Computation of the Network Reliability (Extended Abstract) Kyoko Sekine3 and Hiroshi Imai Department of Information Science, University of Tokyo
Abstract It is known that a certain case of the all-terminal network reliability can be computed via the Tutte polynomial which is an invariant in graph theory. Recently, we have proposed a new approach of computing the Tutte polynomial of a graph of moderate size [9]. In this paper, by using it we analyze computation method of the general all-terminal network reliability and two-terminal network reliability with di erent edge deletion probability. In addition, to obtain the upper bound for the all-terminal network reliability, the polynomial-time algorithm of computing the Tutte polynomial for a complete graph [1] is applied to compute that reliability for a complete graph.
1 Introduction The Tutte polynomial is a fundamental invariant in graph theory, and has many applications in various elds. For example, it is related to the chromatic polynomial which denotes the number of vertex colourings. It also gives the number of spanning trees, the number of forests, the number of acyclic orientations, etc. [12]. Moreover, it is also related to the well-known Jones polynomial in knot theory, network reliability and statistical physics. However a problem of computing the Tutte polynomial is #P-hard in general. Therefore it is dicult to compute the Tutte polynomial in practice. Recently, we have proposed a new algorithm of computing the Tutte polynomial [9]. This algorithm is based on the recursive formula. When the recursive formula is applied to a given graph, there comes a number of isomorphic minors. The algorithm becomes ecient by representing these isomorphic minors by one of them. On the other hand, ecient algorithm for testing the isomorphism of arbitrary graphs has not been found. Therefore, rst, this algorithm orders the edges of a given graph, then restricts itself to nd isomorphic minors with the same edge ordering, and computes the Tutte polynomial of one of them. The complexity of the algorithm can be analyzed in terms of Bell numbers and the Catalan numbers in combinatorics.0 By our algorithm, we can compute 1 the Tutte polynomial of any graph with at most 14 vertices and 14 = 91 edges, and that of a planar 2 graph such as 12 2 12 lattice graph with 144 vertices and 264 edges. The network reliability itself has long been investigated (e.g., see [4, 5]), and have been considered many types of reliability and algorithm of computing them. Note that the network reliability is also #P-hard to compute [8] { approximate algorithm of computing upper and lower bounds of the network reliability has been intensively studied [5]. It is known that a certain case of the network reliability is related to the Tutte polynomial. Here, it is shown that the above algorithm of computing the Tutte polynomial can be applied to the general allterminal network reliability and two-terminal network reliability with di erent edge deletion probability. Then, the network reliability for a complete graph is considered. The reliability for a complete graph of n vertices is an upper bound for the other simple connected graphs with the same number of vertices. However, since it has extra edges, its computation by the above algorithm becomes more dicult than the computation of the other graphs. Here, we propose another method of computing the reliability for a complete graph with some of its computation results. 3
E-mail:
[email protected]
1
2 Computing the Network Reliability 2.1
Relation with the Tutte Polynomial
Let G = (V; E ) be a simple connected undirected graph with vertex set V and edge set E . Consider a network (graph) G = (V; E ). The canonical all-terminal network reliability R(G; p) is de ned as the probability that G remains connected after each edge is deleted with the same probability p. R(G; p) can be computed by enumerating the spanning sets of G. Actually, it is related to the Tutte polynomial which is an invariant in graph theory. The Tutte polynomial of a graph G is a two-variable polynomial T (G; x; y) de ned by T (G; x; y ) =
X AE
(x 0 1)(E )0(A) (y 0 1)jAj0(A) :
where : 2E ! Z is the rank function of a graph G. That is, (A) is the rank of a subgraph G0 = (V (A); A): the number of vertices, jV (A)j, minus the number of connected components of G0 . From the de nition of the Tutte polynomial, its values at some speci c points have the following meanings.
T (G; 1; 1) counts the number of spanning trees of G, which is polynomially computable. T (G; 2; 1) counts the number of forests of G, which is #P-hard to compute. T (G; 1; 2) counts the number of spanning sets of G, which is #P-hard to compute.
