The recognition of double Euler trails in series-parallel networks

The Recognition of Double Euler Trails in

Series-Parallel Networks

Tung-Yang Ho

Department of Industrial Engineering and Management, Ta Hwa Institute of Technology, Hsinchu, Taiwan 30085, Republic of China

Ting-Yi Sung

Institute of Information Science, Academia Sinica, Taipei, Taiwan 11529, Republic of China

and

Lih-Hsing Hsu, Chang-Hsiung Tsai, and Jeng-Yan Hwang Department of Computer and Information Science, National Chiao Tung Uni¨ersity,

Hsinchu, Taiwan 30050, Republic of China

Received April 20, 1994; revised March 31, 1998

Ž . X

Given a series-parallel network network, for short N, its dual network N is given by interchanging the series connection and the parallel connection of network N. We usually use a series-parallel graph to represent a network. Let

w x w X_x X

G N and G N be graph representations of N and N , respectively. A sequence Ž w x w Xx. of edges e , e , . . . , e is said to form a common trail on G N , G N1 2 k if it is a

w x w X_{x w x}

trail on both G N and G N . If a common trail covers all of the edges in G N w X_x

and G N , it is called a double Euler trail. However, there are many different graph representations for a network. We say that a network N has a double Euler

Ž . w x w Xx

trail DET if there is a common Euler trail for some G N and some G N . Finding a DET in a network is essential for optimizing the layout area of a

Ž

complementary CMOS functional cell. Maziasz and Hayes IEEE Trans.

Computer-Ž . .

Aided Design 9 1990 , 708]719 gave a linear time algorithm for solving the layout

w x w X_x

problem in fixed G N and G N and an exponential algorithm for finding the optimal cover in a network without fixing graph representations. In this paper, we study properties of subnetworks of a DET network. According to these properties, we propose an algorithm that automatically generates the rules for composition of trail cover classes. On the basis of these rules, a linear time algorithm for recognizing DET networks is presented. Furthermore, we also give a necessary and sufficient condition for the existence of a double Euler circuit in a network. Q 1998 Academic Press

216 0196-6774r98 $25.00

CopyrightQ 1998 by Academic Press All rights of reproduction in any form reserved.

1. INTRODUCTION

Ž .  4

A series-parallel network network, for short N of type tg L, S, P

Žwhich represent leaf, series, and parallel, respectively defined on W is. recursively constructed as follows:

Ž .i N is a network of type L if W< <s 1.

Ž .ii If W< <) 1, N is a network of either type P or type S and consists of kG 2 networks N , . . . , N as child subnetworks parallel or_{1 k} series connected together, where each N is defined on a set W of type ti i i

with ti/ t and the collection of W ’s forms a partition of W.i

A network is often expressed by a tree structure. Networks are useful in practice since they correspond to Boolean formulas with series connection Ždenoted by S implementing logical-AND and parallel connection de-. Ž

.

noted by P implementing logical-OR. For example, the Boolean function

Ž . Ž .

en a k b n c k d can be represented by the network shown in Figure

1. Moreover, networks can be used as a model for electrical circuits. For example, we can use the tree structure shown in Figure 1 to represent the network corresponding to the electrical circuit shown in Figure 2. In the tree representation of N, every node together with all of its descendants forms a subnetwork of N. A node together with some children and their descendants forms a partial subnetwork. The subnetwork of N formed by a child of the root is called a child subnetwork of N. The leaf node is labeled

 4

by x if it is a subnetwork of type L defined on x . Every internal node is labeled by S or P according to the type of the subnetwork it represents. Note that the order of the subtrees in the tree representation is immate-rial, because different orders lead to the same Boolean formula.

On the other hand, every network can be represented by a series-parallel

Ž .

graph s. p. graph for short , which is an edge-labeled graph with two given

distinguished vertices denoted by s and t. We recursively construct an s.p.

FIG. 2. Electrical circuit corresponding to the network shown in Figure 1.

graph to represent a network as described below:

Ž .i Every network N defined on Ws x of type L is represented 4 w x

by an edge-labeled graph G N having only one edge labeled x and the two end points of this edge as distinguished vertices.

Ž .ii Let N be a network having child subnetworks N , . . . , N , and_{1 k} w x

let G N be an s.p. graph representing N with the distinguished vertices_{i i}

s , t for every i. For N of type S, we identify t with s_{i i i iq1} for 1F i F k y 1. w x

The resulting graph G N with the distinguished vertices s , t represents_{1 k} the network N. For N of type P, we identify all s ’s to obtain a new vertex_i w x

s and identify all t ’s to obtain a new vertex t. The resulting graph G N_i

with distinguished vertices s, t represents the network N.

The subgraph G of G induced by the child subnetwork N of N isi i

called a child s. p. subgraph of G. We note that a graph representation for a network is not unique because we can vary the order of the subnetworks and the order of the two distinguished vertices to obtain different graph representations. For example, both the nonisomorphic s.p. graphs shown in Figure 3 represent the network in Figure 1. Although most research has

w x

concentrated on s.p. graphs 2, 5, 8, 13, 14, 16 rather than on networks w6, 7, 9 we believe that studying networks is interesting and practical,x although difficult.

Given a network N on set W, we define its dual network NX on set W by interchanging the types S or P of each node. For example, the network in Figure 4a is the dual network of the network in Figure 1. Figures 4b and 4c show a graph representation and the corresponding circuit, respectively, of the dual network. Note that the Boolean formula corresponding to NX is the dual of the Boolean formula that corresponds to N. It is obvious that ŽNX.Xs N. If two s.p. graphs G, G represent some network N and its dual,X

Ž X.

FIG. 3. Two nonisomorphic s.p. graphs representing the network in Figure 1.

