i c r e c u r r e n c e
P
T(m,n,p) ~ p&w(m ,n,p) + f(m,n,p),
P
w h e r e f(m,n,p) is t h e t i m e r e q u i r e d for t h e work on t h e f i r s t level of r e c u r s i o n , i.e.,
137
the work of Steps i through 4 above, excluding the cost of the recursive calls.
For estimating the overhead f(m,n,p), we first observe that Steps I and 2 are the
most time consuming ones. Now, by Corollary 5 of Chapter Ill, Step i requires at
most O(m2-nZ.log2*(m)) steps, recalhng our assumption that K has size 0(nS). Next,
we note that Step 2 can be accomplished in 0(n4+p2.n 4) steps if we modify the algorithms of Chapter II such that, instead of fixing points, (maximal) sets of imprimitivity are stabilized. This leads to a subgroup tower of width p (cf. Theorem i2 of
Chapter Ill; see also Proposition 3 in Subsection 4.1 below). Since p -< m, we obtain
the estimate
f(m,n,p) ~ c.m~.n 4
for a suitable constant c.
Before e s t i m a t i n g t h e r e c u r r e n c e o b t a i n e d t h u s far, we n e e d to e n s u r e t h a t o u r
a s s u m p t i o n a b o u t t h e m a x i m u m size of t h e o c c u r r i n g g e n e r a t i n g s e t s is n o t invalid a t e d b y S t e p 4. By L e m m a 5, we know t h a t S t e p 4 c a n r e s u l t in a t m o s t p - i
addi-
t i o n a l g e n e r a t o r s for Kt We o b s e r v e t h a t t h e b a s e c a s e (Lines 7-I l in A l g o r i t h m 2)
d e l i v e r s a g e n e r a t i n g s e t of size O(n ~) a t m o s t , given t h a t t h e original g e n e r a t i n g s e t is
of t h a t size. E a c h t r a n s i t i v e level in t h e r e c u r s i o n r e c o m p u t e s a n O(n ~) size g e n e r a t i ~ s e t for t h e s u b g r o u p t o be c o n s i d e r e d , a n d t h e m a x i m u m d e p t h of r e c u r s i o n is a t
m o s t Iogp(n). Thus, all i n t e r m e d i a t e g e n e r a t i n g s e t s a r e a t m o s t O(n 2) in size. As a
c o n s e q u e n c e , we o b t a i n t h e r e c u r r e n c e
T(p.m,n,p) -< p2.T(m,n,p) + cfm2.n 4
For the base case, m = I, we obtain
T(l,n,p) g cl.n2
where we m a y choose the constants for the base case and the estimation of f(m,n,p)
uniformly. An elementary induction now gives us
T(pk,n,p) _< c.k.p2k.n4
for a su[itable constant c. Consequently, we have just proved
L~tL~ 6
Let <K> be a transitive p-group of degree n, IKI = 0(nZ). Then a generating set for the
setwise stabilizer of Y in < K > can be found in 0(nS.logp(n)) steps.
138
We now c o n ~ d e r t h e c a s e w h e r e G = <K> is a n i n t r a n s i t i v e p - g r o u p of d e g r e e n
with t h e o r b i t s A1. . . . . h~, a s s u m i n g a g a i n t h a t IKt is O(n~). L e t mi b e t h e l e n g t h of t h e
o r b i t Ai. Then t h e s t a b i l i z a t i o n of YOA i r e q u i r e s no m o r e t h a n T(mi,n,p) s t e p s , t h u s is
O(miZ.n4.1og~(mi)). Since ~ mi ~ is n o t g r e a t e r t h a n n ~, we o b t a i n
i=1
TUF_,O~ 3
L e t G = <K> b e a p - g r o u p of d e g r e e n. Then a g e n e r a t i n g s e t for t h e s e t w i s e s t a b i l i z e r
of Y in G c a n b e d e t e r m i n e d in O( i KI "n ~ + nS'logp(n)) s t e p s .
Proof
Note t h a t in 0 ( I K l ' n 2 + p~'n 4) s t e p s we c a n find an O(n 2) size g e n e r a t i n g
s e t for G. The t h e o r e m follows t h e r e f o r e f r o m L e m m a 6 a n d t h e a b o v e d i s c u s s i o n , s
We a p p l y T h e o r e m 3 to t h e r e s u l t s of S e c t i o n 2, a n d o b t a i n t h e following i m p r o v e m e n t s over C o r o l l a r i e s 4 a n d 5:
COROLLARr6
Let X b e a c o n n e c t e d t r i v a l e n t g r a p h with n v e r t i c e s . Then g e n e r a t o r s for Aute(X ) m a y
be f o u n d in O(nlS.log(n)) s t e p s .
COROLLARY7
Let X be a b i n a r y cone g r a p h with n v e r t i c e s a n d r o o t v. Then g e n e r a t o r s for Autv(X)
m a y b e f o u n d in 0(nl~.log(n)) s t e p s ,
This i m p l i e s , in p a r t i c u l a r , t h a t i s o m o r p h i s m of t r i v a l e n t g r a p h s m a y be t e s t e d in
0(nl~.log2(n)) s t e p s .
H e r e we f i r s t s p l i t t h e g r a p h s i n t o c o n n e c t e d c o m p o n e n t s , fol-
lowed b y classifying t h e c o m p o n e n t s into i s o m o r p h i s m c l a s s e s using C o r o l l a r y 6 a n d
t h e r e s u l t s of S e c t i o n 1. The d e t a i l s a r e s t r a i g h t f o r w a r d .
4.
An O(n 4) I s o m o r p h i s m T e s t ~or T r i v ~ e n t Graphs
in t h i s s e c t i o n , we wilt d e v e l o p a n 0 ( n 4) i s o m o r p h i s m t e s t for t r i v a l e n t g r a p h s .
More p r e c i s e l y , we give a n 0 ( n 3) m e t h o d for d e t e r m i n i n g s u f f i c i e n t i n f o r m a t i o n a b o u t
Aute(X ), X a c o n n e c t e d t r i v a l e n t g r a p h , t o p e r m i t t e s t i n g i s o m o r p h i s m in t h e m a n n e r
d e s c r i b e d in S e c t i o n i of this c h a p t e r .
The d e s i g n d e p a r t s slightly f r o m t h e o v e r a l l
a p p r o a c h of t h e p r e c e d i n g two m e t h o d s in t h a t t h e a l g o r i t h m d o e s n o t r e l y on s e t w i s e
s t a b i l i z a t i o n in 2 - g r o u p s as t h e c e n t r a l step.
I n s t e a d , o t h e r t e c h n i q u e s a r e used,
including t h e i n t e r s e c t i o n of two Z-groups. The following a r e t h e k e y t e c h n i c a l p o i n t s
of t h e design:
139
(i)
Many of the previously given algorithms for computing with p-groups are
improved. We give these algorithms in Subsection 4. I.
(2) A new problem for 2-groups, called the imprimi2ivity problem, is defined and
solved efficiently. The problem asks for the largest subgroup of a given 2-group
which has a prescribed (minimal) system of imprimitivity. This problem can be
understood as a paradigm for testing isomorphism of trivalent graphs with an
especially simple graph-theoretic structure. The O(n3) solution given in Subsection 4.2 is the critical component of the isomorphism test.
(3)
The notion of restricted izomorphism gadget is developed and is exploited in
order to simplify the graph-theoretic structure of trivalent graphs.
Conse-
quently, the O(n3) solution to the imprimitivity problem can be used for trivalent
graphs in general. This technique is explained in Subsection 4.3.
4. I. Improved Algorithms for p-Groups
We develop a n u m b e r of new or improved algorithms for computing in p-groups
which are given as permutation groups. As we will show, the previously given algorithms for p-groups can be improved significantly. In certain cases, however, the
improvement seems to depend on the availability of a special type of generating set.
While the algorithms are developed for arbitrary primes p, we will later apply them in
the special case of p = 2.
We begin with the problem of determining systems of imprimitivity for p-groups.
Let G = <~rI . . . . .
Irk> be a transitive p-group of degree n. Let x and y be two distinct
points in the permutation domain, and recall Algorithm 2 of Chapter Ill. The algorithm determines the smallest set of imprimitivity of G c o n t ~ i n g both x and y and
requires, by the proof of Theorem i5 of Chapter IIt, O(k.n-log~*(n)) steps. If G is an
imprimitive p-group and if we could guarantee that the smallest set of imprimitivity
containing both x and y is not the entire permutation domain X, then we could
dispense with the trial-and-error step of Corollary 4 of Chapter III, thereby possibly
eliminating a factor of n in the time bound.
W e n o w show that it is possible to find points x and y with this property.
As a
consequence, we can achieve an O(k.n-logs*(n)) algorithm for finding a nontrivial sys-
140
tern of i m p r i m i t i v i t y for t h e p - g r o u p G. Note t h a t this is n o t a n unqualified i m p r o v e m e n t over Corollary 4 of Chapter Ill, since we have no assurance that the sets of
imprimitivity are of eardinality p. However, in only O(k.n-log2*(n).logy(n)) steps we
can determine a complete imprimitivity structure for the p-group G (and hence a
Sylow p-subgroup of the symmetric group which contains G). This second result is a
strict improvement over Corollary 5 of Chapter III.
In order to understand the approach to selecting suitable points x and y, we need
to introduce the following concepts and facts:
DEFINITION I
Let ~ and ~/be elements of the group G. Then [~,@] = ~ - I ~ - I ~ is called the e o m ~ z t=toz of ~ by %
The t e r m c o m m u t a t o r derives f r o m the i d e n t i t y 7r~ = ~n[Tr,~].
Let KG be t h e s u b s e t of G c o n t a i n i n g all c o m m u t a t o r s of e l e m e n t s of G. T h e n KG
g e n e r a t e s a n o r m a l s u b g r o u p G' of G which is called t h e c o ~ r ~ u t ~ t o r s ~ 6 g r o ~ p or the
d e r ~ w d g r o u p of G. We will explore algebraic p r o p e r t i e s of c o m m u t a t o r s a n d d e r i v e d
g r o u p s in S u b s e c t i o n 5.2. of C h a p t e r VI in f u r t h e r detail. For the p r e s e n t , we s t a t e
w i t h o u t proof t h a t G' is the m i n i m a l n o r m a l s u b g r o u p of G whose f a c t o r g r o u p G / G ' is
abe]Jan. H e n c e e v e r y n o r m a l s u b g r o u p N of G whose f a c t o r g r o u p G/N is abe]Jan cont a i n s t h e d e r i v e d ~roup Gt We will m a k e use of this c h a r a c t e r i z a t i o n of G' below.
Let G be a t r a n s i t i v e group, N a n o n t r i v i a l , i n t r a n s i t i v e a n d n o r m a l s u b g r o u p of G.
Then the o r b i t s of N are a s y s t e m of i m p r i m i t i v i t y for G. To see this, l e t A be one of
the o r b i t s of N, n a n e l e m e n t of G. Since N is a n o r m a l s u b g r o u p of G, N~, t h e c o n j u g a t i o n of N by ~r, is N. Now An is a n o r b i t of N~ = N, h e n c e A~ m u s t be o n e of the o r b i t s of
N, a n d t h e r e f o r e A is a n o n t r i v i a t s e t of i m p r i m i t i v i t y for G.
We now r e t u r n to the p r o b l e m of finding two p o i n t s x a n d y which lie i n a n o n trivial s e t of i m p r i m i t i v i t y of t h e t r a n s i t i v e p - g r o u p G. Assume given a g e n e r a t i n g s e t
~r 1. . . . .
wk~ for the t r a n s i t i v e p-group G of d e g r e e n > p. We c o m p u t e t h e k - 1 com-
m u t a t o r s [7r1,~2] ..... [wl,Trk] i n O(k.n) steps. There are two possibilities: We e i t h e r find
a n o n t r i v i a l c o m m u t a t o r ~ = [~rlJri] g 0 , or ~1 c o m m u t e s with e v e r y e l e m e n t of G,
h e n c e is i n t h e c e n t e r of G (of. Definition 9 of C h a p t e r III).
If wl is in the c e n t e r , t h e n ~ = ~rl or ~ = 7fly g e n e r a t e s a n i n t r a n s i t i v e n o r m a l subg r o u p of G; h e n c e , the cycles of ~ m u s t be t h e o r b i t s of this s u b g r o u p <~> a n d are
sets of i m p r i m i t i v i t y for G. Therefore, i n this ease, we find a n o n t r i v i a l s y s t e m of
i m p r i m i t i v i t y for G i n O(k.n) steps. (Note t h a t n i p c a n be c o m p u t e d in O(n) steps.)
141
If !~ = [~i,~i] # 0 is a n o n t r i v i a l c o m m u t a t o r , t h e n we a r g u e t h a t a n y two d i s t i u e t
p o i n t s x a n d y in a c y c l e of t~ m u s t lie in a n o n t r i v i a l s e t of i m p r i m i t i v i t y of G. To s e e
this, c o n s i d e r t h e i n t r a n s i t i v e n o r m a l s u b g r o u p H of G stabilizing t h e s e t s in a m a x i m a l s y s t e m of i m p r i m i t i v i t y of G. Then t h e f a c t o r g r o u p G / H is a b e l i a n of o r d e r p,
h e n c e H c o n t a i n s t h e c o m m u t a t o r s u b g r o u p G' of G. Consequently, t h e c o m m u t a t o r
s u b g r o u p G' is a n i n t r a n s i t i v e n o r m a l s u b g r o u p of G whose Orbits a r e n o n t r i v i a l s e t s of
i m p r i m i t i v i t y for G. Using A l g o r i t h m 2 of C h a p t e r III we t h e r e f o r e h a v e
PROPOBITION1 (Hoffmann, Sims)
Given a generating set ~ i ..... ~k~ for the p-group G of degree n, where 7rI ~ O. Then
one can determine whether G is imprimitive, and if so, find a nontrivial system of
i m p r i m i t i v i t y for G in O(k.n.togz*(n)) s t e p s .
Proof
If G is p r i m i t i v e , t h e n e v e r y c o m m u t a t o r [~l,Tri] is t r i v i a l a n d ~I is a p -
cycle. The c o n c l u s i o n now follows f r o m t h e above d i s c u s s i o n . -
We
now show how to find efficiently a c o m p l e t e i m p r i m i t i v i t y s t r u c t u r e for t h e p -
g r o u p G. Without loss of g e n e r a l i t y , we m a y a s s u m e t h a t G is i m p r i m i t i v e a n d t r a n s i tive.
The i n t r a n s i t i v e
case
is h a n d l e d b y s e p a r a t e i y p r o c e s s i n g
the
transitive
c o n s t i t u e n t s of G.
S u p p o s e we i t e r a t e t h e following two s t e p s u n t i l t h e g r o u p G* h a s b e c o m e t r i v i a l
or cyclic:
(a)
C o m p u t e a s y s t e m S = IS 1. . . . . St{ of n o n t r i v i a l s e t s of i m p r i m i t i v i t y for G u s i n g
P r o p o s i t i o n I.
