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 n5 n 1 | CFn1 (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