In this section we propose an algorithm that generates the rules for the composition of each trail cover class. These rules are used to find all trail cover classes of a network in all graph representations. For the rules for trail cover composition, it suffices to consider series connections of trail cover classes. For a network of the parallel type, we take the duals of its child subnetworks and apply the rules for series type. Then we again take the duals of the resulting trail cover classes to obtain the trail cover classes for the network.
On the basis of the analysis in Section 3, we use an array A of size 8 to represent all trail cover classes. Elements of A are labeled by a , a , c ,1 2 1 d , e , e , f , and f . According to Lemma 3.5, subnetworks with trail1 1 2 1 2 covers in TA are represented by at most two connected components in
w x
G M , or else they cannot form a trail cover in TC. Therefore, we use a1
and a to represent all possible representations of trail covers in T ; each2 A element corresponds to a connected component. a is used to indicate1 three possibilities regarding trail covers in T , i.e., none in T , a trail coverA A in class a, and a trail cover in class b. Lemma 3.1 states that class a
w x w x
of 2, 2P for the second connected component. When subnetworks with trail covers in TA are represented by only one connected component in
w x
G M , we define a2s 0.
By Lemma 3.4, a network can have at most two child subnetworks with trail covers in T . We write cC 1s 0 for no trail cover in T , c s 1 for oneC 1
in T , and cC 1s 2 for two.
By Lemma 3.4, we define d1s 0 for no trail cover in T , d s 1 for oneD 1
in T .D
As in the case of trail covers in T , each of e and e corresponds to aA 1 2 connected component. For the first component, we use e1s 0 to indicate no trail cover in T and eE 1s 1 to indicate at least one in T . We define eE 2
similarly.
According to Lemma 3.3, a network contains at most two child subnet-works with trail covers in T . Therefore, we use f and f to represent theF 1 2 trail cover classes in T , excluding p, since it cannot be concatenated withF any other trail cover class. Each of f1 and f2 takes one of 11 possible values: 0 for no trail cover in T , and 1, 2, . . . , 10 for the 10 trail coverF
Ž .
classes in TF see Table 1 .
We consider all of the combinations of trail cover composition in array A. For each combination, take all permutations to obtain a series connec-tion of these trail cover classes with different orders of concatenaconnec-tion, since each permutation corresponds to a fixed order of concatenation of corresponding graph representations. Since there are only eight elements and each takes of a small number of different values, there are a finite number of permutations. Using Table 1, we find all possible trail cover classes for each permutation and store the results. Finally, we sort these results to obtain the composition of each trail cover class.
We summarize the above discussion in the following procedure:
Step 1. Set up values for A as discussed above. For each combination of these eight arrays, get all permutations and find the resulting trail cover classes according to Table 1.
Step 2. Sort these results to obtain the composition for each trail cover class.
It can be easily verified that there are at most Ž8!r Ž2!2!2!..Ž32.Ž31.Ž21.Ž22.Ž112.permutations in the above procedure. There-fore, this procedure can be carried out in constant time. Since some permutations cannot yield a trail cover in classes a to p, the number of rules can be greatly reduced. In Table 2 we list a total of 156 rules generated from this procedure. The trail cover classes listed in each rule are arranged in order from top to bottom in the graph representation, so it is convenient to check Table 2 to determine the appropriate
representa-w x
tion for each trail cover class. In Table 2, by zy, y P we mean series
w x
connection of zy, y ’s, where z is a nonnegative integer.P
We note from Table 2 that series connectin of some specific trail cover classes results in trail covers in more than one class, one in group TAj T j T j T , and the others in T . For example, suppose that NC D E F 1
and N have trail covers in class a. According to Table 1, trail covers for2 N1s N can be in classes b, k, l where b g T and k, l g T . We want to2 A F
network with a trail cover in class a as in the first case. It follows from
4
Table 1 that N1s N s N has trail covers in a, k, l . Comparing the2 3
4 resulting trail covers of N1s N s N in both cases, we find that k, l in the2 3
4
first case is contained in a, k, l of the second case. Therefore, we can
4
write as a s b instead of as a s b, k, l when considering the composi-tion of DET or DEC networks. This fact can also be observed from Example 2.1.
On the other hand, we will show that the trail covers in b resulting from
4
as a can be obtained from the trail covers in k, l and vice versa. Let M Ž X. Ž X. We assume without loss of generality that L is the beginning section of1 L, L is the ending section of L, and L can be written as L2 s L QL .1 2
w x Obviously, L begins at the vertex that serially connects GM N1 with
w x X
GM N2 and terminates at the same vertex. Furthermore, L begins at one
w Xx w Xx
X X
of the distinguished vertices, which connects GM N1 and GM N2 in Ž X. parallel and terminates at the same vertex. It follows that G, G has
U Uw x
another DET trail, namely, L s QL L . Moreover, L N s L L is in2 1 G 2 1
w x
b. We can provide a similar proof for the case where LG N is in l. Now w x
suppose that LG N is in b. By reversing the above operation, we can
4
derive the trail covers in k, l from the trail cover in b that results from as a.
TABLE 2
Rules for trail cover class composition and refinement
TABLE 2}Continued
TABLE 2}Continued
Using similar arguments for as b, bs b, cs c, cs e, ds e, and es e, we have the following lemma.
LEMMA 4.1. Let N and N be two networks. If N1 2 1s N results in trail2
co¨ers in more than one class, one in group TAj T j T j T , and theC D E
others in T , then the trail coF ¨er in TAj T j T j T is preferred oC D E ¨er the trail co¨ers in T .F
The rules corresponding to the latter case can be considered redundant.
Those corresponding to the former case are called dominating rules. This idea can also be extended to the trail cover classes in T . By the discussionF after Lemma 3.3, a trail cover in TF with fewer nondistinguished trails is preferred over ones having more nondistinguished trails. For example, we
4
know from Table 1 that as f s f, k, l . Since f contains only one nondis-tinguished trail, whereas k and l contain two nondisnondis-tinguished trails, f is
4
preferred over k, l . To be specific, trail cover classes in T excluding pF
4 4
can be distinguished as two sets S1s f, g, h, i and S s j, k, l, m, n, o ,2
based on the number of nondistinguished trails in each trail cover class.
For any two networks N and N , if N1 2 1s N results in trail covers all in T ,2 F
one in S , and the others in S , then the resulting trail covers in S are1 2 2 considered to be redundant.
In Table 2, each redundant rule is marked by a) if its graph topology is the same as that of the dominating rule, and by aU otherwise.