For other meanings and many applications of the Tutte polynomial, see [12]. The relation between the canonical network reliability and the Tutte polynomial is as follows: R(G; p) =
X AE; (A)=(E )
(1 0 p)jAj pjE 0Aj
= pjE j0(E ) (1 0 p)(E )
X AE; (A)=(E )
(1 0 p) 1
1 jAj0(A)
p
= pjE j0(E ) (1 0 p)(E ) T (G; 1; 1p )
Note that when p = 21 it exactly corresponds to counting up the number of spanning sets. Since the Tutte polynomial has the following recursive formula, the canonical all-terminal network reliability can be obtained by using this relation. 8 > >
> : T (Gne; x; y ) + T (G=e; x; y ) otherwise
Here, for an edge e in E , we denote by Gne the graph obtained by deleting e from G, and by G=e the graph obtained by contracting e from G. A loop is an edge connecting the same vertex, and a coloop is an edge whose removal decreases the rank of the graph by 1. By de nition, the Tutte polynomial of a loop is y and that of a coloop is x. The Tutte polynomial of an empty graph is 1. 2.2
All-terminal network reliability
Let p(e) be a given deletion probability of an edge e 2 E . Then, the all-terminal network reliability is de ned as the probability that the graph remains connected after each edge e is deleted with the probability p(e). This reliability will be simply denoted by R(G). In this general case, we cannot nd any relation with the Tutte polynomial. However, it still can be computed by enumerating the spanning sets. Then, like the Tutte polynomial, the following edge deletion/contraction formula holds. 2
Lemma 1
For an edge
e,
R(G) =
8 > > < > > :
(1 0 p(e))R(G=e) e: coloop R(Gne) e: loop p(e)R(Gne) + (1 0 p(e))R(G=e) otherwise
Proof: When e is a coloop e must not be deleted if G is connected, and when e is a loop e can be ignored. When e is neither a loop nor a coloop, in the case that e is deleted, if Gne is connected then G is still connected. Furthermore, in the case that e is not deleted, if G=e is connected then G is connected. 2
The computation process by using this formula can be represented as a binary tree such that the root corresponds to the original graph G, and each parent has at most two children. Nodes in this binary tree correspond to minors of G. Here, a graph obtained from G by deletions and contractions of edges is called a minor of G. For each path from a root to a leaf, a set of contracted edges corresponds to a spanning tree of G one-to-one. Then the number of leaves equals the number of spanning trees. Moreover, for each path edges consist of deleted loops correspond to the externally active edges for the corresponding spanning tree. Here, we say ej , whose ends are a and b, is externally active for a spanning tree T , if each ek which is an edge in a simple path in T connecting a and b satis es k < j (see [11]). For example, for the graph of Figure 1 the edges e4 ; e5 and e7 are externally active for a spanning tree fe1 ; e2 ; e3 ; e6 g. We can enumerate the spanning sets by adding the externally active edges for each spanning tree. Since p(ej ) + (1 0 p(ej )) = 1, by the above recursive formula, both cases (delete ej and not delete ej ) are summed up for each externally active edge ej . 2.3
Computation by the Edge Contraction and Deletion Formula
We proposed a new approach to compute the Tutte polynomial of a graph by using a fact that many isomorphic minors appear in that computation by the recursive formula (Sekine, Imai and Tani [9]). It is easily seen that this algorithm can be used to compute the network reliability in this general case. By sharing the isomorphic minors an expansion tree is modi ed as a rooted acyclic graph with the single source (root) and single sink (edges are oriented from the root to the sink). For example, Figure 1 shows the computation of the all-terminal network reliability for a given graph. In this gure, pi denotes the deletion probability of an edge ei and qi = 1 0 pi . This acyclic graph is closely related with the BDD (Binary Decision Diagrams: a tool for Boolean function manipulation [3]) representing all the spanning trees of a given graph. This is because, as mentioned above, each path corresponds to a spanning tree of the graph one-to-one. From this, the all-terminal network reliability in this general case can be computed by the following algorithm. R(source) := 1; fcomment: source is a unique node in the 0-th levelg for i := 1 to m do begin for all nodes u in the i-th level do R(u) := 0; for each node v in the (i 0 1)-th level do begin if v has an edge to a child u corresponding to a coloop e then R(u) := R(u) + (1 0 p(e))R(v); if v has an edge to a child u corresponding to a loop e then R(u) := R(u) + R(v ); if v has an edge to a child u corresponding to deletion of e and an edge to a child w corresponding to contraction of e then R(u) := R(u) + p(e)R(v); R(w) := R(w) + (1 0 p(e))R(v); end; end; R(sink) is R(G).
