On Sizes of Mincuts in a Fibonacci Graph
MARK KORENBLIT and VADIM E. LEVIT
Department of Computer Science
Holon Academic Institute of Technology
52 Golomb Str., P.O. Box 305, Holon 58102
ISRAEL
{korenblit, levitv}@hait.ac.il
Abstract: - The structure of mincuts of a Fibonacci graph is investigated. We show that an n-vertex Fibonacci
 n  1
graph has mincuts of all sizes from 2 to 
.
 2 
Key-Words: - cut, Fibonacci graph, mincut, probabilistic graph, reliability, size of mincut, st-dag
The problem of revealing and enumerating all
cuts or mincuts in a probabilistic graph is discussed
in [1], [2], [5], [6], [7], [8], [9], [10]. In many cases,
this problem is the first step in the solution of
network reliability problems (specifically, the twoterminal reliability). Also, there are papers in which
the enumeration of cuts is considered as an
independent problem. For instance, a linear (per
mincut) algorithm for enumerating all mincuts of a
graph is proposed in [10]. The problem of
generation of various types of cuts in directed and
undirected graphs is studied in [8]. The paper [7]
investigates the problem of finding a maximum
weight exact cut (a set of edges intersecting each
path from s to t in exactly one edge).
In this paper, we compute numbers of mincuts of
all sizes in a special st-dag called a Fibonacci graph
(FG), and consequently, determine the size of a
maximum mincut in an n-vertex FG. In [4] we
proved that the total number of mincuts (irrespective
1 Introduction
Let G=(V,E) be a probabilistic graph, where V is a
set of vertices and E is a set of edges, representing
pairs of vertices. If the pairs are ordered (i.e., the
pair (v,w) is different from the pair (w,v)) then we
call the graph directed (digraph). All edges of a
probabilistic graph can fail randomly and
independently of one another, according to certain
known probabilities.
We say that a graph G’=(V’,E’) is a subgraph of
G=(V,E) if V’  V and E’  E. A two-terminal
directed acyclic graph (st-dag) has only one source s
and only one target t. In an st-dag, every vertex lies
on some path from s to t.
For a probabilistic graph G and specified vertices
s and t of G, we define the two-terminal reliability to
be the probability that there exists an operating path
(a path of operating edges) between s and t. We call
such a state a system operation and corresponding
event is EP(s,t). A state when no operating path
exists between s and t is said to be a system failure.
In the directed case, the problem of computing the
probability Pr[ EP( s, t )] is usually called stconnectedness.
We define a cut to be a set of edges whose failure
implies system failure. A size of a cut is a number of
edges in the cut. A mincut is a minimal cut. A set of
all mincuts of an st-dag is denoted C(s,t).
1
2
3
. . .
 n2 
,
4
of their sizes) in an n-vertex FG is equal to 
where we also established the complexity of the stconnectedness problem for a Fibonacci graph.
i
i+1
Fig.1. A Fibonacci graph
1
i+2
. . .
n–1
n
2 Preliminaries
CF(n–1,n) =
The notion of a Fibonacci graph (FG) was
introduced in [3]. In such an st-dag, two edges leave
each of its n vertices except the two final vertices
(n–1 and n). Two edges leaving the i vertex (1 ≤ i ≤
n–2) enter the i+1 and the i+2 vertices. The single
edge leaving the n–1 vertex enters the n vertex. No
edge leaves the n vertex. This graph is illustrated in
Fig. 1.
Suppose that all vertices of the certain FG are
numerated successively by increased order from the
source to the target. We identify vertices by their
ordinal numbers. We denote FG enclosed between a
source numbered i and a target numbered j (i<j) as
FG(i,j). Therefore, FG(i,j–1) is a subgraph of
FG(i,j), FG(i,j–2) is a subgraph of FG(i,j) and
FG(i,j–1), etc. We define a mincut of FG(i,j) that
causes also the system failure of its subgraph
FG(i,j–1) as a strong mincut of FG(i,j). We define a
mincut of FG(i,j) that does not cause the system
failure of its subgraph FG(i,j–1) as a weak mincut of
FG(i,j). We denote a set of all mincuts of FG(i,j) as
CF(n–2,n–1)  CF( n  2 ,n–1)  (n–2,n)
and
CF( n  1 ,n) =
{(n–2,n), (n–1,n)}  CF( n  3 ,n–2)  (n–1,n). (3)
3 The Main Result
It is clear that a 2-vertex FG is a single-edge st-dag
and its single mincut consists of this edge itself.