A walk Ws¨0, e ,1 ¨1, e , . . . , e ,2 k ¨k is a finite non-null sequence of

Ž .

vertices and edges where eis ¨iy1,¨i for 1F i F k. Furthermore, we

Ž .

call W a walk from_¨0 to _¨k or a _¨0,_¨k -walk. The vertices _¨0 and_¨k are called terminals of W, and _¨1, . . . ,_¨ky1 are called internal _¨ertices. We

sometimes express a walk ¨₀, e ,₁ ¨₁, e , . . . , e ,_{2 k} ¨_k as e , e , . . . , e for con-_{1 2 k} venience. A section of a walk W is a walk that is a subsequence

Ž . Ž .

FIG. 4. a The dual network of the network in Figure 1. b A graph representation

Ž . Ž . Ž .

¨_i, e_iq1,¨_iq1, e_iq2, . . . , e ,_j ¨_j in W. If all edges of walk W are distinct, W is called a trail.

w x w X_x

Given a network N, let G N and G N be graph representations for the network N and its dual network NX, respectively. A sequence of edges

Ž w x w Xx.

e , . . . , e1 m is said to form a common trail in G N , G N if Ls

X X X X _{w x}

¨₀, e ,₁ ¨₁, e , . . . , e ,₂ m ¨mand Ls¨0, e ,1 ¨1, e , . . . , e ,2 m ¨mare trails in G N w X_x

Ž X. Ž X

and G N , respectively. We call L, L a trail pair. We always use L to

. Ž X.

denote a trail for the dual network. We sometimes write L, L as L for short. If ¨₀s¨_m and ¨X₀s¨_mX , we say that e , . . . , e_{1 m} form a common

w x

circuit. Furthermore, if e , e , . . . , e are all of the edges in both G N and_{1 2 m}

w X_{x Ž .}

G N , we say that e , . . . , e1 m form a common Euler trail circuit in ŽG N ,G Nw x w Xx.. A network N has a double Euler circuit DEC if there is aŽ .

w x w X_x

common Euler circuit for both some G N of N and some G N of the

X _{Ž .}

dual network N . A network N has a double Euler trail DET if there is a

w x w X_x

common Euler trail for both some G N of N and some G N of the dual

X _{Ž w x w} X_{x. Ž} X_.

network N . We say that G N , G N realizes a DET pair L, L for N

X X _{w x}

and N , where L and L are the corresponding Euler trails in G N and w X_x

G N , respectively. We say that a network N is DET if N possesses a

DET. For example, the s.p. graphs in Figures 3a and 4b do not have a common Euler trail, but the ones in Figures 3b and 4b have a common Euler trail abecd. Thus the network in Figure 1 is DET.

The problem of DET networks arises from a more general problem Ž . Ž w x w Xx.

called DCT N . Let DCT G N , G N be the minimum number of w x w X_x

disjoint common trails that cover all of the edges in G N and G N . We

Ž . Ž w x w Xx.

define DCT N as the minimum of DCT G N , G N among all

possi-w x w X_{x w x}

ble graph representations G N and G N . Uehara and vanCleemput 15 proposed a solution method for the layout of cells in the style shown in Figure 5. Assuming the height of each cell is fixed by technological considerations, the width of the cell, and therefore the area of the cell, can be minimized by ordering the transistors in the layout so that chains of transistors can share a common diffusion region. Uehara and vanCleemput defined a graph model for functional cells on two dual multigraphs ŽG N , G Nw x w Xx. and proposed a heuristic method for finding a small

num-Ž w x w Xx.

ber of common trails that cover the given G N , G N . Maziasz and

w x Ž w x w Xx.

Hayes 11 gave a linear time algorithm for solving DCT G N , G N Ž .

and an exponential algorithm for finding the DCT N . Several other papers have also explored the use of graph models to find solutions for

w x

layout 4, 10, 11, 12 . However, using this approach, the choice of graph model becomes a critical issue. We thus choose to work on networks

Ž .

instead. Solving DCT N is useful but difficult. Therefore, we start by Ž . solving the DET problem on networks with the hope of solving DCT N

Ž .

Ž . Ž . Ž . FIG. 5. CMOS functional cell. a Gate-level scheme. b An electric-level scheme. c

Ž . Ž . Ž .

Geometric layout corresponding to the scheme in b . d Another electric-level scheme. e Ž .

Geometric layout corresponding to the scheme in d .

In this paper we study the properties of DET networks and give a linear time algorithm for recognizing DET networks. The paper is organized as follows. In Section 2, we classify the trail cover classes. In Section 3, we study properties of subnetworks of DET networks. On the basis of the analysis in Section 3, we present an algorithm in Section 4 to generate the rules for trail cover class composition. Using these rules, we give a necessary and sufficient condition for DEC networks in Section 5. A linear time algorithm for recognizing DET networks is also presented in Section

5. An example for illustration of the algorithm is given in Section 6. Finally, we give concluding remarks in Section 7.

2. COMMON TRAIL COVER CLASSES AND NETWORK CLASSES

We first informally use an example to introduce our terminology. We Ž w x w Xx.

illustrate in Figure 6 a graph representation G N , G N for a DET Ž

network N shown in Figure 7. At this moment, Figure 7 is used only for .

its tree representation for network N. In this graph pair, there are 10 Ž X.

child s.p. subgraph pairs G , G , each corresponding to a child subnet-i i

work N of N. N , N , . . . , Ni 1 2 10 are ordered from left to right of the root Ž w x w Xx. node in Figure 7. The trail Ls 1, 2, . . . , 46 is a DET trail in G N , G N . Let the distinguished vertices of G be s and t , where s is on the top of_{8 8 8 8}

G . Let the corresponding distinguished vertices of G₈ X₈ be denoted by sX and tX. The trail L induces two disjoint common trails L₁s 31, 32 and

Ž X.  4

L₂s 39, 40, . . . , 46 that cover all of the edges in G , G . We call L , L_{8 8 1 2}

Ž X. Ž X. Ž X.