(b)
R e p l a c e t h e g r o u p G with t h e h o m o m o r p h i e i m a g e G* t h a t h a s t h e k e r n e l H,
w h e r e H is t h e setwise s t a b i l i z e r of t h e s e t s Si, i ~ i-~ r. G e n e r a t o r s for G* c a n
be o b t a i n e d f r o m t h e g e n e r a t o r s for G b y r e p l a c i n g t h e p e r m u t a t i o n lr with t h e
p e r m u t a t i o n ~r* a c t i n g on t h e seL~ Si in a c c o r d a n c e with t h e way t h e s e s e t s a r e
m a p p e d b y ~r.
After i t e r a t i n g t h e two s t e p s , we have d e t e r m i n e d p a r t s of an i m p r i m i t i v i t y s t r u c t u r e
for G and, in p a r t i c u l a r , know p m a x i m a l s e t s of i m p r i m i t i v i t y for G. ~ e e x p l a i n how
to close t h e "gaps" and c o m p l e t e this p a r t i a l i m p r i m i t i v i t y s t r u c t u r e .
Since P r o p o s i t i o n 1 g u a r a n t e e s finding a ~ o ~ r ~
s y s t e m of i m p r i m i t i v i t y if o n e
exists, i t suffices t o show how to o b t a i n g e n e r a t o r s for Gr, w h e r e F is a m a x i m a l s e t of
i m p r i m i t i v i t y of G. F u r t h e r m o r e , s i n c e f r o m a single s e t A of i m p r i m i t i v i t y we earl
d e t e r m i n e t h e r e m a i n i n g s e t s in t h e s y s t e m c o n t a i n i n g A in O(k.n) s t e p s , we m a y
142
restrict attention to a transitive constituent of Gr. We will apply the following wellknown
7 (Sehreier)
Let G = <~I . . . . .
~k> be a t r a n s i t i v e p - g r o u p of d e g r e e n, F 1. . . . . Fp a m a x i m a l s y s t e m
of i m p r i m i t i v i t y for G~ Let ~ be a g e n e r a t o r which does n o t s t a b i l i z e F~, a n d l e t
@l. . . . .
~p be s u i t a b l e p o w e r s of 7ri s u c h t h a t %~jm a p s F1 o n t o Fj f o r 1 -< j -< p. F o r a n y
e l e m e n t ~ e G, l e t 4~(~) d e n o t e t h e p e r m u t a t i o n @j-1 w h e r e ~ m a p s F 1 o n t o Fj. Then
the setwise s t a b i l i z e r GF1 is g e n e r a t e d b y t h e s e t
( L e m m a 7 is u s u a l l y s t a t e d in a m o r e g e n e r a l form).
most (k-1).p+ 1 nontrivial permutations.
Note t h a t K' c o n t a i n s a t
Since e a c h g e n e r a t o r in K' s t a b i l i z e s F 1, we
o b t a i n a c o r r e s p o n d i n g g e n e r a t i n g s e t for t h e t r a n s i t i v e c o n s t i t u e n t H = Gr, (r') b y
deleting in each generator in K' the cycles containing points not in Ft. Consequently,
we o b t a i n a g e n e r a t i n g s e t for H c o n s i s t i n g of p . k p e r m u t a t i o n s of d e g r e e n__ H e n c e ,
P
t h e t o t a l l e n g t h in s y m b o l s of t h e g e n e r a t i n g s e t s o o b t a i n e d f o r H is n o t l a r g e r t h a n
k.n. We now s t a t e t h e r e s u l t i n g a l g o r i t h m in m o r e detail:
I.
Determine the orbits of G.
2.
For each transitive constituent G (A) of G, h an orbit of G, set H to G (A) and execute
Steps 3 to 7.
3.
Using Proposition I, determine a nontrivial system of imprimitivity for H.
4.
If the system lust determined is not maximal, then replace H with the group H*
acting on the r sets of imprimitivity just found for H and go to Step 3.
5.
L e t F b e a m a x i m a l s e t of i m p r i m i t i v i t y for H. Using L e m m a 7, d e t e r m i n e a g e n e r a t i n g s e t K" for t h e t r a n s i t i v e c o n s t i t u e n t H (~) = Hr (r).
6.
R e p l a c e H with H (~). If a m a x i m a l s y s t e m of i m p r i m i t i v i t y for H is known, t h e n
r e t u r n to S t e p 5. If H is t r a n s i t i v e a n d of o r d e r p, t h e n c o n t i n u e w i t h S t e p 7, o t h erwise r e t u r n to S t e p 3.
7.
C o m p l e t e t h e s y s t e m s of i m p r i m i t i v i t y for e a c h d i s t i n c t s e t size using t h e ori~nal
set of generators for the transitive constituent G Ca) presently processed.
143
We a n a l y z e t h e t i m e c o m p l e x i t y of t h i s a l g o r i t h m in
t'RoPosrrioN 2 ( H o f f m a n n , Sims}
Given a g e n e r a t i n g s e t K = ~n 1..... nk~ for t h e p - g r o u p G of d e g r e e n, t h e n a c o m p l e t e
i m p r i m i t i v i t y s t r u c t u r e for G c a n be d e t e r m i n e d in O(k.n.log~*(n).logp(n)) s t e p s .
Proof
Let A be an o r b i t of G of l e n g t h t-< n. The t r a n s i t i v e c o n s i t u e n t G(a) h a s
d e g r e e t,
In t h e w o r s t case, e a c h a p p l i c a t i o n of P r o p o s i t i o n 1 (in Line 3 above) r e q u i r e s
O(k-t-log~*(t)) s t e p s a n d d e t e r m i n e s a m a x i m a l s y s t e m of i m p r i m i t i v i t y . Observe t h a t
t h e t o t a l l e n g t h of t h e g e n e r a t i n g s e t s d e t e r m i n e d b y Line 5 r e m a i n s b o u n d e d b y k.t.
A c a r e f u l i m p l e m e n t a t i o n of this line c a n avoid c o n s t r u c t i n g t h e s e t K' of L e m m a 7 by
c o m p u t i n g t h e i m a g e s only for p o i n t s in t h e m a x i m a l s e t of i m p r i m i t i v i t y c h o s e n .
Hence, Line 5 r e q u i r e s a t m o s t O(k.t) s t e p s , r a t h e r t h a n O(k-t-p) s t e p s .
Now Lines 3 a n d 5 n e e d to b e done a t m o s t a t o t a l of logp(t) t i m e s e a c h a n d t h i s
work t h e n c l e a r l y d o m i n a t e s t h e r u n n i n g t i m e . H e n c e a t o t a l of 0(k.t.logz*(t)dogp(t))
s t e p s suffices to d e t e r m i n e a c o m p l e t e i m p r i m i t i v i t y s t r u c t u r e for t h e t r a n s i t i v e constituent.
Since t h e s u m of t h e o r b i t l e n g t h s is n a n d s i n c e t -< n, t h e c o n c l u s i o n fol-
lows. •
Note t h a t
P r o p o s i t i o n 2 is a s i g n i f i c a n t i m p r o v e m e n t
over C o r o l l a r y 5 of
C h a p t e r III.
A n u m b e r of a l g o r i t h m s for p - g r o u p s c a n be m a d e m u c h m o r e efficient if t h e pg r o u p is s p e c i f i e d by a g e n e r a t i n g s e t with s p e c i a l p r o p e r t i e s . We now define p r e c i s e l y
t h e p r o p e r t i e s of s u c h u s e f u l g e n e r a t i n g s e t s , a n d review a m o d i f i c a t i o n of A l g o r i t h m
3 of C h a p t e r II for c o m p u t i n g t h e m .
DEFINITION2
Let G be a g r o u p with a s u b g r o u p t o w e r
I = G (m) <~G(m-l) ~ . - • ~ G (0) = G
in w h i c h e a c h f a c t o r g r o u p G(i)/G (i+1), 0 -< i < m, h a s no n o n t r i v i a l n o r m a l s u b g r o u p .
Then t h e s u b g r o u p t o w e r is a composition series of G. If, f u r t h e r m o r e , e a c h g r o u p
G (~+~) setwise s t a b i l i z e s a m a x i m a l s e t of i m p r i m i t i v i t y of G (i), t h e n t h e c o m p o s i t i o n
s e r i e s is c a l l e d an imprirnitivity series.
It is well known t h a t e v e r y p - g r o u p h a s a c o m p o s i t i o n s e r i e s in w h i c h e a c h f a c t o r
g r o u p is cyclic of o r d e r p (cf. T h e o r e m s 9 a n d 10 of C h a p t e r III). Moreover, e v e r y p -
~44
group also has a n i m p r i m i t i v i t y series.
DE~mUTION 3
A s e q u e n c e [~m, ~m-! ..... ~rl] of e l e m e n t s of t h e p - g r o u p G is a c o m p o s ~ @ ~ (an
i m p r i m i ~ i v i t y ) seqz~e~ee for G if t h e groups G(i) = <~m ..... ~ri+1>, 0 -~ i < m, t o g e t h e r
with G(m) = I, f o r m a c o m p o s i t i o n (an i m p r i m i t i v i t y ) series for G.
PROPOSITION 3
Given a g e n e r a t i n g s e t K = t~t ..... ~r~t of a t r a n s i t i v e p - g r o u p G of d e g r e e n a n d o r d e r
p m a n d given a c o m p l e t e i m p r i m i t i v i t y s t r u c t u r e for C.
Then a n i m p r i m i t i v i t y
s e q u e n c e for C c a n be c o m p u t e d in 0 ( k . m . n + ( p - 1 ) 2 . m 3 - n ) steps,
Proof
We d e t e r m i n e the
s e q u e n c e using a m o d i f i c a t i o n of A l g o r i t h m 3 of
C h a p t e r II.
First, we r e n u m b e r the p o i n t s in t h e p e r m u t a t i o n d o m a i n s u c h t h a t
fi ..... Pt, IP + I ..... 2p~ . . . . .
I1 ..... PZt . . . . .
f i .....
P
.....
~ n - p + i ..... n~;
~n-P 2+1 ..... nt;
t
.....
P
is t h e d e t e r m i n e d i m p r i m i t i v i t y s t r u c t u r e . This e n u m e r a t i o n r e q u i r e s O(n) s t e p s a n d
is c r u c i a l for the efficiency of the s u b s e q u e n t c o m p u t a t i o n s .
The c o r r e s p o n d i n g
c h a n g e to the g e n e r a t o r s of G r e q u i r e s O(k.n) steps.
We e n u m e r a t e t h e s e t s of i m p r i m i t i v i t y in the above s t r u c t u r e o r d e r e d by
n-1
~.~
d e c r e a s i n g c a r d i n a l i t y f r o m i to s = p. p_--~ H e n c e I1 .....
will be t h e first set, l n l
the last. For i <- i -< s, let G(i) be the setwise s t a b i l i z e r of A i n G(i-1), w h e r e / i is the ith
set of i m p r i m i t i v i t y in the above e n u m e r a t i o n , a n d G(°) = G. The s u b g r o u p tower so
defined c o n t a i n s a n i m p r i m i t i v i t y s e r i e s for G which we d e t e r m i n e with the familiar
sifting and closing u n d e r pair p r o d u c t f o r m a t i o n of Algorithm 3 of C h a p t e r II.
Clearly, t h e i n d e x of G(i+1) i n G (i) is e i t h e r i or p. Given ~ E C (i), we t e s t i n which
c o s e t of G(i+]) the e l e m e n t ~ of G(i) is c o n t a i n e d : Let AI be the i+ 1st s e t of i m p r i m i tivity. In the s y s t e m c o n t a i n i n g A1, t h e r e are also the sets As ..... hp s u c h t h a t t h e
u n i o n of t h e s e sets with ~i is again a s e t of i m p r i m i t i v i t y w h i c h ] s s t a b l e in G(i) since h i
is m a x i m a l .
Now t h e c o s e t of G(i+1) c o n t a i n i n g ~ is d e t e r m i n e d by the s e t Ai onto
which 7r m a p s hi~ This s e t c a n be f o u n d i n c o n s t a n t Lime b y c o m p u t i n g t h e i m a g e of a
single p o i n t x c A1 u n d e r ~ b e c a u s e
of the
e n u m e r a t i o n of the
p o i n t s in the
145
permutation domain.
Hence, in constant time the coset containing ~ can be deter-
mined.
We fill a m a t r i x of c o s e t r e p r e s e n t a t i v e s
tially, all e n t r i e s in t h i s m a t r i x a r e e m p t y .
eventually contain p-t
empty.
coset representatives
n-1
w i t h p. p - - ~ - r o w s a n d p - 1 c o l u m n s .
If G (I+1) h a s i n d e x p in G (0, t h e n r o w i will
for G (i+1) in G(i); o t h e r w i s e , row i r e m a i n s
( N o t e t h a t for t h e e o s e t G0+l) t h e i d e n t i t y s e r v e s as c o s e t r e p r e s e n t a t i v e
d o e s n o t h a v e t o be s t o r e d ) .
Ini-
S i n c e t h e o r d e r of G is pm t h e r e
and
are eventually m
nonempty rows in the matrix.
C o n s i d e r s i f t i n g a n e l e m e n t g ~ G. F o r a t m o s t e a c h row we h a v e to d e t e r m i n e
the proper eoset in which the partially sifted ~ lies. If the corresponding coset
representative
is already known, then we must multiply with its inverse from the
right requiring 0(n) steps. This can happen at most m times. Hence the cost of sifting ~ is O(m.n).
We have to sift k generators.
nonempty
entries, hence
at most
Furthermore,
the completed
(p-l)2.m ~ pair products
table has m.(p-l)
are sifted.
ConseqenUy,
the matrix can be fully determined in O(k.m-n+(p-l)2.mS.n) steps.
Finally, since the composition factors G(i)/G (i+l) arc cyclic of prime order, it follows that the nonempty entries in the first column of the matrix form an imprimitivity sequence for G. In the intransitive case, we stabilize all the sets of imprimitivity in one orbit
before stabilizing any set in subsequent orbits. A routine extension of the proposition
establishes that the same time bound holds for intransitive p-groups.
The reason for determining an imprimitivity sequence rather than a composition
series is that we lack efficient membership tests in the groups arising. For example,
one would like to determine a p-step central series for G by the s a m e means, but this
does not s e e m to be possible.
Given a complete imprimitivity structure for the p-group G and a composition
sequence, we now show h o w to construct new imprimitivity sequences for G in substantially less time. The crucial observation is the following
TItF~P~M 4 (Hoffmann)
L e t G = H + H = + H n 2 + • • • +H% p-1 b e a p - g r o u p w i t h a s u b g r o u p H of i n d e x p in G. L e t A
b e a n o r b i t of G on w h i c h H is t r a n s i t i v e , a n d l e t }F, F 2 . . . . . Fpl b e a m a x i m a l s y s t e m of
I46
imprimitivity contained in A. Let H = Hp+Hp3~+, • ° H p ~ p-I. Then G~ = <Hr,@>. where
= ~r~/-J a n d ~/J, 0 ~ j < p, is the u n i q u e power of ~ m a p p i n g t h e s e t F o n t o P~.
Proof
Since ~ s t a b i l i z e s F, the g r o u p G(1) = <Hr,~> is a s u b g r o u p of Gp. Since Hp
has i n d e x pZ in G a n d Gr has index p, it suffices to show t h a t ~ is n o t in Hr. Now if
= 7r~-J ~ Hp, t h e n ~ = ~ J E H a n d so H = G, a c o n t r a d i c t i o n . Hence the i n d e x of Hr
in G0) m u s t be p, i.e., GO) = Gp. ".