Theorem 1. For n ≥ 3, the mincuts of an n-vertex
FG are characterized as follows:
1. A mincut of the maximum size in an n-
 n  1
vertex FG has 
edges.
 2 
2. An n-vertex FG has mincuts of all sizes in
_
CF( i , j), a set of all strong mincuts of FG(i,j) as
 n  1
the range from 2 to 
and no other
 2 
mincuts.
3. The number of mincuts of all sizes in an nvertex FG is described by the following
formulae:
_
CF( i , j–1, j), and a set of all weak mincuts of
_
FG(i,j) as CF( i , j  1 , j).
The n-vertex FG depicted in Fig. 1 is FG(1,n).
The source of the initial FG is supposed to be
numbered 1. We reveal the subgraphs from the FG
in such a way that all the subgraphs, including the
initial FG, have the same source. For this reason, the
source number may be omitted when denoting sets
of mincuts, strong mincuts, and weak mincuts. In
such a case, CF(n), CF(n–1,n), and CF( n  1 ,n)
denote a set of all mincuts, a set of all strong
mincuts, and a set of all weak mincuts, respectively,
in an n-vertex FG and CFk(n), CFk(n–1,n), and
CFk( n  1 ,n) denote a set of all mincuts of the size
k, a set of all strong mincuts of the size k, and a set
of all weak mincuts of the size k, respectively, in an
n-vertex FG.
We continue our denotation in the following way.
Let S be a set of sets of edges. In such a case, the set
composed by adding an edge (x,y) to each set of
edges of S will be denoted S  (x,y).
It is clear that a set of all mincuts in an n-vertex
FG can be presented as
|CF2(3)| = 2,
|CF2(4)| = 3, |CF3(4)| = 1,
|CF2(6)| = 2, |CF3(6)| = 6, |CF4(6)| = 1,
for odd n ≥ 5:
|CF2(n)| = 2,
|CFk(n)| = n – 1 – 2(k – 3) :
k = 3, 4…,
n 1 n 1
,
,
2
2
for even n ≥ 8:
|CF2(n)| = 2,
|CFk(n)| = n – 1 – 2(k – 3) :
k = 3, 4…,
|CF n (n)| = 6,
CF(n) = CF(n–1,n)  CF( n  1 ,n).
(2)
2
(1)
|CF n (n)| = 1.
2
As shown in [4], in the general case, when n > 3,
2
1
n
n
 2 , 1,
2
2
Proof. The theorem is proved by induction.
Suppose that for even n, a weak mincut of the
maximum size in an n-vertex FG has
and there are no other weak mincuts in the FG.
Initially, a 3-vertex FG has the single weak mincut
{(1,3), (2,3)}, and | CF2 ( 2,3) |  1 , i.e., (5) holds
here.
Suppose that for even n, a strong mincut of the
n
edges,
2
n
| CFk (n  1, n) |  1 : k  2, 3...,  1,
2
maximum size in an n-vertex FG has
n
 1 edges,
2
(4)
| CF2 (n 1, n) |  1,
| CFn (n  1, n) |  2,
2
n
| CFk (n  1, n) |  n  2  2(k  3) : k  3, 4..., ,
2
and there are no other weak mincuts in the FG.
Consider an n+2-vertex FG. As follows from (3),
the new single 2-edge weak mincut in this graph is
{(n,n+2), (n+1,n+2)}. Each k-edge (k = 2, 3…,
(6)
| CFn (n  1, n) |  1 ,
2
n
 1) weak mincut in an n-vertex FG receives the
2
1
an n+2-vertex FG. Therefore, if n2 = n + 2 then
and there are no other strong mincuts in the FG. As
follows from (2), there are two groups of strong
mincuts in an n+1-vertex FG. The first of them
includes all strong mincuts of an n-vertex FG
presented in (6). The second one consists of all the
weak mincuts of an n-vertex FG (see (4))
supplemented by the edge (n–1,n+1). We denote a
set of all mincuts of the size k in the second group
of strong mincuts in an n+1-vertex FG as
CFk (n  1, n, n  1). In such a case,
n
n
 1  2 , and for an n2-vertex FG
2
2
| CFk (n  1, n, n  1) |
additional edge (n+1,n+2) and becomes an l-edge (l
n
) weak mincut in an n+2-vertex
2
n
FG. Two weak mincuts of the size
in an n-vertex
2
n
FG change to two weak mincuts of the size  1 in
2
= k + 1 = 3, 4…,
n
| CFk 1 (n  1, n)  (n  1, n  1) |  1 : k  3, 4..., ,
2
(7)
| CFn (n  1, n, n  1) |
n
| CFk (n2  1, n2 ) |  1 : k  2, 3..., 2  1,
2
| CFn2 (n2  1, n2 ) |  2.