the trail co¨er induced by L, L in G , G . On the other hand, L, L_{8 8}

can be treated as the concatenation of all of the disjoint trail covers Ž X.

induced by L, L in G for ii s 1, 2, . . . , 10. To define trail cover types, we

first need to define trail types. For example, the trail L in G begins at s1 8 8

and terminates at t , whereas L8 X1 begins at sX and terminates at sX. We say

Ž X. Ž X. Ž . Ž . Ž .

that the trail type of L , L1 1 in G , G8 8 is s, s r t, s , where x, y r

Žz, w.s the beginning vertex of L , the beginning vertex of L r theŽ ₁ X₁. Ž

X_.

ending vertex of L , the ending vertex of L . All of the subscripts and1 1

superscripts are omitted for simplification. For the same reason, the trail

L in G begins at t and terminates at s, whereas L2 8 X2 in GX8 begins at tX and terminates at an internal vertex of GX₈. We say that the trail type ŽL , L₂ X₂. in G , GŽ ₈ X₈. is t, tŽ . Žr s, I , where I denotes an internal vertex.. Since the terminal vertex of LX₂ is an internal vertex, L₂ can only concatenate with another trail from its beginning vertex. For this reason,

Ž . Ž .

we change the trail type of L into t, t₂ r I, I . The trail cover type of L , L_{1 2}4 in G is then determined by the trail types of L and L and is_{8 1 2}

Ž . Ž . Ž . Ž .4 given by s, s r t, s q t, t r I, I .

Now we formally define the term trail co¨er type. Let N be a network

Ž w x w Xx.

with graph representation G N , G N . Let LL be a family of disjoint Ž w x w Xx. Ž X. common trails that cover all of the edges in G N , G N . Let L, L be a trail pair in LL with ¨0,¨n and ¨X0,¨Xn as terminals in L and LX, respectively. If the terminals of both L and LXare distinguished vertices of ŽG N , G Nw x w Xx. Ž, L, L is called a distinguished trail, and a nondistinguishedX. Ž X. Ž w x w Xx.

trail otherwise. To be specific, the type of L, L in G N , G N , Ž X. Ž Ž . Ž X.. Ž Ž . Ž X.. denoted by T L, L , has the form T ¨0 , T ¨0 r T ¨n , T ¨n , where

Ž . Ž . Ž

FIG. 7. The tree representation of the network with an s.p. graph pair shown in Figure 6.

.

by I . The order of the terminals in a trail type is irrelevant; e.g., Žt, t. Žr I, I and I, I r t, t are considered to be equivalent. A nondistin-. Ž . Ž . guished trail can concatenate with at most one other trail. Since we are Ž . Ž . Ž . considering the concatenation of common trails, types t, I , s, I , I, t ,

ŽI, s.4 of TŽ Ž¨0., TŽ¨X0..or TŽ Ž¨n., TŽ¨nX..can be represented by I, I ; e.g.,Ž . Žt, s. Žr t, I is denoted by t, s r I, I , and I, t r t, I is denoted by. Ž . Ž . Ž . Ž . ŽI, I.r I, I .Ž .

Ž X. Ž X . Ž X .4

Let LLs L , L , L , L , . . . , L , L1 1 2 2 k k be a set of disjoint common Ž w x w Xx.

trails that cover all of the edges in G N , G N for some network N.

Ž . Ž .  Ž X. Ž X .

The trail co¨er type t LL is defined as t LL s T L , L q T L , L_{1 1 2 2}

Ž X .4

q ??? qT L , L . Throughout this paper, ‘‘q’’ is commutative. We de-k k

Ž . X

fine dual trail co¨er type for t LL , i.e., a trail cover type for LL in

Ž w x w Xx. Let M be a network with graph representation G M , G M and LL

be a set of disjoint common trails that cover all of the edges in ŽG M , G Mw x w Xx.. For a partial subnetwork R of M, we denote by GMw xR

w x

the subgraph of G M induced by R. Let N be a partial subnetwork of M

w x _Xw X_{x w x}

such that GM N and GM N are connected subgraphs of G M and

w X_{x Ž w x Xw} X_x.

G M , respectively. The edges of GM N , GM N constitute a set of Ž w x Xw Xx. Ž . maximal sections of LL that cover edges of G_M N , G_M N . Lett N, LL_G

Ž w x Xw Xx.

denote the trail cover type on GM N , GM N derived from LL. To

Ž .

classify different types in t N, LL , we note that any common trail isG

Ž . Ž . Ž . Ž .4

constructed from t, t r s, s ; t, s r s, t , which represents the trail cover

type of a leaf, by a sequence of series connection, parallel connection, and

w x

taking the duals. Maziasz and Hayes 10, 11 considered the closure of all series connection, parallel connection, and taking the duals of the trail

Ž . Ž . Ž . Ž .4 Ž .

covers generated by t, t r s, s ; t, s r s, t . They classified t N, LLG

into 42 types according to the directions of each trail. In their classifica-tion, there is no common trail that begins and ends at the same

distin-w x w X_{x Ž . Ž .}

guished vertex in both G N and G N , e.g., t, t r t, t . Furthermore, Ž . Ž . Ž . Ž . Ž . nondistinguished trails are represented by t, t r I, I , t, s r I, I , s, t r ŽI, I , s, s. Ž . Žr I, I , and I, I r I, I .. Ž . Ž .