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6 7
Figure 1: Computation process of the all-terminal reliability For given two graphs since there is no ecient algorithm to decide that they are isomorphic or not, we restrict ourselves to nding isomorphic minors with the same edge ordering as follows. First for a given graph order the edges e1 ; e2 ; . . . ; em (m = jE j). Then we apply contraction deletion formula for edges in this order. Suppose that Ei = fe1 ; e2 ; . . . ; ei g, and Ei = fei+1 ; ei+2 ; . . . ; em g. Then the minors of G in the i-th level have the edge set Ei . For i = 1; . . . ; m, de ne the i-th level elimination e to be a vertex subset consisting of vertices v such that v is incident to some edges in E and front V i i some edges in Ei . By the edges contracted in this process, we can de ne an equivalence relation on Vei such that two vertices are in the same equivalence class if and only if, in the process of obtaining the minor, they are uni ed into one vertex by the contractions. Then consider a partition of Vei into the equivalence classes by this relation. We call this partition the i-th level elimination partition of the minor. For example, in Figure 1 the third level elimination front is fv2 ; v3 ; v4 g, since all incident edges 4
of v1 are contracted or deleted. When e1 and e2 are contracted and e3 is deleted v2 and v3 are uni ed to one vertex, then the elimination partition of this minor is ffv2 ; v3 g; fv4 gg. By using these de nitions, the following holds. Theorem 1
Let
H1
H2
and
be two minors of
G with the same edge set Ei . H1 and H2 are isomorphic i-th level elimination partition are identical.
with the same edge ordering if and only if their
Proof: ()) By the de nition, every vertex of the i-th level elimination front Vei is incident to some edges in Ei . Now H1 and H2 are isomorphic with the same edge ordering. The vertices are decided uniquely by their incident edges. Then i-th level elimination partition of H1 and H2 are identical. (() See [9]. 2
The size of an acyclic graph is de ned to be the number of minors in it. The width of the acyclic graph is de ned to be the maximum among the number of minors at each level. It should be noted that with respect to a working space this algorithm can be implemented proportionally to the size of the acyclic graph. The depth of the acyclic graph is the number of edges. Then the width of the acyclic graph is important. We demonstrated some numerical practice of the size and width of the acyclic graph for complete graphs and lattice graphs [9]. Theorem 2
Let
l
be the maximum size of the elimination front. Then, the width of an acyclic graph of
a computation process is bounded by of a set of
2.4
l
Bl ,
where
Bl ,
called the Bell number, is the number of partitions
elements.
Two-terminal network reliability
Given a pair of two special vertices s and t, the two-terminal network reliability R(Gjs; t) is de ned as the probability that there exists a path connecting s and t in the remaining graph after each edge e is deleted with the probability p(e). First, we modify the given graph G into G0 by introducing two new vertices s0 and t0 and connecting them with s and t by edges (the reason is mentioned below). The deletion probability of these two edges are de ned to be 0. Moreover suppose that the ordering of these two edges is the last for the following recursive formula. Then, the two-terminal reliability between s and t is equivalent to that between s0 and t0 . We call an edge e active if there exists a simple path connecting s0 and t0 and containing e. Otherwise we call it inactive. By this de nition, the following recursive formula holds. Lemma 2
For an edge
e
besides the newly added two edges,
8 > >
> : p(e)R(G0 nejs0 ; t0 ) + (1 0 p(e))R(G0 =ejs0 ; t0 ) otherwise
Proof: When e is a loop or inactive e is irrelevant in calculating the probability R(G0 js0 ; t0 ), and we can simply reduce it to R(G0 nejs0 ; t0 ). When e is an active coloop e the deletion of e makes s0 and t0 are disconnected. The remaining case is that e is active and neither a loop nor a coloop; this is similar to the general case of the all terminal network reliability. 2
By considering a decomposition tree for two-connected components, we can decide active or inactive for each edge in G0 . Here, each node of the decomposition tree corresponds to a two-connected component or an articulation point. If an articulation point is contained by some two-connected component, there is an edge between corresponding two nodes. The both components which include s0 or t0 are called active. The other components are de ned to be active if their corresponding nodes in the decomposition tree is on a path between two nodes corresponding to the two components containing s0 and t0 ; the other components are inactive. Then, an edge e in G0 is active if it is in active component; the other edges are inactive. This gives the following Lemma. 5
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Figure 2: Computation process of the two-terminal reliability Lemma 3
Given the two-connected component decomposition of
0 path between s
0 and t ,
G0
with the marked components on the
whether an edge is active or inactive can be checked in a constant time.
