Internat. J. Mah. & Math. Sci
737
Vol. 5 No 4 (1982) 737-744
ON THE ACYCLIC POINT-CONNECTIVITY OF THE n-CUBE
JOHN BANKS
Department of Mathematics
University of California, Santa Cruz
Santa Cruz, California 95064 U.S.A.
JOHN MITCHEM
Department of Mathematics and Computer Science
San Jose State University
San Jose, California 95192 U.S.A.
(Received October 5, 1981 and in revised form May 21, 1982)
ABSTRACT.
The acyclic point-connectivity of a graph G, denoted (G), is the minimum
number of points whose removal from G results in an acyclic graph.
(Qn 2
n-I
() -< 2
Harary stated erroneously that
that for n > 4, 7
2
n-4 -<
n-I
2
i where
n-y-2,
Qn
In a 1975 paper,
denotes the n-cube.
where y
[log2(n-
We prove
We show
i)].
that the upper bound is obtained for n -< 8 and conjecture that it is obtained for all
n.
KEY WORDS AND PHRASES.
n-cube, acyic point-connectivity.
1980 THEMATICS SUBJECT CLASSIFICATION CODES.
1
05C05, 05C40.
IN TRODUCT I ON.
Definitions and examples for this paper can be found in Harary [i], which has a
For
readable survey of the general area to which our results and conjectures apply.
current work, consult especially the Journal of Graph Theory and the Journal of Com-
binatorics, Seri’’s B.
More specifically, in [2] Harary introduced the acyclic point-
connectivity of a graph G, denoted (G), as the minimum number of points whose removal
from G results in an acyclic graph.
(Km,n
rain{re,n}
i and
(%)
He then stated without proof that (K
n-I
2
P
i, where
Qn
denotes the n-cube.
two equalities are indeed trivial; the last one is incorrect.
p
2,
The first
In this paper we obtain
upper bounds for (Qn), show that the upper bound is obtained for n -< 8, and conjecture
J. BANKS AND J. MITCHEM
738
that the upper bound is obtained for all n.
Before beginning the proof we introduce the following notation and definitions.
The dis-
For any real number x, [x] denotes the greatest integer not greater than x.
d(v,w).
tance between two points v and w is denoted by
O’s
with point set consisting of all n-tuples of
Qn
I < i
with fixed ith coordinate,
the subgraph of
<
Qn
is the graph
and l’s, and two points are adjacent
We observe that
iff they differ in exactly one coordinate.
points of
The n-cube
Qn
is bipartite and all
n, induce the subgraph
In fact,
Qn-l"
induced by all points with nth coordinate 0(i) is denoted
Qn
Q-I (Q-l)"
MAIN RESULTS.
2.
For n > 4, 7
THEOREM 1.
2 n-4 <
(Qn)
< 2 n-I
2 n-y-2
We begin the proof of Theorem I with a number of lemmas.
[log2(n-l)]
where y
Some of the easy proofs are
omitted.
LE[A i.
Let v,w
LEMMA 2.
For u,v
V(); d(v,w)
V(, n > i, if
j iff v and w differ in exactly j coordlnate
d(u,v)
2, then u and v are contained in
exactly one 4-cycle.
LEMMA 3.
()
t, then
(Qn+l
> 2t.
This follows imediately from the observation that
PROOF.
copes
joint
If
of
LEMMA 4.
Qn"
(Q2
i and
(Q3
Qn+l
contains two dis-
3.
The first equality is clear and with Lema 3 implies that
PROOF.
u,v
V(Q3).
(Q3
> 3.
If d(u,v) < 2(=3), then
Q3- {v,u}
> 2.
(Q3
contains a 4-cycle (6-cycle).
Let
Thus
Equality follows from the fact that the removal of (i, i, 0), (0,
I, i),
results in a disconnected graph consisting of K and
1
which we
(i, 0, i) from
Q3
denote by K
KI, 3.
I
0
We also note that the five point path
P5
KI,3,
results from the
removal of (0, O, 0), (i, l, 0), and (0, 0, i) from
LEMMA 5.
K
I
o
The only induced acycllc subgraphs
Q3"
of Q3
with order 5 are
P5
and
K1, 3PROOF.
Exactly one of
,
three points induce
P5
r
El
0
3.
P3’
Q2’
and
Q2"
has three points of such a subgraph H.
These
and since H contains no cycles and is induced, it follows that
739
ACYCLIC POINT-CONNECTIVITY OF THE n-CUBE
LEMMA 6.
u(Q4
6 and the resulting induced acyclic subgraph H of
consists
KI, 4.
of two disjoint copies of
From Lemmas 4 and 3 it follows that
PROOF.
Q4
that six points may be removed to form
(Q4
>
6.