To simplify the exposition of this 42-type classification, we define an equivalence relation of trail cover types, which renders the concept of trail

co¨er class. We use the example of Figure 6 to informally introduce this

concept. A network may have different graph representations. Given a graph representation of a network, we can obtain other graph representa-tions by a sequence of interchanging the distinguished vertices of their s.p. Ž . Ž . Ž X. Ž X. subgraphs. For this reason, the trail type s, sr t, s of L , L in G , G_{1 1 8 8}

Ž . Ž . Ž . Ž . Ž . Ž . in Figure 6 may change into s, t r t, s , t, s r s, s , or t, t r s, t . All of

Ž X.

these types represent a common trail L , L_{1 1} in which L begins at a₁ distinguished vertex and terminates at the other distinguished vertex, whereas LX1 begins at one distinguished vertex and terminates at the same vertex. It is observed that the subgraph of G induced by 31, 32 has exactly₈ two vertices of odd degree, namely, the distinguished vertices of G . We8

use 2 to indicate the trail class of L . Similarly, we use 0 to indicate the1

trail class of LX₁ because none of its distinguished vertices are of odd Ž X. Ž X. w x

degree. We say that the trail class of L , L1 1 in G , G8 8 is 2, 0 . We use x to indicate a trail in G in which one end point is a distinguished vertex of

G and the other an internal vertex, and y to indicate a trail in G in which L begins and terminates at internal vertices. In our example, the trail class

Ž X . w x w x

of L , L_{2 2} is 2, x . However, since 2, x can concatenate with another

w x w x

common trail only at one end, we change trail class 2, x into x, x . In

Ž . Ž . Ž . Ž .

other words, as t, sr t, I is represented by t, s r I, I , we can write the

Ž X . w x w x  4

trail class of L , L_{2 2} as x, x instead of 2, x . Since L , L_{1 2} forms a trail  4 wŽ . Ž .x cover of G , we say the trail cover class of L , L_{8 1 2} is x, x q 2, 0 .

To formally define the term trail co¨er class, we first define an

equiva-lence relation of trail cover types as follows. For a graph representation ŽG M , G Mw x w Xx. of a network M, a trail cover typet in G M , G MŽ w x w Xx. is equivalent tot if and only if t can be obtained from t by permuting the

X

w x w x

distinguished vertices oft in G M or G M or both, i.e., by reversing the

X

w x w x

direction of t in G M or G M or both. This equivalence relation enables us to define a trail cover class that induces all trail cover types that are equivalent to each other. To consider the equivalence of trail cover types, we need to compare the trail sets in primal and dual graphs, respectively. Now we are motivated to introduce new definitions of trail classes in an s.p. graph without considering the direction of a trail.

DEFINITION 2.1. Given an s.p. graph G with distinguished vertices s Ž

and t, let L be a trail of G. Note that this trail does not necessarily .

contain all of the edges of G. We say that L is in class 2 if L begins at a distinguished vertex and terminates at the other distinguished vertex, in class 0 if L begins at a distinguished vertex and returns to the same distinguished vertex, in class x if one end point of L is a distinguished vertex and the other is an internal vertex, and in class y if L begins and terminates at internal vertices.

Ž .

As shown in Figure 6, the trail 23, 24, 25 of G is in class 2, the trail₄ Ž17, 18 of G in class 0, the trail 40, 41, . . . , 46 of G in class x, and the. ₁ Ž . ₈

Ž .

trail 1, 2, . . . , 46 of G in class y.

 4

Given an s.p. graph G, let Ts L , L be a set of two disjoint trails in_{1 2}

G. We say that T is in class u if both L and L are in class 0 and begin1 2

and end at different distinguished vertices. We use u to symbolize a trail cover consisting of two class 0 trails as top and bottom parts. If L and L1 2

are in class 2 and begin and end at the same distinguished vertex, T is said to be in class f, which is used to symbolize the left and right parts of T. Similarly, we define classes u x and f x for the case where L and L are1 2

in class x. We say that T is in classu x if L and L begin or terminate at1 2

different distinguished vertices, and in class f x if L and L begin or1 2

terminate at the same distinguished vertex. Suppose that if L is in class 01

and L is in class x, we say that T is in class 02 q x. Other trail cover

classes can be similarly defined for trail sets with more than two trails of other combinations.

Again we use Figure 6 as an example. We can consider a trail set in G ,i

which does not necessarily contain all of the edges in G , but contains alli

of the edges in a subgraph of G that corresponds to a partial subnetwork.i

Ž . Ž .4 Ž

The trail set 15, 16 , 19, 20 in G is in class2 f, the trail set 1, 2, . . . ,

. Ž .4

6 , 9, 10, . . . , 14 in G is in class3 u. In the example of Figure 6, we cannot

find a trail set induced by L which is in class u x or f x. Since a graph can have many different trail sets, for illustration purposes we define other

trail sets in G₈ and G . New trail sets in G_{3 8} and G₃ are given by Ž41, 40, 39 , 42, 43, 44, 45, 46 , and. Ž .4 Ž3, 2, 1, 7, 8 , 11, 10, 9, 21, 22 , respec-. Ž .4

Ž . Ž .4

tively. The trail set 41, 40, 39 , 42, 43, 44, 45, 46 in G is in class₈ u x, and Ž3, 2, 1, 7, 8 , 11, 10, 9, 21, 22. Ž .4 in class f x.

Ž X. Ž w x w Xx. Let M be a network that realizes a DET L, L in G M , G M . For

Ž w x Xw Xx.