We apply t h e t h e o r e m to c o m p o s i t i o n s e q u e n c e s , and o b t a i n
COROLLAIVf 8 (Hoffmann)
Let [~m ..... ~I] be a composition
sequence
for the p-group G, F a m a x i m a l
set of
imprimitivity of G, and a s s u m e that q/= ~i is the leftmost generator in the sequence
not stabilizing F. That is, I~ # F
where
~J,
0<-j <p,
is
the
[~m ..... 7ri+1,~i-I..... ~I, Tri]
is
and F n k = F for all k >i.
unique
also
power
a
of
~
For k < i
mapping
composition
F
sequence
let ~k =Trk~-j,
to
F nk.
for
Then
G
and
[~Tm..... ~ri+i, ~i-i..... ~i] is a composition sequence for GI~.
Proof
Since F is maximal, the definition of the ~k, i -- k < i, m a k e s sense. The
conclusion n o w follows with a simple induction from T h e o r e m 4. "
The corollary is a significant i m p r o v e m e n t
over (the m o r e
general m e t h o d
of)
L e n l m a 7. It exploits e s s e n t i a l l y t h a t t h e c o m p o s i t i o n f a c t o r s of p - g r o u p s are cyclic of
p r i m e order. Corollary 8 now implies the following i m m e d i a t e and useful corollaries:
CORO~
9 (Hoffmann)
Given a composition sequence [~m, -:-,~I] for the p-group G of degree n, and given a
complete imprimitivity structure for G. Then an imprimitivity sequence for G can be
d e t e r m i n e d in 0(ma.n) steps.
Proof
In O(m.n) s t e p s t h e p o i n t s i n the p e r m u t a t i o n d o m a i n c a n be e n u m e r a t e d
as in the proof of T h e o r e m 4, a n d t h e g e n e r a t o r s wk i n t h e c o m p o s i t i o n s e q u e n c e c a n
be a d j u s t e d accordingly.
We r e p e a t e d l y apply Corollary 8, s u c c e s s i v e l y stabilizing m a x i m a l sets of i m p r i m i tivity. If a s e t is n o t y e t stable, we o b t a i n a c o m p o s i t i o n s e q u e n c e for t h e stabilizer.
There are O(n) s c a n s of the s e q u e n c e to d e t e r m i n e w h e t h e r the n e x t s e t of
i m p r i m i t i v i t y is a l r e a d y stable, e a c h s c a n r e q u i r i n g O(m) steps. Exactly m t i m e s we
find t h a t in the s e q u e n c e c o n s i d e r e d t h e n e x t m a x i m a l s e t of i m p r i m i t i v i t y is n o t y e t
stable, a n d e a c h t i m e this h a p p e n s , t h e c o m p u t a t i o n of t h e new s e q u e n c e r e q u i r e s at
m o s t 0(re.n) steps.
Therefore, a t o t a l of 0(mS.n) steps suffice to c o n s t r u c t an
147
imprimitivity sequence. -
COROLIARr 10 (Hoffmann)
Given a c o m p o s i t i o n s e q u e n c e [Trm. . . . . ~T1] for t h e p - g r o u p G of d e g r e e n, a n d g i v e n a
c o m p l e t e i m p r i m i t i v i t y s t r u c t u r e for G. Then a n i m p r i m i t i v i t y s e q u e n c e for G x c a n be
d e t e r m i n e d in O(m.n.logp(n)) s t e p s ,
Proof
Without loss of g e n e r a l i t y , we m a y a s s u m e t h a t Gx ~ G. Let P ! . . . . . Fr b e
the s e t s of i m p r i m i t i v i t y in t h e given i m p r i m i t i v i t y s t r u c t u r e of G, s u c h t h a t e a c h s e t
contains
x and
the
sets
are ordered
by d e c r e a s i n g
cardinality.
Observe t h a t
r-< logp(n). S u c c e s s i v e l y s t a b i l i z e t h e s e s e t s using C o r o l l a r y 8. The t i m e b o u n d now
follows r e a d i l y f r o m the p r o o f of Corollary 9. "
The g e n e r a l p r o c e d u r e for finding a g e n e r a t i n g s e t for a p o i n t s t a b i l i z e r in pg r o u p s is given by P r o p o s i t i o n 3 above. The v a s t i m p r o v e m e n t in efficiency in Coroll a r y 10 d e m o n s t r a t e s t h e u s e f u l n e s s of s p e c i f y i n g p - g r o u p s b y c o m p o s i t i o n s e q u e n c e s
and c o m p l e t e i m p r i r a i t i v i t y s t r u c t u r e s .
We develop n e x t an i n t e r s e c t i o n a l g o r i t h m for p - g r o u p s . We first s k e t c h a r e c u r sive p r o c e d u r e for i n t e r s e c t i n g two p - g r o u p s and t h e n r e o r g a n i z e t h e s e q u e n c e of
c o m p u t a t i o n in o r d e r to m i n i m i z e t h e i n c u r r e d o v e r h e a d (and w i t h i t t h e r u n n i n g
t i m e ) , tn p a r t i c u l a r , we will e x p l o i t t h e p r o p e r t i e s of i m p r i m i t i v i t y s e q u e n c e s in o r d e r
to r e d u c e t h e a m o u n t of w o r k to be p e r f o r m e d . Note t h a t for t h e r e s u l t i n g g r o u p t h e
i n t e r s e c t i o n a l g o r i t h m find~s a n i m p r i m i t i v i t y s e q u e n c e a n d a c o m p l e t e i m p r i m i t i v i t y
s t r u c t u r e (the s t r u c t u r e of t h e p - g r o u p s i n t e r s e c t e d is i n h e r i t e d ) .
Let G, H < Sym(X), a n d l e t GxH be t h e i r ( e x t e r n a l ) d i r e c t p r o d u c t a c t i n g on X+X,
two d i s j o i n t c o p i e s of X. The e l e m e n t s of GxH a r e p a i r s ~ = (~l,~ra), w h e r e ~t E G a n d
7r2 E H. Let A be a s u b g r o u p of GxH. Then the first projection, projl(A), is t h e s e t
n l E G i 3 n a e H , Qh,TrDEA~.
The secondpro#eet~on, proj~(A), is the s e t
InzEH[~IEG,(~I,n~)EA~.
It is
easily
shown that
both
projections
are
groups,
hence
p r o j l ( A) < G a n d
proja(A) < H. The two p r o j e c t i o n s a r e also c a l l e d t h e G- a n d t h e H - c o m p o n e n t of A,
respectively.
148
Let A < S)Tn(X)×Sym(X), ~ e Sym(X)xSym(X). A any s u b s e t of X. We define t h e s e t
J(A~,A) = ! ( a l , a ~ ) ¢ A~ I V x ~ h, x al = x a~
That is, J(AW,A) is the set of all ~ in the right coset A~ of A which m a p the points in A
identically in both projections. The following l e m m a is elementary and explains the
basic intuition about the set function d:
T.~u~IA 8 (Luks)
Let A < Sym(X)xSym(X), W = (,rx,~2) e Sym(X)xSym(X),
and G, H < Sym(X).
Then the
following is true:
(ai) If A c X is stable in proj1(A), then J(A,A) is a group.
(a~) If &cX is stable in projl(A), then J(AW,A) is either empty, or it is a right eoset of
J(A,~),
(a3) J ( ( G x H ) ~ , X) = ~ ( ~ , , ) ] ~ e GTrI(~H~2 {. In p a r t i c u l a r , J ( G x H , X) is i s o m o r p h i c to
Gf~H.
Proof
(al): Let (al,a2) and (Bvfl2) be elements of J(A,A). Since A is stable in proj1(A ),
for all x e ~, the image points x a~ and x al are in A, hence x ala1-1 is also in A.
Therefore, x =1~I-I -- X a2~2-I, and so (al,a~)(fll,492)-I is also in J(A,A).
(a2): Let ~ e J(A,~), ~ e J(AW,A).
Since A is stable in proj1(A), we
have
H ~ = (~I~I,a~@~)e J(AW,&). Hence J(AW,A) contains the right coset J(A,A)~. Next, if
-i
< J(AW,A), then, for all x e &, x *lx~
9
e A, h e n c e x ~lXl-I = X ~Z
-1
. B u t ~--i iS in A
and hence in J(A,A), Thus J(A~,h) is the right coset J(A,A)~.
(a3): Trivial. -
The overall design of a recursive intersection algorithm is based on the following
lemma:
LEUMA 9 ( L u k s )
Let A < Sym(X)xSym(X), ~ = (~q,Tr2) c Sym(X)xSym(X). Then t h e following holds:
( b l ) If A = A'+A'xz+ • - • + A ' ~ a n d AcX, t h e n
J (A~',A) = j (A'W,A) + J (A'~:~',A)+ "-" + J (A'xk~,A).
(b~) If proj~(A) setwise stabilizes the disjoint sets F and F', where F, F'cX, then
J ( A ~ , r + r ' ) = J (J (A<F),F').
(bS) If proj~(A) fixes t h e
p o i n t x e X, t h e n
= (~,~0~) e A such that x " ~ - I = x ~z.
J(AW,Ixt)= Az~
iff t h e r e
exists
a
149
Pru~
( b I ) is t r i v i a l a n d (b2) is s t r a i g h t f o r w a r d , so we only show (b3): Here we
s e e k all e l e m e n t s (at,a~) e A s u c h t h a t x al~1 = x a2~. Now x ai = x, h e n c e we n e e d to
solve x %=~-1 = x a2 for some a s e proj2(A). Clearly, if no s u c h a~ exists, t h e n J(AW,lx0
must be empty.
So, let ~ = (~I,~2) be an element of A such that x =1~z-~ = x ~m. Clearly
then, x ~I~1 = x ~2~2, and so, by L e m m a
8 (b2), J(A~,Ix~) = J(A, Ixl)~W. Finally, since
projl(A ) fixes x, we have J(A,~xl) = Az, where A x is the stabilizer of x in both copies of
X on which A acts. Note the close analogy with the rules for computing Sy(GTr,Z) w h e n determining
setwise stabilizers.
We wish to design an efficient algorithm for computing J((GxH)W,X), where both G
and H are p-groups contained in Sym(X). Based on the rules (bl) - (b3) of L e m m a
9,
we can design a recursive algorithm analogous to Algorithm 2 of Section 3 above.
Because of the irnprimitivity structure of p-groups, one alternates rules (bl) and (b2)
working recursively through the imprimitivity structure of G until the base case,
using rule (b3), is reached. Then a point stabilizer has to be found, and this is done
using
Corollary
I0.
The
details are
fairly straightforward,
and
a
careful
implement&lion would result in an O(n4.1ogp(n)) m e t h o d for intersecting two p-groups
of degree n.
We improve the basic recursive algorithm by reorganizing the computation such
that all base case problems J(AW,~x~) are processed together for each point x e X.
This reduces both the overhead during the recursion proper and the time spent for
each base case, since we now determine a point stabilizer A x or a set stabilizer A' only
0(n) times. The overhead computation can be further sped up using Corollaries 8 and
10, and this second improvement becomes a crucial factor in the timing of the resulting
algorithm.
The reorganization of the computation for J((GxH)W,X) is developed as follows:
Since G is a p-group, we m a y visualize the imprimitivity structure of G as a forest of
complete p-ary trees, where the p sons of each interior vertex are ordered cyclically
(cf. Chapter Ill). If G has s orbits, this imprimitivity forest of G consists of s trees, the
leaves of each tree comprising an orbit of G. We assume that the points in the permutation domain have been renumbered
as in Proposition 3 above, Le., the trees are
drawn as planar graphs and the cyclic ordering of the sons of each vertex is
w 1 < w 2 < . . . < w p < w I, where the p sons w i have been enumerated left to right, The
150
leaves of the trees are also e n u m e r a t e d
from left 'to right in ascending order, and
corresponding to each interior vertex v of a tree T there is a set ~v of imprimitivity
for G consisting precisely of the leaves of the subtree T v rooted in v.
For simplicity, assume that G is transitive, i.e., the imprimitivity forest of G consists of a single tree T, and let us visualize the process of computing
With every vertex v of T we associate a subproblem
J((GxH)~,X):
J(Ax,Av) which has to be solved,
where projl(A) setwise stabilizes L~v. If v is a leaf, then ~ = }v~, i.e., we have to solve
the base case using rule (bS).
Let w I..... wp be the sons (from left to right) of the interior vertex v, Fi, I -< i -< p,
the set of leaves of the subtree Twi. Then proj1(A) acts transitively on ~v iff, for at
least one generator ~ of A and every point x c F l , x al ~ F I. In this ease, we have to
apply rule (bl) and find the largest subgroup A' of A such that proj1(A' ) stabilizes F I.
Clearly the index of A' in A is p. So, let
A = A % A ' ~ z + " - " +A'~p,
A p p l y i n g r u l e (b2) also, we now o b t a i n t h e p s u b p r o b l e m s a s s o c i a t e d w i t h wt:
J(AT,ro = B'~,
J ( A ' [ z ~ , F 1 ) = B'~2,
:
J(A'~,F1)
=
B'~p:
n e x t , a s s o c i a t e d w i t h w~, we obtain
/(B'~,F~) = C'~,
J(B'~p,Fz) = C'~p;
and similarly for the remaining sons of v, provided none of these sets is empty.
If proj1(A) is intransitive on Av, then it m u s t stabilize the sets F i, hence here w e
obtain the subproblem
S ( A % r l ) = B'~
a s s o c i a t e d w i t h wl,
J(B'~,F2) = c'~
a s s o c i a t e d w i t h wz, a n d so o n f o r t h e r e m a i n i n g sons.
This situation suggests guiding the computation
of J ( ( G x H ) ~ , X )
by a single
depth-first traversal of the imprimitivity tree (forest) of G. For this traversal, we
151
k e e p two lists, R a n d K, where R is a list of c o s e t r e p r e s e n t a t i v e s (initially R = [~]),
a n d K is a list of g e n e r a t o r s , initially for GxH, a n d l a t e r for s u b g r o u p s A of GxH.
Reaching a v e r t e x v f r o m above with K a n d R shall m e a n t h a t we wish to solve e a c h of
the p r o b l e m s J(A~i,Av), w h e r e A = <K>, proji(A) setwise s t a b i h z e s Av, a n d ~ is in R.
When leaving t h e v e r t e x v to r e t u r n to its f a t h e r with the lists K' and R', this shall
m e a n t h a t J(<K>,Av) = <K'>, a n d t h a t for e a c h ~ e R t h e r e is, c o r r e s p o n d i n g l y , a n
e n t r y ¢ e R' s u c h t h a t J(<K>X,Av) = <K'>¢ or t h a t J(<K>~,hv) is e m p t y , In the l a t t e r
case, the e n t r y ¢ i n R' is "marked".
Now a s s u m e t h a t proh(A ) is t r a n s i t i v e on Av, a n d the p s u b p r o b l e m s J(A'~j~,hv) of
J(A'~,Av) are the n o n e m p t y cosets B'¢j of B' = J(A',hv), 1 _< j _< p. Then, by L e m m a 8
(a2), t h e u n i o n of t h e s e cosets is the c o s e t J(A,Av)¢I.