The decomposition tree should be maintained for the deletion of an active edge, since it is possible that some active edges in the same component become inactive edges. Inactive components are completely irrelevant to the two-terminal network reliability between s0 and t0 . The reason why two new edges must be added is to avoid the following case. When all incident edges of s are contracted or deleted (at most one edge is contracted by the de nition), the vertex s goes out from the elimination front and s is uni ed to another vertex. Then even though two minors have the same i-th level elimination partition, it is possible that the vertices which are uni ed to the vertex s are di erent. At that case, some active edge in one minor may be inactive in the other minor. For example, in the third level of Figure 2, if e8 and e9 are not added, the minor which is obtained by contracting e1 and deleting e2 ; e3 and the minor which is obtained by contracting e2 and deleting e1 ; e3 are isomorphic with the same edge ordering. However, e5 is active in the former minor but inactive in 6
the other minor. The algorithm of computing the Tutte polynomial can be used to compute the two-terminal network reliability as well (see Figure 2). Di erent from the all-terminal case, when e is an inactive coloop, e is deleted by the recursive formula. However one of two components which contain the articulation points joined by e are inactive in G0 . This component is still inactive in G0 ne and G0 =e. Then the following relation holds1 : R(G0 js0 ; t0 ) = R(G0 nejs0 ; t0 ) = R(G0 =ejs0 ; t0 ): In this two-terminal case, since there are inactive edges the whole structure of the acyclic graph computing the all-terminal case may not be used in this computation. Unlike the all-terminal case, each path from the root the sink does not corresponds to a simple path connecting s and t one-to-one, although the set of contracted edges for each path includes such a path. For example, in Figure 2 fe2; e6 g is one of simple path connecting s and t. However there are two corresponding sets consisting of contracted edges fe1 ; e2 ; e6 g and fe2 ; e6 g. It has been left open whether such an acyclic graph can be constructed.2 If such an acyclic graph can be constructed, we can solve this problem more simply. The size of the acyclic graph depends on the ordering of edges. The good edge ordering to make the size of the acyclic graph smaller has strong connection with the existence of a small separators [10]. For planar graphs such an edge ordering exists by the planar separator theorem [6]. Summarizing the above discussion and using the result of [9], we obtain the following theorem. Theorem 3
The all-terminal network reliability and the two-terminal network reliability both with gen-
p
eral edge deletion probability can be computed in operations proportional to the size of the acyclic graph of O ( n) computation process. Especially, for a planar graph with vertices, they can be computed in
O (2
n
)
time.
Finally, the directed network reliability from the root to other vertices has connection with greedoids and their Tutte polynomial (see [2]).
3 Upper Bound of the Network Reliability 3.1
Computing the Tutte polynomial for a Complete Graph
The direct method of computing the Tutte polynomial of the complete graphs recursively is proposed by Annan [1]. It is described as follows. Consider a graph Um;r obtained from Km by adding a new vertex v and connecting it with each vertex of Km by r multiple edges. By de nition, Kn is isomorphic to Un01;1 . T (Um;r ; x; y ) =
m X i=1
!
i m (yr01 + yr02 + 1 1 1 + 1)i y(2) T (Um0i;i ; x; y ) + (x 0 1)T (Um01;1 ; x; y ); i
where, T (U0;r ) = 1. This formula corresponds to applying the deletion/contraction formula recursively to all of the edges adjoining v and merging all isomorphic minors. For example, apply the deletion/contraction formula to n 0 1 edges which are incident to a vertex of Kn . Like the previous algorithm if we merged isomorphic minors with the same edge ordering, there are 2(n01) 0 n + 1 non-isomorphic minors. However, this formula further merging isomorphic minors with di erent edge ordering. In this case, we nd that there are only n 0 1 non-isomorphic minors. Note that T (Kn ; x; y) = T (Un01;1 ; x; y) is obtained by computing all T (Uj;k ; x; y ) such that j + k n 0 1. By the de nition, for a complete graph Kn the highest degree in x is less than n and in y is less than n2 . Then there are at most O(n3 ) terms. T (G; 1; 1) denotes the number of spanning trees and for Kn this is equal to nn02 . Therefore each coecient can be written in at most O (n log n) bits. 1
With respect to the Tutte polynomial, for a coloop
e, G=e
and
Gne
are 2-isomorphic (their sets of cycles are same).