a partial subnetwork R of M, L induces a trail cover in GM R , GM R ,

w x

which is denoted by LG R . Using the notation of trail classes 0, 2, x, y,u,

f, u x, and f x, we can reclassify the 42 trail cover types proposed by

w x

Maziasz and Hayes 10, 11 into the following 18 trail cover classes: w x Ž . Ž . Ž . Ž .4 class a: 2, 2 s t, t r s, s ; t, s r s, t w x Ž . Ž . Ž . Ž .4 class b: 2, 0 s t, t r s, t ; t, s r s, s w x Ž . Ž . Ž . Ž .4 class c: 0, 2 s t, t r t, s ; s, t r s, s w x Ž . Ž . Ž . Ž .4 class d: u, f s t, t r t, s q s, t r s, s w x Ž . Ž . Ž . Ž .4 class e: f, u s t, t r s, t q t, s r s, s w x Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .4 class f : x, x s t, t r I, I ; t, s r I, I ; s, t r I, I ; s, s r I, I wŽ . Ž .x Ž . Ž . Ž . Ž . Ž . Ž . class g: x, x q 2, 2 s t, t r s, s q t, s r I, I ; t, t r s, s Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .4 q s, t r I, I ; t, s r s, t q t, t r I, I ; t, s r s, t q s, s r I, I wŽ . Ž .x Ž . Ž . Ž . Ž . Ž . Ž . class h: x, x q 0, 2 s t, t r t, s q s, t r I, I ; t, t r t, s Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .4 q s, s r I, I ; s, t r s, s q t, t r I, I ; s, t r s, s q t, s r I, I wŽ . Ž .x Ž . Ž . Ž . Ž . Ž . Ž . class i: x, x q 2, 0 s t, t r s, t q t, s r I, I ; t, t r s, t q Žs, s.r I, I ; t, s r s, s q t, t r I, I ; t, s r s, s q s, t r I, IŽ . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .4 w x Ž . Ž . Ž . Ž . Ž . Ž . Ž class j: f x, u x s t, t r I, I q t, s r I, I ; s, t r I, I q s, . Ž .4 sr I, I w x Ž . Ž . Ž . Ž . Ž . Ž . Ž class k: u x, f x s t, t r I, I q s, t r I, I ; t, s r I, I q s, . Ž .4 sr I, I w x Ž . Ž . Ž . Ž . Ž . Ž . Ž . class l: u x, u x s t, t r I, I q s, s r I, I ; t, s r I, I q s, t r ŽI, I.4 wŽ . Ž .x Ž . Ž . Ž . Ž . Ž . class m: u x, u x q 2, 2 s t, s r s, t q t, t r I, I q s, s r ŽI, I ; t, t. Ž .r s, s q t, s r I, I q s, t r I, IŽ . Ž . Ž . Ž . Ž .4 wŽ . Ž .x Ž . Ž . Ž . Ž . Ž . class n: f x, u x q 0, 2 s t, t r t, s q s, t r I, I q s, s r ŽI, I ; s, t. Ž .r s, s q t, t r I, I q t, s r I, IŽ . Ž . Ž . Ž . Ž .4 wŽ . Ž .x Ž . Ž . Ž . Ž . Ž . class o: u x, f x q 2, 0 s t, t r s, t q t, s r I, I q s, s r ŽI, I ; t, s. Ž .r s, s q t, t r I, I q s, t r I, IŽ . Ž . Ž . Ž . Ž .4 w x Ž . Ž .4 class p: y, y s I, I r I, I w Ž .x Ž . Ž . Ž . Ž . Ž . Ž . Ž . class q: 3 x, x s t, t r I, I q t, s r I, I q s, t r I, I ; t, t r ŽI, I.q t, s r I, I q s, s r I, I ; t, t r I, I q s, t r I, I q s, s rŽ . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . ŽI, I ; t, s. Ž .r I, I q s, t r I, I q s, s r I, IŽ . Ž . Ž . Ž . Ž .4 w Ž .x Ž . Ž . Ž . Ž . Ž . Ž . Ž class r: 4 x, x s t, t r I, I q t, s r I, I q s, t r I, I q s, . Ž .4 sr I, I .

w x

Theorem 1 of 11 states that these 42 trail cover types define a complete Ž . Ž . set of trail cover types. Since the nondistinguished trail of type I, I r I, I cannot be concatenated with any other trail, it stands in class p itself. Since the nondistinguished trail with one distinguished vertex as a terminal in each trail cover can concatenate with at most one other trail, it must be the beginning or ending section of the trail cover. Therefore, no DET network can have trail covers that include more than two of such nondis-tinguished trails. In other words, no DET network can have trail covers in classes q and r. Thus we eliminate these two classes from our analysis and focus on the first 16 trail cover classes, a, . . . , p only. We restrict all of the



trail covers in the following discussion to be in a subset of class a, class

4  4

b, . . . , class p . For simplicity, we write a, b, . . . , p .

It is necessary to distinguish trail cover classes on series-type networks from those on parallel-type networks. The dual class of a trail cover class,

w x Žw x . Ž X. w x Žw x . Ž X .

say w, z _S z, w _P in N, N , is defined as z, w _P w, z _S in N , N . The dual class is obtained by reversing the role of primal and dual networks. Since the type S or P of a network is given, we can sometimes omit the subscripts S or P of a trail cover class without ambiguity. We call wz, w the dual trail cover class or simply, dual class ofx Ž . ww, z . Forx

wŽ . Ž .x wŽ . Ž .x example, the dual class of f x, u x q 0, 2 is u x, f x q 2, 0 , since

w x w x w x w x

the dual of f x, u x is u x, f x and the dual of 0, 2 is 2, 0 . In Figure 7 we also show the trail cover class and the dual trail cover class associated with each node derived from Ls 1, 2, . . . , 46. Note that the trail cover class and the dual trail cover class notations in Figure 7 are for the particular graph shown in Figure 6 and the specific DET L. It is possible to obtain other trail cover classes by rearrangement of subgraphs, i.e., different concatenation of s.p. graphs. In other words, a different concate-nation of s.p. graphs can yield a different concateconcate-nation of trail cover

Ž

classes. Later in Lemma 4.1 and Table 2 of Section 4, we will show that some specific concatenations of s.p. graphs or trail cover classes are

. preferred to obtain a DET.