Here we wiI1 show t h a t
J(A,Av) = <B',¢~¢~-1>.
At all t i m e s d u r i n g t h e c o m p u t a t i o n , we will a r r a n g e t h e g e n e r a t o r s in K s u c h t h a t
is a n i m p r i m i t i v i t y s e q u e n c e for A = <K> a n d [~rm,1..... ~1,1] is a n i m p r i m i t i v i t y
s e q u e n c e for projl(A). We call s u c h a g e n e r a t i n g set a set i n canonical form.
The
initial g e n e r a t i n g s e t for GxH is of the f o r m
[(0,1Tr,2)
where r = m+m',
.....
(0,7Tm÷ 1,2), (TTm,1,0) ..... (~'1.1, ()) ]
[TTr,2.... , ~m+l,~] iS an i m p r i m i t i v i t y
sequence for the g r o u p H of
o r d e r pro' and [#m,I ..... Th,1] is an i m p r i m i t i v i t y sequence f o r the group G of o r d e r p m
Clearly the i n i t i a l g e n e r a t i n g set is i n c a n o n i c a l f o r m .
T h r o u g h o u t the p r o c e s s i n g of the i n t e r i o r n o d e s i n t h e i m p r i m i t i v i t y f o r e s t of G,
we show below t h a t this f o r m a t c a n be m a i n t a i n e d d y n a m i c a l l y for all i n t e r m e d i a t e
s u b g r o u p s A of G×H which arise. This i n c l u d e s the p r o c e s s i n g a t the leaves with t h e
base case of the r e c u r s i o n .
The e n t r i e s i n R have to be g r o u p e d r e c u r s i v e l y , so t h a t t h e p r o c e s s of "gluing"
cosets c a n be c a r r i e d out efficiently. Here it m a y be helpful to visualize R as the h s t
of leaves of a c o m p l e t e p - a r y t r e e .
Let A = <K> be a s u b g r o u p of GxH, where G < Sym(X) is a p-group, H < Sym(X),
and K is in canonical form, A s s u m e that h c X is an orbit of A = proj1(A ), and that
F 1. . . . .
Fp is a m a x i m a l s y s t e m of i m p r i m i t i v i t y of A c o n t a i n e d i n A By c a r r y i n g o u t
152
the operations of Corollary 8, it is clear that we can determine the subgroup A' of A
such that A' = projl(A' ) setwise stabilizes Ft. In particular, if (Tri,1,~ri,2)
is the leftmost
generator in K such that F[ i,I ~ FI, then the sequence
is also a canonical sequence for A, and by dropping (Tri,iJr~,2) we obtain a canonical
sequence for A'.
LEM~
I0
For the subgroup A < GxH, where G is a p-group, let A = A ' + A ~ z + - - • +A'~p, where
projl(A' ) # projl(A ). Assume that, for i -< i - p, B~"i is the n o n e m p t y s e t J(A'~i~,A),
where h is setwise stable in projl(A) and ~1 = ( 0 , 0 ) .
If K is a g e n e r a t i n g set for B in
canonical form, t h e n K' = [K, ~e~l-~] is a g e n e r a t i n g set for J(A,A) and is also in c a n o n ical form.
Proof
By L e m m a 9 (bl), J(A~,A) = B{I+ ' • • +B~p. Now A' has index p in A and
the sets B~i are not e m p t y .
C = J(A,A).
Now ~1 = ~ , ~
By L e m m a 8 (a2), therefore, B = J(A',A) has index p in
and
~ = ~Tz~0e~, where
~,
and ~s are
in A'.
Hence
~ i "1 = ~ s ~ { ' ~ e B < A' inaplies t h a t ~ e A', a c o n t r a d i c t i o n to the a s s u m p t i o n t h a t
A' has index p in A, Since B has index p in C, it follows t h a t C = <K,~2~i'l>. Since
proj~(A') e projl(A ), it follows similarly t h a t p r o j l ( ~
1) = Cz,lCfl ¢ proj1(A'). Hence
projl(B) has index p in projl(C), i.e., K' = [ K , ~ i "~] is in canonical form. Having outlined the o p e r a t i o n s r e q u i r e d to i m p l e m e n t rules (hi) and (bS) during
a traversal of the interior vertices of the imprimitivity forest of GxH, we n o w consider
the implementation
of the base case of the reeursion w h e n reaching a leaf v. W e
reach v with a canonical generating set for A < GxH, where projl(A) fixes v, and a list R
of eoset representatives.
The length of R is ph h the height of the tree containing v,
hence is bounded by n, the degree of G and of H. Because of rule (b8), we m u s t do the
following at v:
(a) Determine the orbit of x in proj2(A).
(b) For each orbit point y, find an element ~ e A such that x ~2 = y.
(c)
Using Corollary I0, find a canonical generating set for A r
(d) For each (unmarked) ~ c R, w e test whether x ~I~2-I is in the orbit of x in proj~(A).
If so, we multiply ~ with ~ from the left, otherwise we m a r k W.
153
We now s p e c i f y t h e d e t a i l s of t h e i n t e r s e c t i o n a l g o r i t h m f o r m a l l y . In t h e following, b o t h G a n d H a r e p - g r o u p s of d e g r e e n c o n t a i n e d in Sym(X). We a s s u m e t h a t t h e
g r o u p s a r e s p e c i f i e d by t h e i m p r i m i t i v i t y s e q u e n c e s KG a n d KH, a n d t h a t t h e Lmprimitivity s t r u c t u r e s for G a n d H a r e given.
We t r a v e r s e e a c h t r e e i n t h e i m p r i m i t i v i t y f o r e s t for G in p r e o r d e r .
The r o o t of
t h e f i r s t t r e e is v i s i t e d with K a n d a n initial l i s t R = [ ( 0 , 0 ) ] - If K' a n d R' is t h e r e s u l t
of t r a v e r s i n g t h e t r e e T, t h e n t h e r o o t of t h e n e x t t r e e is v i s i t e d with t h o s e two lists.
The c o m p u t a t i o n for t h e t r a v e r s a l is now as follows:
ALCO~TI~I 3 (Intersection of two p-Groups)
Comment
0 n l y t h e r e c u r s i v e p r o c e d u r e Visit is s p e c i f i e d which is in c h a r g e of
t r a v e r s i n g t h e n e x t t r e e in t h e i m p r i m i t i v i t y f o r e s t of G. The g l o b a l initiali z a t i o n a n d the driving r o u t i n e a r e s t r a i g h t f o r w a r d .
Method
I.
proeedure visit v e r t e x v with lists K and R;
2.
begin
3.
if v is not a leaf
H e n be~in
4.
let x be the leftmost leaf of Ty;
5.
l e t h be t h e h e i g h t of v e r t e x v (leaves have h e i g h t 0);
6.
if t h e r e is a g e n e r a t o r ~ = (~i,~2) s u c h t h a t x ~1 > x + p h-1 t h e n b e g i n
e o m m e n t p r o j l ( A ), A -- <K>, is t r a n s i t i v e on A, t h e Set of l e a v e s of Tv;
7.
using C o r o l l a r y 8, find a g e n e r a t i n g s e t K1 in c a n o n i c a l f o r m for A', t h e l a r g e s t s u b g r o u p s u c h t h a t projl(A' ) s t a b i l i z e s F, t h e s e t of l e a v e s of Tw, w a son
of v;
8.
9.
for each ~ ~ R do
r e p l a c e ~ with t h e p - t u p l e ~, ~ ,
~S~ ..... ~ p - ~ , w h e r e ~ is t h e l e f t m o s t
g e n e r a t o r in K n o t s t a b i l i z i n g F in t h e f i r s t p r o j e c t i o n ;
10.
t 1.
12.
e n d (if t h e r e is a g e n e r a t o r )
else
E l := K;
t54
13.
for i := t to p do begin
[4.
visit son w i of v with lists KI and R, and let K2 and R2 be the returned lists;
!5,
K1 := K2;
t6.
R := R2;
!7.
end;
18.
if < K > is transitive on A t h e n b e g i n
i9.
if t h e r e is a p - t u p l e ~1 . . . . . Cp in R s u c h t h a t e a c h of i t s m e m b e r s
is
unmarked then begin
c o m m e n t new g e n e r a t o r found, glue e o s e t s t o g e t h e r ;
20.
append ~ - i
21,
end;
22.
to Ki;
f o r e a c h p - t u p l e ~1 . . . . . ~p in R do
ff there is at least one u n m a r k e d ¢i then
23.
24.
replace the tuple with ¢i
25,
else
26.
r e p l a c e t h e t u p l e with ¢~;
27.
e n d (if <K> is t r a n s i t i v e ) ;
28.
r e t u r n the lists K1 and R;
29.
e n d (if v is n o t a leaf)
30,
else begin
c o m m e n t v is a leaf;
31,
d e t e r m i n e a c a n o n i c a l g e n e r a t i n g s e t K1 for Av using Corollary 10, w h e r e
A = <K>;
32.
calculate the orbit of v in proj~(A), and for each orbit point u determine ~ ~ A
39.
for e a c h u m r n a r k e d ~ = (Thjr2) in R do
s u c h that v a~ = u;
34.
if w = v ~i~2-1 is in the orbit of v in proj2(A ) then
35.
r e p l a c e ~ with ~ ,
w h e r e ~ is t h e c o s e t r e p r e s e n t a t i v e
corresponding to the orbit point w;
36.
else
37.
m a r k ~;
38.
r e t u r n t h e lists KI a n d R;
39.
end;
40.
end.
of Av in A
155
The correctness of the algorithm is clear from L e m m a t a
9 and i0, except perhaps
for Line 9. F r o m Corollary 8 we k n o w that Kl d ~ is a generating set for A = <K>. N o w
projl(A' ) setwise stabilizes F, the set of leaves in the tree Tw, w the leftmost son of v,
whereas proji(A ) is transitive on A. It follows that ~i, the first projection of ~, cyclically permutes
the sets F = F I.....Fp, where F i is the set of leaves of the subtree
rooted in the ith son of v. Consequently, the permutations ~i, 0 -< i < p are in distinct
cosets of proj1(A' ) in proji(A), hence the ~i are in distinct cosets of A' in A. N o w the
correctness of Line 9 follows f r o m L e m m a 9 (b i).
We a c c o u n t for t h e t i m e r e q u i r e d b y t h e a b o v e p r o c e d u r e b y d e t e r m i n i n g t h e c o s t
of t h e c o m p u t a t i o n p e r f o r m e d a t e a c h n o d e in t h e i m p r i m i t i v i t y f o r e s t .
It is c l e a r
t h a t we c a n c o n s t r u c t e a c h i m p r i m i t i v i t y t r e e for G s u c h t h a t , for e a c h i n t e r i o r v e r t e x
v, its h e i g h t h, t h e q u a n t i t y p h - i and t h e l e f t m o s t leaf in t h e s u b t r e e Tv c a n be d e t e r mined
in constant
time.
The cost of this is charged
does not alter its time bound.
mutations
in R never exceeds
imprimitivity
when
Moreover,
ph
visiting a vertex v, it is reduced
Let v be any interior vertex.
Line 7
as already mentioned,
n, where
forest (since for each
to the preconditioning
step and
the number
of per-
h is the height of the highest tree in the
time the length of R increases
to its previous size when
We execute
by a factor of p
leaving v).
Lines 4 - 89. The dominant
steps are
requiring, 0(n 2) steps, since a canonical generating set for G x H has
contains less than 2n permutations,
Lines 8-9
also requiring 0(n z) steps, and
Lines 22-26
again requiring 0(n z) steps,
Hence, the total work required at the 0(n) interior vertices is at most 0(n3).
W e n o w analyze the n u m b e r
of steps required to do the leaf processing.
observe that Line 32 can be implemented
examine
O(n) generators
to c o m p u t e
Chapter II, it is clear that we m a y
First,
in 0(nZ) 'steps. To see this, recall that we
the orbit of v.
compute
Recalling Algorithm
4 of
a set of Schreier vectors for each orbit
point u. N o w if XIXz" " ' Xk is a Sehreier vector for the orbit point u, it follows that
X,...)(jis a Schreier vector for s o m e other orbit point for each l - < j < k .
Hence ~I
Schreier vectors can be evaluated in a total of O(n z) steps, since k is bounded by n. It
follows, that Lines 32 - 37 can be implemented in O(n 2) steps.
By Corollary i0, Line 31 m a y require O(n2.1ogp(n)) steps. As there are n leaves to
be processed, it would s e e m that the intersection of two p-groups of degree n m a y
156
r e q u i r e up to 0(n3.1ogp(n)) s t e p s . However, b y a m o r e c a r e f u l a n a l y s i s we c a n d e m o n s t r a t e t h a t O(n ~) s t e p s sucffice.
We now d e r i v e t h e 0 ( n 3) u p p e r b o u n d on t h e t i m e r e q u i r e d to i n t e r s e c t two p g r o u p s G a n d H of d e g r e e n. We a s s u m e t h a t G a n d H have o r d e r s pm a n d pro' R a t h e r
t h a n t i m i n g Line 31 for e a c h leaf, we will d e t e r m i n e t h e c o m b i n e d total of o p e r a t i o n s
r e q u i r e d b y t h i s line for art leaves. Recall t h a t t h e i n i t i a l g e n e r a t i n g s e t for GxH is of
the form
E((),'~..2) ..... ((),~+i,e),(~,-())
where
r = m+m',
[~r2 ..... ~m+1,2]
is
an
..... (~1,t.())]
imprimitivity
sequence
for
H,
and
[~m,1 .....7r1,i]is an imprimitivity sequence for G. Let us call generators of the form
(0,~) i~rpe 2 generators and generators of the form (~,()) ~ype I generators.
Finally,
call a generator ol the form (~,~#), ~ ¢ O, ~ ~ O, a generator of type 3. Throughout
t h e c o m p u t a t i o n of t h e i n t e r s e c t i o n a l g o r i t h m , t h e list K of g e n e r a t o r s will c o n s i s t , in
s e q u e n c e , of g e n e r a t o r s of t y p e 2 followed b y g e n e r a t o r s of t y p e 1 which in t u r n a r e
followed b y g e n e r a t o r s of t y p e 3. The t y p e 3 g e n e r a t o r s a r e c o m p u t e d f r o m c o s e t
r e p r e s e . n t a t i v e s (cf. L e m m a 10), a n d t h e i r n u m b e r is a t m o s t e q u a l to t h e n u m b e r of
i n t e r i o r v e r t i c e s in t h e i m p r i m i t i v i t y f o r e s t of G w h i c h have b e e n v i s i t e d c o m p l e t e l y .
We n o t e t h a t proj2(A) is d e t e r m i n e d only b y g e n e r a t o r s of t y p e s 2 or 3. Moreover,
s i n c e p r o j 2 ( A ) < H, t h e c o m b i n e d n u m b e r of g e n e r a t o r s of t y p e s 2 a n d 3 c a n n o t
exceed m' at any time.