However, Figure i shows
,4 u ,4"
FIGt.RE 1
6, and according to Lemma 5 when H is formed each of
(Q4
Thus
have three points removed forming respectively
K u
I
,3"
H’
Assume
PS"
Now
H’
Thus
jacent to the three points removed from
this implies that
H’
contains
Let v i(w i), 1 < i
1 < i
8.
Also let
KI, 3.
Thus
v2, v3,
v
4}
Now it can be checked that if
wi, I
H",
Thus w
then H contains a cycle.
proved.
LEMMA 7.
(Q5
Q3’
H’
8, be the points of
{Vl,
14.
H". Also, H’
and
is adjacent with two points of
H".
must be in different components of
H’
induce
< i <
8
H"
Q3’
u
and
H"
KI, 3
and v
8
Q3"
must
P5
are
or
and these points
H’
are ad-
H". However,
u K
I.
Q3’(Q3 ’’) and let
KI, 3
H"
and points of
KI,3,
and the isolated point of
H"
and
v.z be adjacent
to
be the isolated point in
H’.
7, is the required point of degree three in
must be the degree three point and the Lemma is
740
J. BANKS AND J. MITCHEM
PROOF.
According to Lemmas 6 and 3,
(Q5
>
12.
Figure 2 shows that
(Q5
< 14.
F:r.E 2 65
Assume that
(Q5)
12 or 13, and let H be an acycllc subgraph of
removal of at most 13 points.
M3,
from
and M
4
of
MI, M2,
Q3"
and
Now
Q5
Q5
formed by the
may be viewed as 4 disjoint copies, say
MI, M2,
It follows from Lema 4 that exactly three points have been removed
M3,
and 3 or 4 points have been removed from M
4.
Furthermore, Lema
6 implies that the darkly shaded lines of Figure 3 are contained in H.
ACYCLIC POINT-CONNECTIVITY OF THE n-CUBE
741
VI
FIGURE
However, as shown in Figure 3, points v2, v3, and v lie in 6-cycles with x and y,
4
point v
I
is in a 4-cycle containing x and y, and point u is in a 4-cycle containing w.
(Qs)
Hence
> 14.
Before we prove the main result two additional definitions are required.
1 < k
+
V()
the point which differs from v in exactly the coordinates i, k
i < n
o <
It is clear that
n.
01
Then
k+l ,n ,n
o
denotes the element in
2
n-y-2
{0,i}
denotes the mapping which assigns to each v in
is an automorphism of
Also if
i
no,
{0,i},
and n.
then
i
1
[log2(n-l) ].
Then there exists a set of
points of Q
whose removal results in an acyclic graph G which consists
n
n-I
and 2
n2 n-y-2 isolated points.
mutually disjoint copies K
1 ,n
n
of exactly 2
n-y-2
The set of points of degree n is denoted by
points.
Qn"
+ i,
{.}.
Let n > 2 and y
THEOREM 2.
n-I
2
@i’,k+l,no,n
Let
C(Gn)
C(Gn),
is a subset of one bipartite set of
and these points are called cluster
%
and all points from the other
742
J. BANKS AND J. MITCHEM
y
Then 0 < k < 2
If k
{v e
i
Cn,v
+
2
Furthermore, let nO
y
+
i and n
n
O
+ k.
i.
y
# 2
Gn.
are contained in
bipartite set of
V(Gn):
v
n
(i’
i, let v
( i’
i
C(Gn. Then
i-l’ i’ i+l’
the elements of
E
2’
n0’n0+l’’’’’n)’
"’’’n0-1’
u{v}u{x V(G
C
D
.
n):
n,v
n,v
n D
x is adjacent to v}. If u # v where u,v e C(Gn) then D
n,u
n,v
2,3,4,5. In order to cornPROOF. Lemmas 4, 6, and 7 verify the theorem for n
k
2 < i < n
are isolated points of G
n
1
o
Also let
plete the proof we show that the theorem is true for n, assuming it is true for n
i.
We consider two cases.
CASE I.
By
2Y
n
+
ass u mption the theorem is true for
Indu c time
there exist 2
i.
n-l-(y-l)-2
2
n-2
points whose removal from
n-l-(y-l)-2
which consists of 2
2
(n- i)(2 n-y-2)
n-2
set of
2
n-y-2
n
Qn-l’
C(Gn_ I)
Also
Thus
results in graph
-i
mutually disjoint copies of
0 isolated points.
y.
2
i
Gn_ I
and
Kl,n_ 1
is contained in a bipartite
Qn-l"
Let G’
n-i
where
(i’
C(G_I) and
e
n-i
n
n-I
if u # v where u,v e
2
C(Gn),
Th’s the 2
In this case k
nate, vi
Thus v
V(_l ).
n-y-2
Qn
I
...,n_l,0)
which contains
n-I
and by Leuna 2, v and w
are in acyclic graph G
n
i
i
KI, n
n
O
Let v
are mutually disjoint.
n
has been
Furthermore,
Also G contains
n
V(Gn).
i
From Lemma i, we have that
C
n,v
i
Since v
be the point of
are in a 4-cycle of
-i
i
differs from v in the nth coordi-
adjacent to
-I"
v i.