We define a different concatenation of s.p. graphs as follows. Let G₁ and G2 be two s.p. graphs having distinguished vertices sk and tk for

Ž

ks 1, 2. In this paper, we use s to denote the distinguished vertex on the

.

top of an s.p. graph and t to denote the one on the bottom. When s andk

t , k_k s 1, 2, are specified, we can define a different concatenation of G₁

and G . We use G2 1siG2 and G2siG1 to denote all possible series connections of G and G , where G1 2 j₁siG represents an s.p. graph withj₂

 4

Gj₁ placed on top of Gj₂ for j , j1 2g 1, 2 and j / j . To be specific,1 2

G s1_{G denotes the resulting s.p. graph with distinguished vertices ss s}

1 2 1

and ts t . The remaining three concatenations of G si_{G for is 2, 3, 4}

2 1 2

is 1, 2, 3, 4, and parallel connections Gpi_{G , G p}i_{G as well. We}

1 2 2 1

 i 4  i

use G₁s G to denote the set G s G ¬ i s 1, 2, 3, 4 j G s G ¬ i s 1,_{2 1 2 2 1}

4 Ž .

2, 3, 4 , and G₁p G is defined similarly. Note that G s G s G /_{2 1 2 3}

Ž . Ž . Ž .

G₁s G s G , but G p G p G s G p G p G . Given k s.p. graphs_{2 3 1 2 3 1 2 3} G , G , . . . , G , we use G si_{G s}i_{??? s}i_{G to denote the s.p. graph}

1 2 k 1 2 k

ŽŽŽG₁s G s G ??? s G , and G p G p ??? p G to denote the s.p.i ₂. i ₃. i k. 1 i 2 i i k

ŽŽŽ i . i . i .

graph G₁p G p G ??? p G . Moreover, we define G s G s ??? s G_{2 3 k 1 2 k}

ŽŽŽ . . .

as the union of G_is G s G ??? s G , where i , i , . . . , i form a_{i i i 1 2 k}

1 2 3 k

permutation of 1, 2, . . . , k. We define G₁p G p ??? p G similarly._{2 k}

For each trail cover class of the series type, it is easy to find a corresponding dual trail cover class. Therefore, it suffices to consider a

 4

series connection of trail cover classes only. Let TCs a, b, . . . , p be the set of all trail cover classes. We define an operation s1 _{on k trail cover}

classes as a series connection of k trail cover classes, which maps from Ž .

TC= ??? = TC to P TC , the power set of TC, as follows:

1 1 1 _{ Ž} X_.

x1s x s ??? s x s g g TC ¬ for every s.p. graph pair G , G with2 k i i

is 1, 2, . . . , k having a trail cover T in x , there exists an s.p. graph pairi i

ŽG, G , with GX. g G1s G s ??? s G and G g G p G p ??? p G ,1 2 1 1 k X X1 1 X2 1 1 Xk

having a trail cover T ing such that the induced trail cover of T in G isi

4

T for all i .i

The operation x s1_{x s}1 _{??? s}1_{x can be treated as a concatenation}

1 2 k

of k trail cover classes x , x , . . . , x in specific series connection of s.p._{1 2} k

graphs. For X , X , . . . , X : TC, we define X s1_{X s}1 _{??? s}1_{X as the}

1 2 k 1 2 k

union of x s1_{x s}1 _{??? s}1_{x with x g X for every i.}

1 2 k i i

We define x s x s ??? s x as we define x s1_{x s}1 _{??? s}1_{x , except}

1 2 k 1 2 k

that G and GX in the definition are replaced by Gg G₁s G s ??? s G_{2 k} and GXg GX₁p GX₂p ??? p GXk. It can be treated as a series concatenation of

k trail cover classes without restriction of the order of the series

connec-tion of the corresponding k s.p. graphs. It follows that x s1_{x s}1_{. . .}

1 2

s1_{x : x s x s ??? s x . We define X s X s ??? s X likewise. We are}

k 1 2 k 1 2 k

interested, in particular, in x1s x . Three examples are given below to2

demonstrate the derivation of x1s x , each also illustrated by a figure with2

Ž X. X X

graph pairs G , G . We use s , t and s , t to denote the distinguishedi i i i i i

vertices of G and Gi Xi, respectively.

w x w x

EXAMPLE2.1. We show here the derivation of 2, 2s 2, 2 , i.e., as a. It w x Ž . Ž . Ž . Ž .4

is known that 2, 2 s t, t r s, s ; t, s r s, t . Then

w

x w

x



Ž

.

Ž

. Ž

.

Ž

.

4

j t, t r I, I q s, t r I, I ;



Ž

.

Ž

.

Ž

.

Ž

.

t , s r I, I q s, s r I, I

4

j t, t r I, I q s, s r I, I ;



Ž

.

Ž

.

Ž

.

Ž

.

4

Ž

.

Ž

.

Ž

.

Ž

.

w

x w

x w

x



4

Example 2.1 is illustrated in Figure 8, which contains three graph pairs. ŽG , G1 X1.and G , GŽ 2 X2.possess trails L1s a and L s b, respectively, both2

in trail cover class a. Up to isomorphism, there is only one graph pair ŽG , G₃ X₃.with G₃g G₁s G and G g G p G , as shown in Figure 8c. Let₂ X₃ X₁ X₂

z be the only nondistinguished vertex in G . Let M be a DET network₃

Ž X. Ž w x w Xx. Ž X.

realizing a DET trail L, L in G M , G M that contains G , G_{3 3} as a w x

subgraph pair. Suppose that L begins at any vertex u in G M with u/ z.

Ž .

Then L enters G from s to t_{3 3 3} or t to s , i.e., the trail induced by L in_{3 3}

G is given by ab. On the other hand, L₃ X enters GX₃ either from sX to sX or

X X _Ž X_{. Ž} X_.

from t to t . The trail cover induced by L, L in G , G3 3 is in class b. Suppose that L begins at z. Since z is an internal vertex in G , it follows3

Ž .

that deg z is even and thus L terminates at z. We assume without loss of generality that L begins with a and terminates with b. We can also assume that LX begins at sX. It follows that LX first traverses GX3 by a and leaves at

tX3and reenters GX3 with b. The reentering vertex of LX in GX3 is either sX3 or

X _Ž X_{. Ž} X_.

t . The trail cover of3 L, L induced in G , G3 3 is in class l if the

X _{ 4}

reentering vertex is s and in class k otherwise. Thus a3 s a s b, k, l .

w x w x

EXAMPLE2.2. We show here the derivation of 0, 2s 0, 2 , i.e., cs c. It w x Ž . Ž . Ž . Ž .4

is known that 0, 2 s t, t r t, s ; s, t r s, s . Then

w

x w

x



Ž

.