2
1
| CFn (n  1, n)  (n  1, n  1) |  2.
2
2
An n2-vertex FG has no other weak mincuts.
Therefore, we have the same data as for an n-vertex
FG. Initially, consider a 6-vertex FG. It has three
weak mincuts. They are {(4,6), (5,6)}, {(2,4), (3,4),
If n1 = n + 1 (n1 is odd) then
n
n 1
1  1
. Based
2
2
(5,6)}, and {(1,2), (3,4), (5,6)}. | CF2 (5,6) |  1 and
on (2), (6), (7) we receive for an n1-vertex FG that
| CF3 (5,6) |  2 , i.e., (4) holds here. Hence, (4)
| CF2 (n1 1, n1 ) |  1,
holds for any even n ≥ 6, and there are no other
weak mincuts in such an n-vertex FG. Analogously,
it can be shown that for odd n ≥ 3, a weak mincut of
the maximum size in an n-vertex FG has
(8)
n 1
| CFk (n1  1, n1 ) |  n1  2  2(k  3) : k  3, 4..., 1 .
2
n 1
2
There are no other strong mincuts here and,
therefore, a strong mincut of the maximum size in
edges,
n 1
| CFk (n  1, n) |  1 : k  2, 3...,
,
2
such an n1-vertex FG has
(5)
n1  1
edges. Now, we
2
intend to show that if equations (8) hold for odd n1,
3
then equations (6) hold for even n2 = n1 + 1. As
follows from (2), there are two groups of strong
mincuts in an n1+1-vertex FG. The first of them
includes all strong mincuts of an n1-vertex FG
presented in (8). The second one consists of all
weak mincuts of an n1-vertex FG (change n for n1 in
(5)) supplemented by the edge (n1–1,n1+1). For the
second group,
even
n
we
sum
(4)
and
(6).
By
(4),
n
| CFk (n  1, n) |  2 when k  , and, thus,
2
n

| CFn (n) |  2  n  2  2  3   6.
2

2
For this reason, we get for even n
| CFk (n1  1, n1 , n1  1) |
| CF2 (n) |  2,
| CFk 1 (n1  1, n1 )  (n1  1, n1  1) |  1 :
n 1
k  3, 4..., 1
 1.
2
n
| CFk (n) |  n  1  2(k  3) : k  3, 4...,  1,
2
(9)
(11)
n  1 n2
Since 1
 , using (2), (8), and (9) we
2
2
| CFn (n) |  6,
2
receive for an n2-vertex FG that
| CFn (n) |  1.
| CF2 (n2 1, n2 ) |  1,
2
| CFk (n2  1, n2 ) |  n2  2  2(k  3) : k  3, 4...,
Therefore, a mincut of the maximum size in an n-
n2
,
2
vertex FG (n is even) has
| CFn2 (n 2  1, n 2 ) |  1.
2
n
 n  1
1  
 edges. As
2
 2 
follows from (10) and (11), an n-vertex FG has all
1
 n  1
mincut sizes in the range from 2 to 
 . It has
 2 
no mincut of other size since there are no other
mincuts in FG except the mincuts mentioned in (10)
and (11). As follows from the initial conditions, (10)
holds for odd n ≥ 5. We do not consider equations
(11) for n = 6, because the general equation |CFk(n)|
= n – 1 – 2(k – 3) is never used in such a case.
Therefore, (11) holds for even n ≥ 8. The special
cases when n = 3, 4, and 6 are considered
separately. A 3-vertex FG has two 2-edge mincuts:
{(1,3), (1,2)} and {(1,3), (2,3)}. A 4-vertex FG has
three 2-edge mincuts: {(1,3), (1,2)}, {(2,4), (3,4)},
{(1,2), (3,4)}, and one 3-edge mincut {(1,3), {(2,3),
(2,4)}. A 6-vertex FG has two 2-edge mincuts:
{(1,3), (1,2)}, {(4,6), (5,6)}, six 3-edge mincuts:
{(1,3), (2,3), (2,4)}, {(2,4), (3,4), (3,5)}, {(1,2),
(3,4), (3,5)}, {(3,5), (4,5), (4,6)}, {(2,4), (3,4),
(5,6)}, {(1,2), (3,4), (5,6)}, and one 4-edge mincut
{(1,3), (2,3), (4,5), (4,6)}. Now, the proof of the
theorem is complete. ■
An n2-vertex FG has no other strong mincuts.