Then they also have the same Tutte polynomial by Whitney's 2-isomorphism theorem [7].
2
That is, it is not known how to construct the BDD which counts the number of simple paths directly.
7
3.2
Reliability Function for a Complete Graph
By applying the above formula, we can compute eciently the reliability function of a complete graph with the same edge deletion probability. Theorem 4 R(Um;r ; p) = where
R(U0;r ; p) = 1.
m X i=1
!
m (1 0 pr )i pr(m0i) R(Um0i;i ; p) i
Proof: Um;r has m + 1 vertices labelled v1 ; . . . vm for a clique and vm+1 for the remaining vertex. Both vm+1 and each other vertex are incident vertices of r multiple edges. We call this r multiple edges a bundle. We now apply the deletion/contraction formula to these m 2 r edges. We can choose i bundles from m bundles, and delete all edges of the remaining (m 0 i) bundles. The probability with which these r(m 0 i) edges are deleted is pr(m0i) . Note that if Um;r is connected, we must choose at least one bundle, i.e., 1 i m. For each i bundles, we have at least one edge which is not deleted. That probability is (1 0 pr )i . We contract one of such edges for each i bundles. Then the remaining edges become loops and can be deleted by the formula. Now we obtain Um0i;i and carry on the same procedure for this
graph. 2 Again R(Kn ; p) = R(Un01;1 ; p) and this is obtained by computing all R(Uj;k ; p) such that j + k n 0 1. The highest degree is 12 n(n 0 1) and its coecient is (01)n01 (n 0 1)!. R(Kn ; p) can be divided by (1 0 p)n01 and by this factorization each term has a positive sign in the remaining factor. For example, R(K4 ; p) = (1 0 p)3 (6p3 + 6p2 + 3p + 1). The practical computing results are given in Table 1 and Figure 3. In Figure 3, each curve represents an upper bound for the other simple connected graphs with the same number of vertices. For example, Figure 4 shows the case of k 2 k lattice graphs. Table 1: List of R(Kn ; p) n Reliability Polynomial R(Kn ; p) 2 0p + 1 3 2 p3 0 3 p2 + 1 4 06 p6 + 12 p5 0 3 p4 0 4 p3 + 1 5 24 p10 0 60 p9 + 30 p8 + 20 p7 0 10 p6 0 5 p4 + 1 6 0120 p15 + 360 p14 0 270 p13 0 90 p12 + 120 p11 + 20 p9 0 15 p8 0 6 p5 + 1 7 720 p21 0 2520 p20 + 2520 p19 + 210 p18 0 1260 p17 + 210 p16 0 70 p15 + 210 p14 0 35 p12 + 42 p11 0 21 p10 0 7 p6 + 1 8 05040 p28 +20160 p27 0 25200 p26 +3360 p25 +12810 p24 0 5040 p23 0 1960 p21 +420 p20 +560 p19 0 336 p18 + 336 p17 0 35 p16 0 56 p15 + 56 p13 0 28 p12 0 8 p7 + 1 9 40320 p36 0 181440 p35 +272160 p34 0 90720 p33 0 128520 p32 +90720 p31 0 2520 p30 +15120 p29 0 11340 p28 0 7000 p27 + 5544 p26 0 4536 p25 + 1386 p24 + 1008 p23 0 504 p21 + 378 p20 0 84 p18 + 72 p15 0 36 p14 0 9 p8 + 1 10 0362880 p45 +1814400 p44 0 3175200 p43 +1663200 p42 +1247400 p41 0 1489320 p40 +201600 p39 0 75600 p38 +189000 p37 +65100 p36 0105840 p35 +60480 p34 027930 p33 011970 p32 +5040 p31 + 5040 p30 0 5040 p29 + 1260 p28 + 1680 p27 0 126 p25 0 930 p24 + 720 p23 0 120 p21 + 90 p17 0 45 p16 0 10 p9 + 1 Acknowledgment
Part of this work of the second author was supported by the Grant-in-Aid of the Ministry of Education, Science and Culture of Japan. 8