Ž

.

Ž

.

Ž

.

4



Ž

.

Ž

.

4

w

x w

x

s



4

Hence cs c s d, p .

Example 2.2 is illustrated in Figure 9, which contains four graph pairs. ŽG , G1 X1. and G , GŽ 2 X2. possess trails L1s ab and L s cd, respectively,2

both in trail cover class c. Up to isomorphism, there are exactly two graph

Ž X. Ž X. X X X

pairs G , G3 3 and G , G , with G4 4 ig G1s G and G g G p G for2 i 1 2

is 3, 4, as shown in Figures 9c and 9d. Let z be the only nondistinguished

Ž .

vertex in G3 s G . Let M be a DET network realizing a DET trail4

ŽL, L in G M , G MX. Ž w x w Xx. that contains G , GŽ 3 X3. as a subgraph pair. Sup-w x

pose that L begins at any vertex u in G M with u/ z. It follows that L enters G from either s or t . We can assume without loss of generality3 3 3

that L enters G at s with L . Since L is in class c, L leaves G at s_{3 3 1 1 3 3} after traversing L and reenters G from t with L . On the other hand,_{1 3 3 2}

Ž . Ž .4 X

we can verify that the trail set formed by ab , cd in G is in class₃ f.

Ž X. Ž X.

 4 FIG. 8. Illustration of as a s b, k, l .

that L begins at z. We can assume without loss of generality that L begins with L . Since the trail cover class of L is c, L terminates at z and is1 1 1

immediately followed by L . Since L is in c, L terminates at z. Thus_{2 2 2}

Ls L L is in class p._{1 2}

Ž X. Ž w x w Xx. Let M be a DET network realizing a DET trail L, L in G M , G M

Ž X.

that contains G , G4 4 as a subgraph pair. Suppose that L begins at any w x

vertex u in G M with u/ z. As in the above case, the trail cover class in

 4 FIG. 9. Illustration of cs c s d, p .

z. Similarly, the trail cover in G derived from L is given by L₄ s L L and_{1 2}

 4 is in class p. Thus cs c s d, p .

w x w x

EXAMPLE2.3. We show here the derivation of 0, 2s x, x , i.e., cs f. It w x Ž . Ž . Ž . Ž .4 w x Ž . Ž . is known that 0, 2 s t, t r t, s ; s, t r s, s and x, x s t, t r I, I ;

Žt, s.r I, I ; s, t r I, I ; s, s r I, I . ThenŽ . Ž . Ž . Ž . Ž .4

w

x w

x



Ž

.

Ž

.

Ž

.

Ž

.

Ž

.

Ž

.

Ž

.

Ž

.

Ž

.

Ž

.

Ž

.

Ž

.

4

Ž

.

Ž

.

Ž

.

Ž

.



Ž

.

Ž

.

4

w

x



Ž

.

Ž

.

4

We illustrate Example 2.3 in Figure 10, which contains four graph pairs. ŽG , G₁ X₁. possesses a trail L₁s ab in class c, and G , G possesses aŽ ₂ X₂. trails L₂s cde in class f. Up to isomorphism, there are four graph pairs ŽG, G with GX. g G₁s G and G g G p G . We here only show two graph₂ X X₁ X₂

Ž X. X X X

pairs G , G_{i i} with G_ig G₁s G and G g G p G for i s 3, 4. Let z be_{2 i 1 2 i} the common vertex of edges e, c, and d in G for ii s 3, 4. Let M be a DET

Ž X. Ž w x w Xx.

network realizing a DET trail L, L in G M , G M that contains ŽG , G3 X3. as a subgraph pair. Since deg zŽ 3.s 3, we can assume that L begins at z . Thus L begins with L , which is followed by L . The trail3 2 1

cover in G derived from L is given by L3 s L L and is in class p. Let M1 2

Ž X. Ž w x w Xx. be a DET network realizing a DET trail L, L in G M , G M that

Ž X. Ž .

contains G , G4 4 as a subgraph pair. Since deg z4 s 3, we can assume

that L begins at z by L and leaves G at t . Eventually L must return to4 2 4 4

Ž .

G at s4 4 since e has been traversed, t is ruled out , from which it follows4

L . After traversing L , it will leave G at s . Thus the trail cover class in_{1 1 4 4} G derived from L is class h. We can use similar arguments to discuss the₄

Ž X. X X X

other two graph pairs G, G with Gg G₁s G and G g G p G . Conse-_{2 1 2}  4

quently, we obtain cs f s h, p .

Using arguments similar to that employed in the above examples, we construct Table 1 to illustrate the results of s on TC = TC. If a series connectin of two trail cover classes does not yield a trail cover in TC, we write asb s B, which is indicated by a blank entry in Table 1. Obviously,

Ž .

t N, LL is in trail cover class a if L is a DET and N is of type L. NoteG

that Table 1 is closed under series connection, parallel connection, and taking the duals generated by class a. There are exactly 16 different trail cover classes needed to form DET.

A network N is possible if there exists a trail cover class in TC for some ŽG N , G Nw x w Xx.. We use Q to denote the set of all possible networks. For

Ž . Ž w x w Xx.

any network N in Q, Possible N s G N , G N ¬ there exists a trail

Ž w x w Xx. 4 Ž . 

 4 FIG. 10. Illustration of cs f s h, p .

Ž w x w Xx. Ž .4

cover class for some G N , G N in Possible N . For example,

Ž .  4

Class N s a for any network N of type L. It follows from Example 2.1

Ž .  4

that Class N s b, k, l for any network N that is a series connection of

Ž .  4

two networks of type L. Hence we have Class N s c, j, l for any network N that is a parallel connection of two networks of type L.