Now c o n s i d e r c o m p u t i n g a n i m p r i m i t i v i t y s e q u e n c e for t h e g r o u p s Ax f r o m a n
i m p r i m i t i v i t y s e q u e n c e for A using C o r o l l a r y 10. H e r e we d e t e r m i n e , s u c c e s s i v e l y ,
s e t w i s e s t a b i l i z e r s of m a x i m a l s e t s of i m p r i m i t i v i t y of H c o n t a i n i n g x. In doing so,
t h e r e a r e two p o s s i b l i t i e s : E i t h e r t h e s e t is a l r e a d y s t a b l e .
In this c a s e we p e r f o r m
O(m') s t e p s , s i n c e we only c o m p u t e t h e i m a g e of a single p o i n t for e a c h of t h e (at
m o s t ) m ' g e n e r a t o r s of t y p e 2 o r 3. We e n c o u n t e r t h i s s i t u a t i o n a t m o s t logp(n) t i m e s
at e a c h leaf. H e n c e this p a r t r e q u i r e s a t m o s t 0(n.m'.logp(n)) s t e p s for all leaves.
Otherwise, if t h e s e t is n o t y e t s t a b l e , we d r o p one g e n e r a t o r (of t y p e 2 or t y p e 3) a n d
a d j u s t t h e r e m a i n i n g g e n e r a t o r s in O(m'.n) s t e p s . S i n c e one g e n e r a t o r is d r o p p e d in
t h e o p e r a t i o n , t h i s s i t u a t i o n c a n be e n c o u n t e r e d a t o t a l of m + m ' t i m e s , s i n c e a t m o s t
m' g e n e r a t o r s of t y p e 2 c a n be d r o p p e d mud a t m o s t m g e n e r a t o r s of t y p e 3 c a n be
introduced.
C o n s e q u e n t l y , s u b s t e p (c) r e q u i r e s a c o m b i n e d t o t a l of O ( m ' . ( m + m ' ) ' n )
s t e p s , which is O(nS). In s u m m a r y , t h e above a r g u m e n t e s t a b l i s h e s t h e following
157
TI~'OI~ 5 (Hoffmann)
L e t G, H ~ Sym(X) b e p - g r o u p s of d e g r e e n. Given c o m p l e t e i m p r i m i t i v i t y s t r u c t e s
and i m p r i m i t i v i t y s e q u e n c e s for b o t h g r o u p s , a n d g i v e n ~ c Sym(X)×Sym(X), t h e n
J((G×H)~,X) c a n be d e t e r m i n e d in O(n 3) s t e p s .
COROLL~V~ 11
Let G, H < Sym(X) b e two p - g r o u p s of d e g r e e n. Given c o m p l e t e i m p r i m i t i v i t y s t r u c t u r e s a n d i m p r i m i t i v i t y s e q u e n c e s for G a n d H, t h e n a n i m p r i m i t i v i t y s e q u e n c e for
G(SH m a y be d e t e r m i n e d in O(n s) s t e p s .
A l g o r i t h m 3 c a n be e a s i l y m o d i f i e d to p e r f o r m t h e w o r k of t h e p r o c e d u r e STABILIZE in A l g o r i t h m 2 of S e c t i o n 3 above, i.e., to c o m p u t e s e t w i s e s t a b i l i z e r s . Exploiting
the s t r o n g s i m i l a r i t y in t h e r e c u r s i v e r u l e s for t h e f u n c t i o n s J a n d S¥, we m e r e l y have
to a l t e r t h e leaf p r o c e s s i n g (Lines 31 - 37 of A l g o r i t h m 3 above).
The new leaf p r o c e s s i n g is m u c h s i m p l e r due to the s i m p l e r r u l e for t h e r e c u r s i o n
b a s e case:
Sy(GTr,lvl) = G~
iff
v~ ~ Y w h e n e v e r v E Y
( p r o v i d e d , of c o u r s e , t h a t G fixes v). When visiting t h e leaf v, we MUSt c o m p u t e w=v 1~
for e a c h of t h e (up t o n) e l e m e n t s in R. If v is in Y b u t w is not, or vice v e r s a , t h e n
m u s t b e m a r k e d , o t h e r w i s e no f u r t h e r p r o c e s s i n g is r e q u i r e d .
This t a k e s a t o t a l of
O(n) s t e p s . Now if G fixes x, t h e n Sy(G,Ix~) = G, a n d so t h e list of g e n e r a t o r s is n e v e r
affected, h e n c e it r e m a i n s , in p a r t i c u l a r , an i m p r i m i t i v i t y s e q u e n c e for G. Consequently, t h e t o t a l a m o u n t of work a t t h e l e a v e s is O(n2). In s u m m a r y , we have j u s t
shown
PROPOSITION4 (Hoffmann)
Let G < Sym(X) be a p - g r o u p of d e g r e e n. Given a c o m p l e t e i m p r i m i t i v i t y s t r u c t u r e
for G and a n i m p r i m i t i v i t y s e q u e n c e , t h e n a c o m p o s i t i o n s e q u e n c e for t h e setwise s t a b i l i z e r GT of YcX in G c a n be d e t e r m i n e d in O(n s) s t e p s .
4.2. The Imprimitivity Problem for 2-Groups
We c o n s i d e r t h e p r o b l e m of d e t e r m i n i n g t h e l a r g e s t s u b g r o u p H of a given 2g r o u p G s u c h t h a t H has a p r e s c r i b e d ( m i n i m a l ) s y s t e m of i m p r i m i t i v i t y . G r o u p t h e o r e t i c a l l y t h i s p r o b l e m is of i n t e r e s t in its own r i g h t , b u t i t h a s a g e o m e t r i c
158
interpretation which motivates it as a suitable abstraction of determining the automorphisms of very simple binary cone graphs. "We begin with a formal statement of
the problem and develop its sohition before applying it to such cone graphs.
PROBLEH S (Impri~it~vi£y Problem for 2-Groups)
Given a complete imprimitivity structure and an imprimitivity sequence for the 2group G of degree n, and given a partition S of the permutation domain into blocks of
size 2 and at most one block of size i: Find the largest subgroup H of C such that S is
a minimal system of imprimitivity for H.
Note that it suffices to consider the case where n is even.
For if G has odd
degree, then H m u s t be a subgroup of G x ",'here Ixl is the singleton block in S. Hence
we m a y consider the problem in Gx viewed as a permutation group of even degree. We
will give an O(n s) algorithm for solving the imprin~itivity problem for 2-groups.
Let G < Sym~,Li be a 2-group of even degree, S a partition of L into blocks of size
2, Let L z be the set of all subsets of L of cardinality 2, G ~ the isomorphic image of G in
Sym(LZ). Here the isomorphism ' is provided by the induced action of G on the set L z.
Using the setwise stabilizer algorithm, the imprimitivity problem for G m a y be solved
as f o l l o w s :
!.. Determine an imprimitivity sequence and a complete imprimitivity structure for
G*.
2. Determine G'S and observe that H' = G'S.
3. Find H < Sym(L) isomorphic to G's.
~n)
Since G' has degree k2 , an
implementation
of this procedure seems to require O(n 6)
steps (el. Proposition 4). We plan to improve this bound by considering the imprimitivity problem as an intersection problem of G with a group isomorphic to CzCbtrn,
where m = n_2"
The group-theoretic structure of C 2 % S m is sufficiently complex so that we do not
wish to intersect this group with G directly. Instead, we plan to determine a 2-group
B < Sym(L) such that }3~G = H. It so happens that we can find a suitable 2-group 13 in
O(n2.1oga(n)) steps. Therefore, by Corollary I l, we can determine H in O(n s) steps.
The group 13 will be constructed by considering the induced action of G on the set
L z. However, instead of working with G', we will consider the group G, a Syl0w ~subgroup Of Sym(L a) containing G'. We will show that G can be determined from a
159
complete imprimitivity s t r u c t u r e for G, in O(nz) steps. Next, we determine ~s, the
setwise stabilizer of S in G. Since G is a Sylow 2-subgroup of a symmetric group, a
representation of Gs can be computed in O(n2.1og2(n)) steps (cf. Theorem 17 of
Chapter III). Note that Gs contains as subgroup G's, the isomorphic image of the
sought group H in Sym(L2),
We restrict Gs to S thereby obtaining a 2-group D. More precisely, a composition
sequence for D can be computed in 0(n 2) steps from the (implicit) representation of
Gs. Now recall that S partitions L, hence the blocks in S do not "overlap". Consequently, there must be a 2-group D < Sym(L) with 3 as a minimal system of imprimitivity such that the induced action of D on S is D. We take the largest such group B,
which is B = C2CbD. Since D is the restriction of Gs to S and since G~ contains H', it
follows that B contains H.
L F l i ~ 11
L e t G < S y m ( L ) be a 2-group, w h e r e L = f l ..... nl and n i s even. L e t G b e a S y l o w 2 subgroup of Sym(Ls) containin~ G', the isomorphic image of G in Sym(L2). Let S be a
partition of L into blocks of size 2, and let D be the restriction of Gs to 3 . If H is the
largest subgroup of G such that S is a minimal system of imprimitivity for H, then H,
the homomorphic image of H in Sym(S) (induced by the natural action of H on the
blocks in S), is a subgroup of D.
Proof
Let ~ be an element of H, ~' the corresponding element in H', and ~ the
image of ~ in H. Then ~' ~ Gs, since S is a minimal system of imprimitivity for H.
Hence the restriction of ~' to S is ~. CORO~
12 (Hoffmann)
Assume the hypotheses of Lemma 11. Then B = C~r~D is the largest 2-group in Sym(L)
such that ,.q is a minimal system of imprimitivity for B and B, the homomorphic image
of ]3 in Sym(S) is D. Moreover, B contains H, i.e., H = ]3[~G.
Proof
Let A < Sym(L) be a 2-group such that S is a minimal system of imprimi-
tivity for A and such that A = D. Since S is a minimal system of imprimitivity, the
kernel KA of the homomorphism A-~ A is elementary abelian of order 2r, where
n
r--< m = ~
Let B = C~OoD. Then the kernel KB of ]3 ~ B is elementary abelian of order
2m. Since the degree of A and B is n, KA < KB, hence A < B, i.e., B is maximal. Finally,
160
letTrEH.
By L e m m a !1, t h e i m a g e ~ of 7 r i n H i s
also i n D .
Since KB h a s o r d e r 2 m,
t h e r e f o r e , ~ c B. n
We e x p l a i n how to d e t e r m i n e G f r o m a c o m p l e t e i m p r i m i t i v i t y s t r u c t u r e for G.
The c o n s t r u c t i o n is b a s e d o n a c o r r e s p o n d e n c e b e t w e e n a c o m p l e t e i m p r i m i t i v i t y
s t r u c t u r e of G a n d a c o m p l e t e i m p r i m i t i v i t y s t r u c t u r e of G'.
LE~L~ 12 (Hoffmann)
Let G < Sym(L) be a 2-group, w h e r e L = [1 . . . . . nl a n d n is a p o w e r of 2, a n d l e t F be a
c o m p l e t e i m p r i m i t i v i t y s t r u c t u r e for G. Let h a n d A' b e s e t s of i m p r i m i t i v i t y in F of
equal size, n o t n e c e s s a r i l y d i s t i n c t b u t of c a r d i n a l i t y g r e a t e r t h a n one. A s s u m e t h a t
F1, 1"2, F'I, F'2 a r e ( n o t n e c e s s a r i l y n o n t r i v i a l ) s e t s of i m p r i m i t i v i t y in F s u c h t h a t
h = F I + F ~ and A' = F ' I + F ' ~. If G' is t h e i s o m o r p h i c i m a g e o5 G in Sym(L2), t h e n t h e s e t s
#a(h,A') = [ [x,yt ] (x e F 1, y e F'z) o r (x e F e, y e F'I)~,
~b(A,A') = [ [x,y~ I (x ~ l~, y e r ' l ) o r (x e r z, y ~ r'z) I, a n d
• (a,a') = ~Aa,a') + ~b(a,a')
a r e all s e t s of i m p r i m i t i v i t y for G'.
Proof
L e t 7r e G a n d a s s u m e t h a t # m a p s h a n d A' o n t o ~ a n d h', r e s p e c t i v e l y .
Then ~T m a p s ~I,(A,A') o n t o ~I,(h,h').
Furthermore,
Ca(A,A') a n d vb(a,a') a r e e i t h e r
m a p p e d onto 4~a(~,~') and ~b(~,h'), r e s p e c t i v e l y , or t h e y a r e m a p p e d onto ~b(h,h') a n d
~a(h,A'), r e s p e c t i v e l y . We r e m a r k w i t h o u t p r o o f t h a t t h e s e t s of i m p r i m i t i v i t y d e f i n e d b y L e m m a 12 a r e
t h e only s e t s of i m p r i r r d t i v i t y for G' if G is a transitiv.e Sylow 2 - s u b g r o u p of a s y m metric group.
L e m m a 12 c a n be g e n e r a l i z e d t o 2 - g r o u p s of d e g r e e n o t a p o w e r of 2. H e r e one
n e e d s to c o n s i d e r s e p a r a t e l y t h e s e t s Ix,y] w h e r e x a n d y a r e in d i f f e r e n t t r e e s in F .
While t h e g e n e r a l i z a t i o n is quite s t r a i g h t f o r w a r d , K is c u m b e r s o m e in f o r m u l a t i o n .
H e n c e we will p r o c e e d d i f f e r e n t l y a n d e n l a r g e t h e p e r m u t a t i o n d o m a i n w i t h a d d i t i o n a l
p o i n t s (left fixed b y G) so t h a t G is r e p r e s e n t e d i s o m o r p h i c a l l y as a 2-group of d e g r e e
a p o w e r of 2. Since a t m o s t n - I
a d d i t i o n a l p o i n t s a r e n e e d e d , one verifies t h a t t h i s
d e g r e e c h a n g e d o e s n o t a d v e r s e l y a f f e c t t h e a s y m p t o t i c c o m p l e x i t y of t h e a l g o r i t h m s
to be d e s c r i b e d below. Note also t h a t L e m m a 11 and C o r o l l a r y 12 r e m a i n i n t a c t .
Let G b e a ~-group of d e g r e e 2 m, G a Sylow 2 - s u b g r o u p of Syrn(L a) c o n t a i n i n g G'.
We a s s u m e t h a t G has t h e c o m p l e t e i m p r i m i t i v i t y s t r u c t u r e F , a c o m p l e t e b i n a r y t r e e
161
of h e i g h t m . We d e s c r i b e a n
O(n2) m e t h o d
for c o n s t r u c t i n g a c o m p l e t e i m p r i m i t i v i t y
s t r u c t u r e t7 for G~, t h e r e b y d e t e r m i n i n g G. It is n o t difficult t o v e r i f y t h a t F c o n s i s t s
of b i n a r y t r e e s with t h e h e i g h t s 2 m - 2 , 2 m - 3 . . . . . m - L
m-2+h,
H e r e t h e t r e e of h e i g h t
1 ~ h -< m, has l e a v e s l a b e l l e d with p a i r s ~x,y] s u c h t h a t t h e n e a r e s t c o m m o n
a n c e s t o r of x a n d y in F is a t d i s t a n c e h f r o m t h e leaves.