Therefore, we have the same data as for an n-vertex
FG in (6). Hence, (8) follows from (6) and vice
versa. Initially, consider a 5-vertex FG. It has four
strong mincuts, namely, {(1,3), (1,2)}, {(1,3), (2,3),
(2,4)}, {(2,4), (3,4), (3,5)}, and {(1,2), (3,4), (3,5)}.
Thus, | CF2 (4,5) |  1 and | CF3 (4,5) |  3 , i.e., (8)
holds here. Hence, (6) is true for a 6-vertex FG.
Therefore, (8) holds for any odd n ≥ 5 and (6) holds
for any even n ≥ 6, and there are no other strong
mincuts in such an n-vertex FG in both cases.
Now, based on (1) we combine the
corresponding results for weak and strong mincuts.
For odd n by summing (5) and (8) we obtain
| CF2 (n) |  2,
| CFk (n) |  n  1  2(k  3) : k  3, 4...,
n 1
.
2
(10)
Corollary 2. An n-vertex FG has four mincuts of
the maximum size for odd n  5 and one mincut of
the maximum size for any even n.
Therefore, a mincut of the maximum size in an nvertex FG (n is odd) has
1
n  1  n  1

 edges. For
2
 2 
4
Proof. As shown in Theorem 1 a mincut of the
maximum size in an n-vertex FG has
[6] J. S. Provan and M. O. Ball, Computing
Network Reliability in Time Polynomial in the
Number of Cuts, Oper. Res. 32, 1984, pp. 516526.
[7] J. S. Provan and V. G. Kulkarni, Exact Cuts in
Networks, Networks 19, 1989, pp. 281-289.
[8] J. S. Provan and D. R. Shier, A Paradigm for
Listing (s, t)-Cuts in Graphs, Algorithmica 15,
1996, pp. 351-372.
[9] D. R. Shier, Network Reliability and Algebraic
Structures, Oxford University Press, Oxford,
New York, 1991.
[10] S. Tsukiyama, I. Shirakawa, H. Ozaki, and H.
Ariyoshi, An Algorithm to Enumerate All
Cutsets of a Graph in Linear Time per Cutset, J.
ACM 27, 1980, pp. 619-632.
n 1
edges
2
for odd n. As follows from Theorem 1, for odd
n5
 n 1 
| CFn1 (n) |  n  1  2
 3  4.
 2

2
For even n the result is derived directly from
Theorem 1 and the preceding remark about a 2vertex FG. ■
4 Conclusion and Future Work
It is shown that the size of a maximum mincut in an
 n  1
n-vertex Fibonacci graph is equal to 
 for any
 2 
n ≥ 3. It can be easily proved by induction on n that
the number of edges in an n-vertex FG equals
2n  3. Thus, by Theorem 1, for a large n, the
number of edges in a maximum mincut of a
Fibonacci graph can always be estimated as a
quarter of a number of edges in the graph. An
interesting open problem is to tally up the
coefficients of the reliability polynomial of a
Fibonacci graph using information on the number of
the mincuts of a given size only.
References:
[1] M. O. Ball, C. J. Colbourn, and J. S. Provan,
Network
Reliability,
Network
Models,
Handbooks in OR & MS 7, North-Holland,
Amsterdam, 1995, pp. 673-762.
[2] C. J. Colbourn, The Combinatorics of Network
Reliability, Oxford University Press, Oxford,
New York, 1987.
[3] M. Ch. Golumbic and Y. Perl, Generalized
Fibonacci Maximum Path Graphs, Discr. Math.
28, 1979, pp. 237-245.
[4] M. Korenblit and V. E. Levit, The stConnectedness Problem for a Fibonacci Graph,
Proceedings of the 3rd WSEAS International
Conference on Applied and Theoretical
Mathematics, Miedzyzdroje, Poland, September
2002, pp. 1391-1395.
[5] J. S. Provan and M. O. Ball, The Complexity of
Counting Cuts and of Computing the Probability
that a Graph is Connected, SIAM J. Comput. 12,
1983, pp. 777-788.
5