Table 2: List of R(Lk2k ; p) k Reliability Polynomial R(Lk2k ; p)
2 03 p4 + 8 p3 0 6 p2 + 1 3 79 p12 0 560 p11 + 1668 p10 0 2656 p9 + 2331 p8 0 960 p7 + 96 p5 + 21 p4 0 16 p3 0 4 p2 + 1 4 017493 p24 + 232144 p23 0 1409764 p22 + 5168576 p21 0 12693232 p20 + 21854512 p19 0 26726036 p18 + 22824576 p17 0 12739373 p16 + 3710880 p15 + 139672 p14 0 370176 p13 0 35464 p12 + 63968 p11 + 5912 p10 0 7808 p9 0 1791 p8 + 656 p7 + 204 p6 + 64 p5 0 8 p4 0 16 p3 0 4 p2 + 1 5 32126211 p40 0 681809240 p39 + 6852471548 p38 0 43322118652 p37 + 192968405711 p36 0 642590690400 p35 + 1655933457966 p34 0 3370276114636 p33 + 5476061558391 p32 0 7122774813980 p31 + 7375859530466 p30 0 5981426876044 p29 + 3667377815630 p28 0 1573096624396 p27 + 375423772810 p26 + 9584416484 p25 0 24 23 22 21 20 26112103320 p 0 6268146140 p + 8011274210 p 0 1051500660 p 0 575028980 p 0 53196700 p19 + 139031550 p18 0 2265380 p17 0 10705120 p16 0 3593556 p15 + 1357510 p14 + 394172 p13 + 35042 p12 0 49636 p11 0 10290 p10 0 2036 p9 +1021 p8 + 164 p7 +250 p6 + 64 p5 0 11 p4 0 20 p3 0 4 p2 + 1 Reliability R(Kn , p) 1
0.8 n=2
n=3
n=30
0.6
0.4
0.2
00
0.2
0.4 0.6 Edge deletion probability p
0.8
1
Figure 3: R(Kn ; p) (n = 2; . . . ; 30)
References [1] J. D. Annan: The Complexity of Counting Problems. PhD Thesis, University of Oxford, 1994. [2] A. Bjorner and G. M. Ziegler: Introduction to Greedoids. In \Matroid Applications" (N. White, ed.), Encyclopedia of Mathematics and Its Applications, Vol.26, Cambridge University Press, 1992, pp.284{357. [3] R. E. Bryant: Graph-Based Algorithms for Boolean Function Manipulation. IEEE Transactions on Computers, Vol.C-35, 1986, pp677-691. [4] C. J. Colbourn: The Combinatorics of Network Reliability. Oxford University Press, 1987. [5] D. D. Harms, M. Kraetzl, C. J. Colbourn and J. S. Devitt: Network Reliability: Experiments with a Symbolic Algebra Environment. CRC Press, Inc., 1995. 9
Reliability R(kxk Lattice Graph, p) 1
0.8
0.6
0.4
k=9
k=3
k=2
0.2
00
0.2
0.4
0.6
0.8
1
Edge deletion probability p
Figure 4: R(Lk;k ; p) (k = 2; . . . ; 9) [6] R. J. Lipton and R. E. Tarjan: A Separator Theorem for Planar Graphs. SIAM J. on Appl. Math., Vol.36, No.2 (1979), pp. 177{189. [7] J. G. Oxley: Matroid Theory, Oxford Science Publications, 1992. [8] J. S. Provan: The Complexity of Reliability Computations in Planar and Acyclic Graphs. SIAM Journal on Computing, Vol.15, No.3 (1986), pp.694{702. [9] K. Sekine, H. Imai and S. Tani: Computing the Tutte Polynomial of a Graph of Moderate Size. Proceedings of the 6th International Symposium on Algorithms and Computation (ISAAC'95), Lecture Notes in Computer Science, Vol.1004 (1995), pp.224{233. [10] S. Tani: An Extended Framework of Ordered Binary Decision Diagrams for Combinatorial Graph Problems. Master's thesis, University of Tokyo, 1995. [11] W. T. Tutte: A Contribution to the Theory of Chromatic Polynomials. Canadian Journal of Mathematics, Vol.6, 1954, pp. 80{91. [12] D. J. A. Welsh: Complexity: Knots, Colourings and Counting, London Mathematical Society Lecture Note Series, Vol.186, Cambridge University Press, 1993.
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