Consider the set
~(hl,h2),
w h e r e hi a n d h 2 a r e s e t s of i m p r i m i t i v i t y in F of e q u a l
size. 5 i is t h e leaf s e t of a s u b t r e e of F r o o t e d in t h e i n t e r i o r v e r t e x v l, a n d h 2 of a
s u b t r e e r o o t e d in v2, h e n c e we c a n r e p r e s e n t ~(hl,h2) by t h e p a i r (vl,v2). Now
~(hl,h2) = ¢~(hl,ae) + ~b(hl,he).
If h i = A1,1 + A1,2 a n d h 2 = h2,1 + 42,2, t h e n
=
+
and
Cb(hl,h2) -= ~(AI,i,~2A ) ÷ ~(A1,2,A2,2).
So, if vl, 1 a n d vl, 2 a r e t h e sons of v 1 a n d v2,1 a n d v2,~ t h e s o n s of v~, t h e n we o b t a i n t h e
following p a r t o f / ~ shown in F i g u r e 5:
(vl,l,v2,2)
(vl,2,v2,D
(vl,.v2,1)
(vl,z,~2,2)
Figure 5
H e r e t h e i n t e r m e d i a t e v e r t i c e s r e p r e s e n t t h e s e t s @a(A1,A2) a n d ~ b ( A l , ~ ) which c o u l d
have
been
labelled
with
the
pairs
of
pairs
((vl,l,v2,2), (vl,2,v2,1))
and
((vl.i,v2,1), (Vl,z,v22)), r e s p e c t i v e l y .
F i g u r e 5 s u g g e s t s a r e c u r s i v e p r o c e d u r e for c o n s t r u c t i n g t h o s e s u b t r e e s of 1~
whose r o o t s r e p r e s e n t s e t s #(A1,A2), w h e r e 41 + A2 is again a s e t of i m p r i m i t i v i t y in F .
We i l l u s t r a t e t h i s in
182
EXAMPLE 5
Let G have d e g r e e 23. We wilt c o n s t r u c t t h e t r e e i n /~ of h e i g h t 3 - 2 + 3 = 4 which c o n sists of all p a i r s x,y s u c h t h a t t h e n e a r e s t c o m m o n a n c e s t o r of x and y in F is the r o o t
of F (which is a t d i s t a n c e 3 from the loaves). We a s s u m e a v e r t e x labelling of F as
shown in F i g u r e 6 below.
1
2
3
4
5
6
?
8
\J\ /V V V
Figure 6
The r o o t of t h e t r e e to be c o n s t r u c t e d is labelled e,f. The p r i n c i p l e for c o n s t r u c t ing it r e c u r s i v e l y f r o m this v e r t e x p a i r is e v i d e n t f r o m Figure 7 below. E~
1,8 2,7 1,7 2,8 3,6 4,5 3,5 4,6
1,6 2~5 1.5 2,6 3,8 4,? 3,7 4,8
Figure 7
Now l e t v be a v o r t e x in F a t d i s t a n c e h f r o m t h e loaves, v I a n d v~ its sons, a n d
a s s u m e t h a t Ai a n d h~ a r e t h e leaf sets of the s u b t r e e s r o o t e d in v 1 a n d v 2, r e s p e c tively. We have to e x p l a i n how the set ~I,(hl,h~) fits into l a r g e r sets of i m p r i m i t i v i t y
w h e n v is n o t t h e root. Here we o b s e r v e t h a t this s e t c a n be c o n s i d e r e d e q u i v a l e n t to
the s e t h i + A2 of i m p r i m i t i v i t y for G, so, if we "equate" the v e r t e x v with t h e pair
(vl,v~) of its sons, t h e n the t r e e in F c o n t a i n i n g the s u b t r e e with root (vl,v2) c a n be
c o m p l e t e d b y a t t a c h i n g e a c h of t h e s e s u b t r e e s to t h e v e r t e x i n F with which t h e r o o t
p a i r has b e e n e q u a t e d . This s i m p l e idea is i l l u s t r a t e d i n
163
Ex~.
8
Consider the group G of Example 5. The tree in /~ of height 3 is shown in Figure 8
below. Note that the root and its sons come from the tree F.
1,4
2,3
1,8
2,4
5,8
6,7
5,7
6,8
Figure 8
The correctness of the construction is easily verified, and it is clear, moreover,
that F can be constructed in time proportional to its size, i.e., in O(n~) steps. Hence
we obtain
PRoPosrrioN 5 (Hoffmann)
Given a complete imprimitivity structure for the 2-group G of degree n = 2m, a complete imprimitivity structure for G' can be found in O(n ~) steps. Hence a representation for G, a Sylow 2-subgroup of Sym(L~) containing G', can be found in 0(n 2) steps.
We now describe how to obtain a representation of G£ and how to determine from
it a composition sequence for the group D, the restriction of Gs to S. It is crucial to
c o n s t r u c t a composition sequence for D directly since a composition sequence for GB
might require O(n4) symbols.
As a first step, we label those leaves in the imprimitivity forest F that are pairs in
S. Clearly this can be done in O(n~.log2(n)) steps. Next, we run the tree isomorphism
algorithm separately for each tree in F. The object here is to label isomorphic subtrees of each tree T and to rearrange those subtrees such that isomorphic subtrees
are equal, i.e., an isomorphism exists which exchanges corresponding leaves in the
164
two subtrees.
Since we deal with complete binary trees, it is clear that this part
requires only O(n 2) steps.
Having
determined
all isomorphic
subtrees
and
having
rearranged
them
appropriately, we n o w have an implicit representation for the groups G 8 and for D. In
o r d e r to r e a d out f r o m this r e p r e s e n t a t i o n a c o m p o s i t i o n series for D, we now have to
a u g m e n t the t r e e v e r t i c e s with t h e following p o i n t e r s a n d labels:
(a) Those leaves in .,~which are pairs in S are linked together into a list, from left to
right.
(b)
For e a c h i n t e r i o r v e r t e x v, a l a b e l is a d d e d to i n d i c a t e w h e t h e r t h e s u b t r e e
rooted in v contains a leaf belonging to S, and if so, a pointer to the ieftmost
such leaf in the subtree is also added.
Clearly this i n f o r m a t i o n c a n be g a t h e r e d in O(n s) steps.
Let us d e n o t e t h e s u b t r e e r o o t e d i n the i n t e r i o r v e r t e x v with Tv, a n d t h e s e t of
leaves in t h e s u b t r e e with Av. Let v be an i n t e r i o r v e r t e x of a t r e e i n /~ a n d a s s u m e
that u and w are the sons of v. We associate with v the subgroup K (v) of G~ which is
the pointwise stabilizer of all points in L 2 which are ~o~ in A v. Then K (V) setwise
stabilizes the m a x i m a l s~ts of imprimitivity A u and A w if[ T u and T w are not isomorphic.
Hence,
either
K (v) :I~U)xK (w), and
K (v) = <I((U)xK (w), 7[>, and
then
then
T u and
Tu
and
Tw
are
T w are isomorphic
not
and
isomorphic,
~r exchanges
or
the
corresponding leaves of T u and Tw. Moreover, in that case the index of K(U)xK (w) in K (v)
is 2.
Now l e t Eu a n d E~ be c o m p o s i t i o n s e q u e n c e s for K(u) a n d K<w), r e s p e c t i v e l y . If Tu
a n d T~ are n o t i s o m o r p h i c , t h e n t h e i r c o n c a t e n a t i o n Eu, Ew is a c o m p o s i t i o n s e q u e n c e
for K(v). Otherwise, Eu, Ew, [~r] is a c o m p o s i t i o n s e q u e n c e for K(v). H e n c e a composition s e q u e n c e for Gs c a n be d e t e r m i n e d b y a simple p o s t - o r d e r t r a v e r s a l of the t r e e s
in F .
The d e t e r m i n a t i o n of a c o m p o s i t i o n s e q u e n c e for D is quite similar. Here we m u s t
c o n s t r u c t only t h o s e g e n e r a t o r s ~ which e x c h a n g e the leaf sets Au a n d Aw c o n t a i n i n g
at l e a s t one leaf which is a pair in S .
Since t h e c o r r e s p o n d i n g i n t e r i o r v e r t i c e s a r e
m a r k e d , this s i t u a t i o n is r e a d i l y r e c o g n i z e d .
So we m u s t modify t h e p r o c e d u r e
d e s c r i b e d above by adjoining to the c o n c a t e n a t e d s e q u e n c e E u, Ew t h e g e n e r a t o r W iff
T u and T w are isomorphic ~r~d both u and w are m a r k e d vertices. Furthermore, W is in
165
sxn(s).
The c o m p u t a t i o n of a c o m p o s i t i o n s e q u e n c e for D c a n be i m p l e m e n t e d i n 0 ( n z)
steps: We n o t e t h a t we c a n c o n s t r u c t W d i r e c t l y with the help of t h e p o i n t e r s a t t h e
sons of the v e r t e x c o n s i d e r e d and the p o i n t e r s linking all leaves which are pairs i n S .
Now c o n s i d e r c o m p u t i n g t h e c o m p o s i t i o n s e q u e n c e for D. If v is m a r k e d a n d has t h e
i s o m o r p h i c sons u and w (which are t h e r e f o r e also m a r k e d ) , t h e n h u and ~v c o n t a i n an
equal n u m b e r of leaves which are pairs in S .
the
n+
2
number
n+
4
...
of
+
marked
vertices
with
There are n s u c h leaves total. Hence
isomorphic
sons
is
at
most
n < n. Therefore, no m o r e t h a n n g e n e r a t o r s are c o n s t r u c t e d , each
2 TM
in a t m o s t O(n) steps. Thus, a total of O(n~) s t e p s suffice to c o n s t r u c t a c o m p o s i t i o n
s e q u e n c e for D.
We now give the final steps for computing H, the largest subgroup of the given 2group G with the prescribed minimal system S of imprimitivity.
Having shown how to obtain a composition sequence for.D, we must now determine a composition sequence for B, the largest 2-group in Sym(L) with S as a system
of imprimitivity and homomorphie image D in Sym(S).
Let S = IIxl,yll.....Ixr,yr~I
and let ~ be a composition sequence for D. (We now consider the elements in
equivalently as permutations in Sym(L). This is justifiable since S is a partition). We
claim that A = [(Xl,yl).....(Xr,Yr)], ~ is a composition sequence and generates B.
It is clear that A is a composition sequence. Let C be the group generated by A.
Surely the homomorphic image C is D. N o w the kernel of the h o m o m o r p h i s m from C
n hence B = C is the
to C is an elementary abelian 2-group of order 2r, where r = -~
desired group.
By Proposition 2, we can find a complete imprimitivity structure for 1~ in
O(n2.1og2(n).log2*(n)) steps. Hence, by Corollary 9, the composition sequence for B
can be converted into an imprimitivity sequence for B in O(n s) steps. Summarizing
the results of the previous sections, we have now established
TH~.o~ 6
Let G < Sym(L) b e a 2-group of e v e n d e g r e e for which a c o m p l e t e i m p r i m i t i v i t y s t r u c t u r e a n d a n i m p r i m i t i v i t y s e q u e n c e is given. Let S be a p a r t i t i o n of L i n t o blocks of
size 2, a n d let H be the l a r g e s t s u b g r o u p of G s u c h t h a t S is a m i n i m a l s y s t e m of
i r n p r i m i t i v i t y for H. Then a n i m p r i m i t i v i t y s e q u e n c e for H c a n be f o u n d i n O(ns) s t e p s .
166
W e n o w turn to the interpretation of the ~mprimitivity problem as an automorphism problem for certain graphs. For the remainder of the section, come ~rmph shall
always m e a n 6inamj cone graph. The Lemf set of the cone graph X is the set of leaves
of the underlying tree. The subgraph consisting of the leaf set and the cross edges
will be c a l l e d t h e l ~ f g r ~ p h of X.
We r e c o n s i d e r P r o b l e m 2, t h e d e t e r m i n a t i o n of Autv(X ) of t h e c o n e g r a p h X. Since
we a r e ultimately- i n t e r e s t e d in t h e a p p l i c a t i o n of this p r o b l e m to t r i v a l e n t g r a p h isom o r p h i s m , we will only c o n s i d e r cone g r a p h s whose leaf g r a p h s have a p a r t i c u l a r
structure.
We o b s e r v e t h a t t h e g r o u p of a u t o m o r p ~ s m s
of t h e ( r e g u l a r , b i n a r y ) c o n e g r a p h
X which fix t h e r o o t c a n be f a i t h f u l l y r e p r e s e n t e d as a p e r m u t a t i o n g r o u p a c t i n g on
the leaf set.
Consequently, we m a y c o n s i d e r t h e p r o b l e m of d e t e r m i n i n g this g r o u p
e q u i v a l e n t l y as t h e p r 0 b l e m of i n t e r s e c t i n g t h e a u t o m o r p h i s m g r o u p of t h e leaf g r a p h
with t h e Sy!ow 2 - s u b g r o u p of t h e s y m m e t r i c g r o u p of t h e leaf s e t whose i m p r i m i t i v i t y
s t r u c t u r e is t h e u n d e r l y i n g t r e e . We e x p e c t t h a t t h e difficulty of t h e a u t o m o r p h i s m
p r o b l e m is r e l a t e d to t h e g r a p h - t h e o r e t i c s t r u c t u r e ( a n d t h e c o n s e q u e n t s t r u c t u r e of
t h e a u t o m o r p h i s m g r o u p ) of t h e leaf g r a p h .
Let us c o n s i d e r t h e a u t o m o r p h i s m p r o b l e m for c o n e g r a p h s in which t h e leaf
g r a p h has v a l e n c e one, i.e., t h e leaf g r a p h c o n s i s t s of i s o l a t e d e d g e s a n d i s o l a t e d v e r tices.
S u c h c o n e g r a p h s a r e c a l l e d simple s i n c e t h e y have a s i m p l e g r a p h - t h e o r e t i c
structure.
A s e t of two v e r t i c e s c a n b e c o n s i d e r e d a n u n d i r e c t e d e d g e , a n d we n o t e t h a t t h e
i m p r i m i t i v i t y p r o b l e m is e s s e n t i a l l y t h e a u t o m o r p h i s m p r o b l e m for a s i m p l e cone
g r a p h whose leaf g r a p h d o e s n o t have a n y i s o l a t e d v e r t i c e s . This s u g g e s t s a p p r o a c h ing t h e a u t o m o r p h i s m p r o b l e m for s i m p l e cone g r a p h s as follows:
Let X be a s i m p l e c o n e g r a p h with leaf s e t L a n d u n d e r l y i n g t r e e T, a n d l e t n be
t h e n u m b e r of l e a v e s of T. Let G be t h e Sylow 2 - s u b g r o u p of Sym(L) i n d u c e d b y t h e
a u t o m o r p h i s m s of T. It is c l e a r t h a t we c a n c o m p u t e an i m p r i m i t i v i t y s e q u e n c e for G
in O(n z) s t e p s . L e t J c L be t h e s e t of all l e a v e s of X i n c i d e n t to a c r o s s edge, J ' c L t h e
s e t of all i s o l a t e d v e r t i c e s in Lhe leaf g r a p h . We c o m p u t e Gj, t h e s e t w i s e s t a b i l i z e r of J
in G, using P r o p o s i t i o n 4, a n d n o t e t h a t t h e s o u g h t a u t o m o r p h i s m g r o u p is a s u b g r o u p
of O~. We r e s t r i c t Gj to J o b t a i n i n g a g r o u p A. Now the c r o s s e d g e s of X define a p a r t i tion of J into b l o c k s of size 2. We call t h i s p a r t i t i o n S a n d solve t h e i m p r i m i t i v i t y
p r o b l e m for S a n d A using T h e o r e m 6. Let H be t h e s u b g r o u p of A so o b t a i n e d , a n d l e t
167
}3 be the restriction of Gj to the set J'. Then G C~ H × B is the desired automorphism
group of the simple cone graph. Note that all groups are 2-groups for which we have
(or can compute efficiently)imprimitivity sequences. In summary, we therefore have
PROPOSITION6 (Hoffmann)
Let X be a simple cone graph with n vertices. Then an imprimitivity sequence (with
respect to the imprimitivity structure defined by the underlying tree) for Autv(X) can
be d e t e r m i n e d in 0 ( n s) steps.