Now
d(wl,v)
However, all points of B n
V(Gn) and v i is an isolated point.
C(Gn)
C(G_I). Then d(u,v) > 4. Thus
Thus w
Let u # v be elements of
<
Now
Qn"
G
n(2 n-y-2) are isolated points.
B and v
Let w
C(G_I). Note
C(G_ I) C(Gn )"
induced by B u
points from
copies of
0 and n
i
Qn
the points adjacent to v are disjoint from the points ad-
n-y-2
all points of B and thus 2
3.
Let A be the bipartite set of
V(Gn_I).
as the subgraph of
formed by removing 2
d(ci,v)
induced by all points of the form (e
let B be the other bipartite set.
Now define G
jacent to u
Qn
be the subgraph of
i
n D
for otherwise, d(u,v)
2. It now follows that D
n,v
n,u
y
y
CASE II. n
2 + 1 + k, 1 < k < 2
i.
.
C
n,u
2,
V(_ 1)
n,v
ACYCLIC POINT-CONNECTIVITY OF THE n-CUBE
Again we apply the inductive assumption on n
Xn_ I, 0)
form (xI,
Qn’
Gn_I.
automorphism of
isomorphic to
C(G"
follows that
formed from
C(G_ I)
tite set of
u
C(Gn_I)
%
.
n C
n,v
we suppose that u,v
n,u
Similarly C
C(%)
C
n,v
and
Ol,k+l,n0,n(V),
Hence, v
.
consists of all but one point of
v.1
C
n,v
o C
n,v
Then k
and x be the point of
and d(x,)
i
0
2.
we first
C(Gn_ 1
consists of all but one point, namely
+
2 < i -< n
o
-i which
V(Gn)
-Thus x
is adjacent to v
is an isolated point of G
n
or
n G
-i
G"n_l
Vk+l,
C(G_I)
i and v
for otherwise,
So
Let
n
in
i
C(G i),
Qn"
2(2
n-2
Gn consists
of 2(2
(n- i)2 n-y-3)
2
n-y-3)
n-y-2
2
2
n-y-2
n-I
would contain a 4-
nD
n,u n,v
It now follows easily that D
disjoint copies of
n2
-
Now
when u # v.
Thus
of
Thus it follows from
Cn_l, u.
By induction, v was an isolated point of G’
i
n-I
say the former.
cycle.
and the other bipar-
C(G_
n,u
C(G_I)
Qn
o D
In order to show that D
n,v
n,u
and
The latter is clear if u
v
I)
the inductio assumption that C
Let v
4 and the inductive assumption, if follows that
V(%).
C(G i).
n,u
is
u V(G"n-+/-4). It
n-y-2
2 n-y-3)
2 n-I
2
n-2
C(Gn) where u # v.
show that C
9
by the removal of 2(2
G"n_l,
5=@l,k+l,no,n(U)
0)
is a subset of one bipartite set of
is contained in
Let u,v
with the
which we now denote by
C(G_I) hen
% induced by V(G 1
8n_l,
as the subgraph of
From the fact that d(u,5)
points.
C(%)
@l,k+l,n0,n(G’n-i)
(81,
Thus if u
Gn was
Gn_ I
Qn induced by the points of the
V(Gn_l). Since 81,k+l,n0,n is an
Xn_l)
it follows that
n
i and obtain graph
be the subgraph of
where (xI,
We now define G
n-l"
G’n_l
Let
prescribed properties.
743
n-y-2
Kl,n
and
This completes the
isolated points.
proof of Theorem 2.
Now the upper bound of Theorem I follows iunediately from Theorem 2and Theorem
l’s lower bound from Lemmas 4 and 7.
We close by stating two conjectures.
6, 7, and Theorem 1 show the first conjecture is true for 2
show the second conjecture is true for m
1 and 2.
_< n _<
8.
Lemmas 4,
Lemmas 4 and 6
Our long proof for m
3 is
It seems possible that Conjecture 2 would be useful in proving Conjecture i.
n-y-2
n-I
CONJECTURE i. For n > i, e(Qn
2
2
where y
[log2(n- i)].
omitted.
744
J. BANKS AND J. MITCHEM
CONJECTURE 2.
sists of exactly 2
If n
n-m-l
2
m,
m > i, any maximum acyclic induced subgraph of
disjoint copies of KI
con-
n
REFERENCES
i.
HARARY, FRANK.
2.
HARARY, FRANK.
Graph Theory, New York, Addison-Wesley, 1969.
On Minimal Feedback Vertex Sets of a Digraph,
on Circuits and Systems_, 1975, 839-840.
I.EEE
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