We r e m a r k w i t h o u t proof t h a t t h e b o u n d of P r o p o s i t i o n 6 r e m a i n s i n t a c t when t h e
cross edges of X are labelled.
4.3. Gadgets for Trivalent Graph Isomorphism
In o r d e r to apply T h e o r e m 6 to the p r o b l e m of d e t e r m i n i n g Auto(X), X a c o n n e c t e d
t r i v a l e n t graph, we n e e d to devise a m e c h a n i s m for r e d u c i n g t h e p r o b l e m of finding,
in Aute(Xk), t h e s u b g r o u p B of all a u t o m o r p h i s m s which m a y be e x t e n d e d to a n a u t o m o r p h i s m i n Aute(Xk+l), to d e t e r m i n i n g Autv(Y), w h e r e Y is a s i m p l e cone graph.
Roughly speaking~ we will use a c o n s t r u c t i o n by which c e r t a i n s u b g r a p h s a r e
s u b s t i t u t e d which p e r m i t viewing the t r i v a l e n t g r a p h as a simple cone graph.
This
i n t u i t i v e n o t i o n is i m p r e c i s e in the s e n s e t h a t i n s t e a d of s u b g r a p h s we r e a l l y s u b s t i t u t e c e r t a i n 2-groups for p o i n t s o n which Aute(Xk) acts (this is done via w r e a t h products), with the effect t h a t t h e d e t e r m i n a t i o n of the s u b g r o u p B r e d u c e s to a
s e q u e n c e of a p p l i c a t i o n s of T h e o r e m 6, plus a few setwise s t a b i l i z a t i o n a n d i n t e r s e c tion o p e r a t i o n .
To g r a s p t h e t h r u s t of this a p p r o a c h , let us first c o n s i d e r cone g r a p h s whose leaf
g r a p h s have v a l e n c e 2. We plan to r e d u c e this case to simple cone g r a p h s by using
i s o m o r p h i s m g a d g e t s in a n a b s t r a c t sense. Let X be a cone g r a p h whose leaf g r a p h
has v a l e n c e 2. We wish to c o n s t r u c t a simple cone g r a p h X~ f r o m X which has, r o u g h l y
speaking, the s a m e a u t o m o r p h i s m g r o u p as X.
Let T be the u n d e r l y i n g t r e e of X of h e i g h t m. The g r a p h X2 will have an u n d e r l y ing t r e e T2 of h e i g h t m + i .
Hence X2 has (about) twice as m a n y v e r t i c e s as X. We p u t
the v e r t i c e s v of T into i - i c o r r e s p o n d e n c e with the i ~ e r i o r v e r t i c e s v' of T~ s u c h t h a t
the t r e e s t r u c t u r e is p r e s e r v e d . Now c o n s i d e r the cross edge (u,v) of X, a n d l e t u l, u 2
and v l, v2 be t h e sons of u' a n d v', r e s p e c t i v e l y . C o r r e s p o n d i n g to t h e edge (u,v), t h e r e
will be exactly one cross edge (ui,vi) in X2. Here the i n d i c e s i a n d j m u s t be c h o s e n
168
s u c h t h a t X~ h a s a leaf g r a p h of v a l e n c e 1. Since t h e leaf g r a p h of X h a s v a l e n c e 2,
this is always p o s s i b l e . Let us call t h e g r a p h Xs a n e x t e n d e d s i m p l e cone g r a p h of X.
Note t h a t X d o e s n o t u n i q u e l y d e t e r m i n e )(2.
EXAMPLg 7
Let X be t h e cone g r a p h shown in F i g u r e 9 below. Its leaf g r a p h h a s v a l e n c e 2. F i g u r e
i0 shows a n e x t e n d e d s i m p l e cone g r a p h of X o b t a i n e d b y t h e c o n s t r u c t i o n d e s c r i b e d
above. Note t h a t this g r a p h has a leaf g r a p h of v a l e n c e i.
Figure 9
F i g u r e 10
The c o n s t r u c t i o n of t h e e x t e n d e d s i m p l e c o n e g r a p h is a g a d g e t c o n s t r u c t i o n in
t h e following s e n s e : E a c h v a l e n c e 2 leaf v in t h e leaf g r a p h of X h a s b e e n r e p l a c e d w i t h
t h e p a i r v 1, v~, a n d t h e two e d g e s i n c i d e n t to v a r e c o n n e c t e d (one each) to t h e s e new
vertices.
Now a n a u t o m o r p h i s m
of X m a y
locally i n t e r c h a n g e
the
two e d g e s
169
e m a n a t i n g f r o m v. H e n c e t h e g a d g e t v 1, v~ m u s t p e r m i t the t r a n s p o s i t i o n (vz,v~). This
is done b y a w r e a t h e x t e n s i o n with C~ of the u n d e r l y i n g a u t o m o r p h i s m g r o u p of t h e
t r e e as realized by t h e i n c r e a s e in the t r e e height.
The c r u c i a l p r o p e r t y of the c o n s t r u c t i o n is s u m m a r i z e d by the following
THEOREM7 (Hoffmann)
L e t X be a regular, binary cone graph with root v whose leaf graph has valence 2, and
let X2 be an extended simple cone graph of X constructed as described above. Then
Autv(X), the group of all automorphisms of X fixing the root v, is the homomorphic
image of Autv(X2), where the h o m o m o r p h i s m is provided by the induced action of
Autv(X~) on t h e i n t e r i o r v e r t i c e s of X~.
Proof
In t h e following, we will i d e n t i f y the v e r t i c e s of X with the i n t e r i o r v e r t i c e s
of X2. Hence the leaf set L of X m a y be c o n s i d e r e d the set of i n t e r i o r v e r t i c e s of X~
whose d i s t a n c e from t h e leaves of X2 is 1. L2 will be t h e leaf s e t of X2.
Let o~ ~ Auto(X2), ~ its h o m o r p h i c image.
Since the b r o t h e r s in I~ f o r m sets of
i m p r i m i t i v i t y for Autv(X2), it follows t h a t ~ E Aut(X).
Conversely, let ~ E Aut(X), a a n e x t e n s i o n of a to t h e v e r t e x s e t of X2. Consider
w E L a n d the e 0 r r e s p o n d i n g b r o t h e r s w l, w2 E L2. Let (u,w) be a cross edge of X,
(uj,Wl) t h e
c o r r e s p o n d i n g cross edge in X~.
If a
maps
(ui,wl) to (uj,w~), t h e n
a' = a(wl,w~) m a p s the edge to (uj,wl). H e n c e t h e r e exists a p e r m u t a t i o n 7r c o n s i s t i n g
e n t i r e l y of t r a n s p o s i t i o n s of b r o t h e r s in Lz, s u c h t h a t fl = aTr is an a u t o m o r p h i s m of
Xz.
Since ~ = ~, it follows t h a t to every ~ E Autv(X) t h e r e exists a n e x t e n s i o n in
Aub(X2).
•
We have r e d u c e d the a u t o m o r p h i s m p r o b l e m for cone g r a p h s whose leaf g r a p h s
have v a l e n c e 2, to the a u t o m o r p h i s m p r o b l e m of simple cone graphs.
It r e m a i n s to
show how to o b t a i n a n i m p r i m i t i v i t y s e q u e n c e for Autv(X) f r o m a n i m p r i m i t i v i t y
s e q u e n c e for Autv(X2). Now it is trivial to c o m p u t e t h e h o m o m o r p h i c image ~' of a n
i m p r i m i t i v i t y s e q u e n c e ~. for Autv(X~), However, £' m a y c o n t a i n r e d u n d a n t g e n e r a tors. Hence we n e e d t h e following
1.~M_A_ 13
Let [~m ..... ~ ] be a s e q u e n c e of g e n e r a t o r s for t h e 2-group G s u c h t h a t , for I ~- i -~ m,
the group G(i) = <~m ..... ~i+i> has index I or 2 i n G0-1) . T h e n a n i m p r i m i t i v i t y
s e q u e n c e for G c a n be d e t e r m i n e d in 0 ( m - n 2) steps.
Proof
By P r o p o s i t i o n 2, we c a n find a c o m p l e t e i m p r i m i t i v i t y s t r u c t u r e for G i n
less t h a n O(m.n 2) steps. We will sift t h e g e n e r a t o r s in the s e q u e n c e urn..... 7rI, as for
t70
Proposition 3, but zu/~ho~z~ ~orming or sifting pair products. We claim that the resulting table defines an imprimitivity sequence for G.
Assume
inductively that the table has r ~ m - i
nonempty
entries after having
sifted Trm ..... hi+ i, and that these entries constitute an imprimitivity sequence for G (i).
Consider sifting 7ri. If ~i is redundant, then no new- table entry results, the index of
G (i)in G 0-1) is i, and we n o w have an imprimitivity sequence for G (i-l). Otherwise, the
n e w entry 7# is made.
N o w the index of G (i) in < G (i),~ > = Q (i-i) is at least 2, hence it
m u s t be exactly 2 by the hypothesis of the lemma.
Hence every pair product formed
with ~/and with existing table entries m u s t be redundant, and so the n e w table defines
an imprimitivity sequence for G (i-I). As consequence of L e m m a
13 and T h e o r e m 7, we obtain immediately
C O R O L L m ~ 13
Let X be a regular, binary cone graph with root v whose leaf graph has valence 2. If X
has n leaves, then an imprimitivity sequence for Autr(X ) can be determined in O(n 3)
steps.
W e r e m a r k that the corollary generalizes to cone graphs whose leaf graphs have
valence 2 and whose cross edges are labelled.
Recall that the a u t o m o r p h i s m
problem for cone graphs m a y be understood as an
intersection problem of the a u t o m o r p h i s m
arising from the a u t o m o r p h i s m
group of the leaf graph with the 2~group G
group of the underlying tree. W e observe that the
solutions in Corollary 13 and Proposition 6 do not m a k e use of the Sylow structure of
G, hence we m a y speak of "abstract" cone graphs, that is, leaf graphs a u g m e n t e d
by
an underlying 2-group G instead of the underlying complete binary tree. Thus we m a y
g e n e r a l i z e t h e s e r e s u l t s as follows:
COROI/ARY 14 (Hoffmann)
Let X = (V,E) be an edge labelled graph of valence 2, and let G < Sym(V) be a 2-group
with given imprimitivity sequence.
Then an imprimitivity sequence for Aut(X)C~G m a y
be determined in O(n s) steps, where n is the n u m b e r
of vertices of X.
Reflecting on the gadget construction above, we also observe that there is really
no need to replace a vertex v with a pair v i, v~ if the valence of v (in the leaf graph) is
less than 2, However, we now m u s t setwise stabilize in G the set of all vertices of
valence 0 and the set of all vertices of valence I. The corresponding extension of the
group G to the n e w permutation d o m a i n is straightforward.
171
We now have d e v e l o p e d all t h e e s s e n t i a l i d e a s a n d t h e m a j o r tools n e e d e d for a n
O(n 4) i s o m o r p h i s m t e s t for t r i v a l e n t g r a p h s . This t e s t follows t h e g e n e r a l a p p r o a c h of
t h e o t h e r two i s o m o r p h i s m t e s t s in this c h a p t e r b u t differs s i g n i f i c a n t l y in t h e d e t a i l s .
In p a r t i c u l a r , n o t e t h a t i t is n o t n e c e s s a r y to d e t e r m i n e Aute(X ) for t h e p u r p o s e of
t e s t i n g i s o m o r p h i s m : If h is t h e maximum d i s t a n c e of any v e r t e x f r o m t h e e d g e e,
t h e n knowing t h e h o m o m o r p h i e i m a g e Ae(X) of Aute(X ) in Sym(Vh) is sufficient. (Ae(X)
is t h e r e s t r i c t i o n of Aute(X ) to t h e s e t of v e r t i c e s at d i s t a n c e h f r o m e). Consequently,
we will d e t e r m i n e t h e g r o u p s Ae(Xg), where, for 0 ~ k - h, Ae(X~) is t h e h o m o m o r p h i e
i m a g e of Aute(Xk) in Sym(Vk), a n d Ae(Xh+l) = Ae(X).
Recall S e c t i o n s I a n d 2. We develop a n a l g o r i t h m for d e t e r m i n i n ~ t h e g r o u p B of
all t h o s e p e r m u t a t i o n s in Ae(Xk) which c a n be e x t e n d e d to an a u t o m o r p h i s m in
Aute(Xk+l). More p r e c i s e l y , given a c o m p l e t e i m p r i m i t i v i t y s t r u c t u r e a n d an i m p r i m i tivity s e q u e n c e for Ae(Xk), we will d e t e r m i n e an i m p r i m i t i v i t y s e q u e n c e for B in 0(n~)
s t e p s . Here n k is t h e c a r d i n a l i t y of V~, t h e s e t of all v e r t i c e s at d i s t a n c e k f r o m e.
F o r e v e r y e d g e t y p e , we e x a m i n e t h e s u b g r a p h s p a n n e d by t h e s e e d g e s and s t u d y
how i t c o n t r i b u t e s to t h e definition of B. We p l a n to c o m p u t e , for e a c h e d g e t y p e t,
the s u b g r o u p B (t) c o n s i s t i n g of t h o s e p e r m u t a t i o n s in Ae(Xk) which could be e x t e n d e d
to a n a u t o m o r p h i s m in Aute(Xk+i) , pTov~ed E k coTt,t;o,i~ed ord,y edges of type t. Havi~qg
so d e t e r m i n e d up to s e v e n s u b g r o u p s of Ae(:(k), e a c h arising f r o m a p a r t i c u l a r e d g e
type, we o b t a i n t h e g r o u p B a s t h e i n t e r s e c t i o n of t h e s e g r o u p s .
Note t h a t t h e
c o r r e c t n e s s of t h i s s t r a t e g y 1ollows f r o m L e m m a t a 1 and 2.
Type to,s: The s u b g r a p h s p a n n e d b y t h e s e e d g e s h a s v a l e n c e 2. H e n c e t h e g r o u p
B (°2) c a n b e d e t e r m i n e d in 0(n~) s t e p s using Corollary 14.
Type t1,1: The s u b s e t of t h e v e r t i c e s in Vk i n c i d e n t to a n e d g e of t h i s t y p e h a s to
be s t a b i l i z e d in Ae(Vk). By P r o p o s i t i o n 4, B (I,1) is found in 0(n~) s t e p s .
Type t a t : P r o c e e d as for t y p e tl, 1.
Type t l 2 : Let vl, v 2 be t h e two v e r t i c e s in Vk i n c i d e n t to a f a m i l y of this t y p e . We
m a y r e p l a c e t h e f a m i l y b y t h e c r o s s e d g e (vl,va). (Note t h a t t h e s e new c r o s s e d g e s
a r e d i s t i n g u i s h e d f r o m t h e " n o r m a l " c r o s s e d g e s of t y p e to, ~ s i n c e we c o n s i d e r a single
edge t y p e at a time).
The s u b g r a p h so o b t a i n e d has v a l e n c e 2, h e n c e B (l'g) c a n be
d e t e r m i n e d using C o r o l l a r y 14.
Type ta,a: We p r o c e e d as for t y p e tl, ~ a n d r e p l a c e e a c h f a m i l y b y a single c r o s s
edge. Note t h a t t h e s e c r o s s e d g e s a r e i s o l a t e d . H e n c e we c a n a p p l y C o r o l l a r y 14.
172
Type te,s: We r e p l a c e e a c h f a m i l y of t h i s type b y t h e t h r e e c r o s s e d g e s (v!,ve),
(v2,v3), a n d (v3,vl). Here v~, v2, a n d v 3 a r e t h e t h r e e v e r t i c e s in Vk i n c i d e n t to t h e f a m ily. Note t h a t t h e s u b g r a p h s p a n n e d by t h e s e new c r o s s e d g e s c o n s i s t s of i s o l a t e d
c y c l e s of l e n g t h 3, h e n c e is a g r a p h of v a l e n c e 2. C o r o l l a r y 14 a p p l i e s .
We have now c o n s i d e r e d all e d g e t y p e s e x c e p t for t y p e tl,a. This c a s e is m o r e
c o m p l i c a t e d a n d we c o n s i d e r it now in d e t a i l . F i r s t , we r e p l a c e e a c h f a m i l y of t y p e tl, s
with a cycle of t h r e e s p e c i a l c r o s s edges, j u s t as for t y p e t2,3. Since t h e s u b g r a p h
s p a n n e d b y t h e s e c r o s s e d g e s m a y have v a l e a c e 4, however, we n e e d t o f u r t h e r
a n a l y z e its s t r u c t u r e .
C o n s i d e r a c y c l e with v e r t i c e s u, v, a n d w o b t a i n e d f r o m a f a m i l y of e d g e s of t y p e
tl, 3. A s s u m i n g a s u i t a b l e e x t e n s i o n of t h e p e r m u t a t i o n d o m a i n of Ae(Xk), t h e i m p r i m i tivity s t r u c t u r e of this g r o u p is a c o m p l e t e b i n a r y t r e e . Since t h e c y c l e [u,v,w,u] has
odd l e n g t h , it follows t h a t t h e h e i g h t of t h e n e a r e s t c o m m o n a n c e s t o r (in t h e i m p r i m i t i v i t y t r e e ) of one of t h e t h r e e p a i r s (u,v), (u,w), a n d (v,w) m u s t differ f r o m t h e h e i g h t
of t h e n e a r e s ~ c o m m o n a n c e s t o r for t h e o t h e r two p a i r s (see F i g u r e 11).
Conse-
quently, one of t h e t h r e e c y c l e e d g e s is d i s t i n g u i s h e d f r o m t h e o t h e r two. We call t h e
\
!
\
/
i
/
\
l
\
\
/
\
/
/
k/
Figure t I
d i s t i n g u i s h e d e d g e t h e b-edge of t h e cycle, a n d call t h e o t h e r two e d g e s a-edges. It is
clear
that
an
automorphism
cannot
map
a
b-edge
to
an
a-edge,
hence
B(1,3) = B(1,s-a)QB (l's-b), w h e r e B (Ls-a) is t h e s u b g r o u p of Ae(Xk) a r i s i n g b y only c o n s i d e r i n g a - e d g e s , a n d B O's-b) t h e s u b g r o u p a r i s i n g b y only c o n s i d e r i n g b - e d g e s .
C o n s i d e r i n g t h e a - e d g e s o b t a i n e d f r o m f a m i l i e s of t y p e tl, s, we n o t e t h a t t h e s e
edges
c a n be
paired
since
e a c h f a m i l y r e s u l t s in e x a c t l y two a - e d g e s in o u r
173
construction. Furthermore, the a-edges can be oriented, from the vertex incident to
both edges of a pair towards the b-edge of the same family. These two observations
are crucial for analyzing the local action of B(1'3-~) on a-edges incident to a c o m m o n
vertex, and are used in the design of suitable isomorphism gadgets coping with
valences higher than 2.
Subject to the pairing and orientation of the a-edges, the group B(1'3-a) is equal to
B(1,3), since the mapping of a directed a-edge pair entails the c o r r e c t mapping of the
b-edge belonging to the same family. This observation saves constructing B(1,3-b) and
one group intersection.
We now describe how to obtain B O's-a) (assuming that the a-edges have been
paired and oriented). First, we reduce the essential valence to i by observing the
edge pairing, then we account fully for the orientation of the a-edges.
In the subgraph spanned by the a-edges, a vertex can have valence up to four. So
we classify the vertices according to their valence and setwise stabilize each of the
five classes in Ae(Xk), t h e r e b y obtaining the 2-group D.
We wish to extend D to a larger 2-group acting on a larger p e r m u t a t i o n domain by
forming wreath products with various 2-groups. These 2-groups should be the gadgets
reducing the valence of the subgraph spanned by the a-edges to one.
For vertices of valence 2, the wreath product is with the cyclic group C2 of order
2. This is exactly as for Corollary 13.
Consider the vertices of valence 3. Due to the implicitly p r e s e n t b-edge, a vertex
of valence 3 is always incident to a pair and a single a-edge. Because of the pairing,
the single a-edge cannot be mapped to one of the pair edges, hence our gadget is a
Sylow 2-subgroup of Sa, the symmetric group of degree 3, as shown graphically in Figure 12.
v'~e3
The Valence 3 Gadget
Figure i2
174
Finally, consider the valence 4 vertices. Here the vertex must be incident to two
pairs of a-edges. It is clear thai each pair is a set of imprimitivity for the action of
B (l's-a)on the a-edges, hence the required gadget is a Sylow 2-subgroup of S 4 as shown
in Figure 13.
The Valence 4 Gadget
Figure 13
With this gadget construction we obtain a new 2-group D' of degree at most 4
times the degree of D. D ~ reduces the essential valence of the subgraph of a-edges to
one, but we still need to fuily account for the orientation of the a-edges. We do this
now by setwise stabilizing, in the group D', the class of all vertices incident to the
o~r~
of an a-edge. To the resulting 2-group D" we m a y apply Corollary 14 and so
obtain B (l's-a) = B (1's). It is clear that the application of the corollary dominates the
timing, and in s u m m a r y -we have now established
THEOREM8 (Hoffmann)
Given a complete irnprimitivity structure and an imprimiiivity sequence for Ae(Xk), an
imprimitivity sequence for the subgroup B, consisting of those permutations which
may
be extended to an automorphism
in Aute,(Xk+l), can be determined in O(nk s)
steps, where n k is the n u m b e r of vertices at distance k from the edge e.
We n o w construct Ae(Xk+~) from B < Ae(Xk) and E k in two stages: First, B is lifted
to a homolnorphie image B in Sym(Vk+1). By L e m m a t a
i and 2 and the definition of }3
this is always possible. Second, we adjoin generators for the group C consisting of
those permutations of Vk+ 1 which m a y be extended to an automorphism of Xk+1 fixing
every vertex not in Vk+ I.
By Corollary 2, the group C is generated by transpositions (u,v), where u and v are
vertices in Vk+ 1 incident to a family of type t&3, tz,2, or t&1. Clearly C is elementary
abelian.
An
imprimitivity
sequence
[(ul,vl)..... (ur,vr)] for C
can
be
found
in
175
O(n~+nk+1) steps by inspectir~ E k.
Let [~I ..... ~,] be the imprirnitivity sequence determined for B. W e construct a
sequence
[W 1..... ~s] of permutations
I -< i- s. W e a s s u m e
generating B, where ~i is obtained from %,
a fixed but arbitrary enumeration
of the vertices in Vk+ I and
explain h o w to determine the W i.
Let ~ be the next generator of B. For each v e Vk+ I determine its ancestry U in
V k, and let U' be the image o f U u n d e r
~. B y L e m m a
i, U'is an ancestry and the fam-
ily types for U and U' are equal. If w c Vk+ I has the s a m e ancestry U as v, then there
are exactly two vertices v', w' 6 Vk+ 1 whose ancestry is U'. In this case we set v W = v'
and ~
= w', assuming, without loss of generality, that v precedes w and v' precedes w'
in the enumeration of Vk+ I. If there is no other vertex w e Vk+ I whose ancestry is U,
then there is exactly one v' e Vk+ I with ancestry U'.. In this case we define v W = v'.
it is clear that this construction determines a permutation W e Sym(Vk+1) which
is unique due to the fixed enumeration of Vk+ I. Moreover, given ~ there is an automorphism
in Aute(Xk+1) whose restriction to Vk+ 1 is W. Hence, if B = < W I..... Ws>, then
[~i ..... Ws] contains a composition
sequence
for B.
Note that [W 1..... Ws~ can be
determined in 0(nk2+nk+12) steps.
Finally,
the
sequence
[(ul,vl)..... (ur,Vr),~ 1..... Ws]
contains
a
composition
sequence for Ae(Xk÷1). Note that r+s -< nk+nk÷ I. Hence, by Froposition Z and L e m m a
13, w e
can determine
a complete
imprimitivi%y
structure a n d
an imprimitivity
sequence for Ae(Xk+1) in 0(nkS+nk+l s) steps. Consequently, we have established
THEOI~:M 9 (Hoffmann)
Given a c o m p l e t e i m p r i m i t i v i t y s t r u c t u r e a n d a n i m p r i m i t i v i t y s e q u e n c e for Ae(X~),
t h e n a c o m p l e t e i m p r i m i t i v i t y s t r u c t u r e a n d an i m p r i m i t i v i t y S e q u e n c e for Ae(Xk÷l)
c a n b e d e t e r m i n e d in O(nkS+nk+l 3) s t e p s , w h e r e n k is t h e c a r d i n a l i t y of Vk.
Proof
T h e o r e m 8 and t h e above d i s c u s s i o n . -
We s u m m a r i z e t h e r e s u l t s d e r i v e d above as follows:
COROLLARY15
Let X be a connected, trivalen% graph with n vertices, e an edge of X, and h the m a x imum
distance in X of any vertex from e. Then Ae(X ) can be determined in O(n s)
steps.
Proof
This follows from a straightforward induction on h using T h e o r e m 8. -
176
C ~
16 (Hoffmann)
L e t X b e a t r i v a l e n t g r a p h with n v e r t i c e s . Then i s o m o r p h i s m of X c a n be t e s t e d in
O(n 4) s t e p s .
Proof
X is s p l i t into c o n n e c t e d c o m p o n e n t s which a r e classified into i s o m o r -
p h i s m c l a s s e s . An e l e m e n t a r y c o m p u t a t i o n e s t a b l i s h e s t h e t i m e b o u n d with t h e h e l p
of C o r o l l a r y 15. E
5.
Notes and References
Miller [i979] h a s i n v e s t i g a t e d t h e p r o b l e m of r e d u c i n g t h e v a l e n c e of a g r a p h
while p r e s e r v i n g i s o m o r p h i s m .
He defines an isomorphism m-gadget as a g r a p h Z of
v a l e n c e a t m o s t m with a d i s t i n g u i s h e d s e t F of m + l v e r t i c e s of v a l e n c e a t m o s t m - l .
Z h a s to have t h e a d d i t i o n a l p r o p e r t y t h a t t h e s e t w i s e s t a b i l i z e r of F in Aut(Z) a c t s on
F as t h e s y m m e t r i c g r o u p Sym(F). Given a g r a p h of v a l e n c e m + t , one first r e p l a c e s
e v e r y v e r t e x v of v a l e n c e m + i with a c o p y of Z, c o n n e c t i n g t h e m + [ e d g e s f o r m e r l y
i n c i d e n t to v to t h e v e r t i c e s in F. Then, t h e g a d g e t e d g e s a r e l a b e l l e d b y a d d i n g t a i l s
of s u f f i c i e n t l e n g t h to each. It is n o t h a r d to s e e t h a t two d i f f e r e n t ways of r e p l a c i n g
e v e r y v e r t e x of v a l e n c e m + t m u s t r e s u l t in two i s o m o r p h i c g r a p h s , i.e., t h e c o n s t r u c t i o n m u s t p r e s e r v e i s o m o r p h i s m . Miller [ 1979] p r o v e d t h a t t h e only p o s s i b l e g a d g e t is
t h e 4 - g a d g e t d i s c o v e r e d b y C a r t e r [1977].
T h e o r e m I is d u e to T u t t e [1947]. T u t t e ' s f o r m u l a t i o n s t a t e s t h a t t h e o r d e r of t h e
s t a b i l i z e r of e v e r y vertex in a t r i v a l e n t g r a p h is e i t h e r 2r or 3.2 r. A v a r i a n t h a s b e e n
p r o v e d in B a b a i a n d Loves [1973]. The b a s i c a p p r o a c h t o t h e t r i v a l e n t c a s e is due to
F u r s t , H o p c r o f t , a n d Luks [ 1980a], e x c e p t for t h e r e d u c t i o n to t h e setwise s t a b i l i z e r in
2 - g r o u p s b y a c l a s s i f i c a t i o n of t h e e d g e s in Ek, wl~ich is d u e to Luks [t980].
Furst,
H o p c r o f t , a n d Luke [1980a] c o n t a i n s sufficient t e c h n i q u e s to d e r i v e t h e e q u i v a l e n t of
C o r o l l a r y 4. _This was n o t e d i n d e p e n d e n t l y by us a n d b y M. F u r s t .
The m e t h o d s of S e c t i o n 3 a r e d u e to Luks [1980]. R e c a l l t h a t we p a r t i t i o n t h e s e t
W c o n s i s t i n g of all n o n e m p t y s u b s e t s of Vk of size a t m o s t 3 into up to 9 b l o c k s . Luks
[1980] u s e s only 8 b l o c k s . This s e e m i n g d i s c r e p a n c y is e x p l a i n e d b y t h e f a c t t h a t for
i > 0 we m a y c o n s i d e r ti,1, tl,m a n d t~,s to b e t h e s a m e label, s i n c e s u b s e t s of Vk of
d i f f e r e n t c a r d i n a l i t y m u s t lie in d i f f e r e n t o r b i t s of G. Thus, only six b l o c k s a r e
required.
177
The material of Section 4 is due to Hoffmann [1981b]. Propositions I and 2 are
the result of joint work with C. Sims. Lemmata 8 and 9 are hinted at in Luks [1980].
The isomorphism test takes the same general approach as the subexponential algorithm of Furst, Hopcroft, and Luks [1980a]. The previous best bound for trivatent
graph isomorphism testing seems to have been 0(n 8) as announced in Luks [i980].