TABLE 1

Ž . The functions that maps TC = TC to P TC

EXAMPLE 2.4. Let N be the network in Figure 1, N₁ be the child  4

subnetwork of N defined on e , N be the child subnetwork defined on₂

a, b , and N4 ₃ be the child subnetwork of N defined on c, d . Then 4

Ž .  4 Ž . Ž .  4

Class N₁ s a , Class N s Class N s c, j, l . It follows from the dis-_{2 3}

Ž . 1 Ž . 1 cussion in Examples 2.1, 2.2, and 2.3 that Class N1 s Class N s2

Ž .  4 Ž . Ž . Ž .  4

Class N3 s h and Class N1 s Class N s Class N s h, p . Therefore,2 3

the network in Figure 1 has a DET trail in class p.

Let N be a possible network of type S with child subnetworks N , N ,1 2

Ž . . . . , N . It can be observed from the above example that Class Nk s

Ž . Ž . Ž .

Class N₁ s Class N s ??? s Class N . Thus we need methods to compute_{2 k}

Ž .

Class N from its child subnetworks.

3. PROPERTIES OF DET NETWORKS

Using Table 1, we can derive the generation of classes a, b, c, d, and e. LEMMA 3.1. The classes a, b, c, d and e of type S are constructed as

follows:

Ž .i w2, 2x_S¤ m 2, 2 w x 4  w_Ps n 2, 0 , where m is odd and m q n G 2.x 4_P Ž .ii w2, 0x_S¤ m 2, 2 w x 4  w_Ps n 2, 0 , where m is ex 4_P ¨en and mq n G 2. Žiii. w0, 2xS¤ 0, 2w xPs m f, u w x 4P , where mG 1. Živ. wu, f ¤ 0, 2 s 0, 2 s m f, uxS w xP w xP  w x 4P . wu, f ¤ m f, u s u, f , where m G 1.x_S  w x 4 w_P x_P Ž .v wf, u ¤ m f, ux_S  w x 4_P , where mG 2. w x w x

In these rules, by ‘‘zy, y ’’ we mean a series connection of z y, y ’s_{P P}

Ž .

where z is a nonnegati_¨e integer. The subscript S P is used to indicate a trail

Ž .

co_¨er for some networks of type S P .

Proof. Every network is recursively constructed from edges and subnet-works that correspond to leaves and subtrees, respectively. Each edge has

w x

a trail cover 2, 2 only of type P or S, depending on the type of its parent. From Table 1, we know that as as b, b s as a or bs b, c s cs e,

ds ds e or cs c, and e s es e. Repeatedly applying these relations, we

obtain the rules as stated in the lemma.

Examining Lemma 3.1 and its proof, we find that the classes a, b, c, d, and e are generated by concatenation of these five classes only, without involving the remaining 11 trail cover classes. Hence we call classes a, b, c,

d, and e primary trail co¨er classes, and any trail cover in classes a, b, c, d,

and e a primary trail co¨er.

LEMMA3.2. The classes a, b, c, d, and e are mutually exclusi¨e. In other

 4

words, let network N ha_¨e a trail co_¨er in a g a, b, c, d, e . Then N cannot

 4

Proof. According to Lemma 3.1, the generation of a, b, c, d, and e are mutually exclusive.

To simplify our exposition, we divide the 15 trail cover classes a, b, . . . , o, excluding p, of type S and of type P into five groups:

w

x w

x w

x w

x



4

w

x w

x



4

w

x w

x



4

w

x w

x



4

w

x



Ž

.

Ž

.

Ž

.

Ž

.

Ž

.

Ž

.

w

x

Ž

.

Ž

.

Ž

.

Ž

.

Ž

.

Ž

.

w

x w

x w

x

Ž

.

Ž

.

w

x w

x w

x

Ž

.

Ž

.

Ž

.

Ž

.

Ž

.

Ž

.

4

Ž

.

Ž

.

Ž

.

Ž

.

It follows from Lemma 3.2 that trail covers in T , T , T , and TA C D E are primary trail cover classes and are mutually exclusive. As shown in Figure

Ž  4.

6, the trail cover types oft N , L for i s 1, 10 are in T . The trail coverG i C

Ž  4.

types of t N , L for i s 2, 3, 9 are in T . The trail cover types ofG i E

Ž  4. Ž  4.

t N , L for i s 4, 5, 6, 7 are in T . The trail cover type of t N , L isG i A G 8

Ž w x w Xx.

in T . In general, let G M , G MF be any graph representation that possesses a DET trail L for some network M. Let N be any type S subnetwork of M. In the following, we will show that those child s.p.

w x Ž .

subgraphs of G_M N with their derived trail cover classes from L in T_A

induce at most two connected components. There are at most two child w x

s.p. subgraphs of G_M N with their derived trail cover classes in T . There_C

w x

is at most one child s.p. subgraph of G_M N with its derived trail cover

w x

class in T . Moreover, if there is a child s.p. subgraph of G_{D M} N with its

derived trail cover class in T , then there is no child s.p. subgraph of_C w x

G_M N with its derived trail cover class in T . Those child s.p. subgraphs of_E

w x

GM N with their derived trail cover classes in TE induce at most two connected components. There are at most two child s.p. subgraphs of

w x

GM N with their derived trail cover classes in T .F

Ž X. Ž w x Let M be a DET network that possesses a DET L, L in G M , w X_x.

G M . Let N be a subnetwork of M. Since M is DET, it suffices to consider properties of trail cover classes of M and of N only, without

considering MX and NX. In the following sections, we use a DET L in

w x Ž X. Ž w x w Xx.

G M to represent a DET L, L in G M , G M without ambiguity. We would like to study the properties of subnetworks of a DET network.

Ž X.

LEMMA3.3. Let G, G realize a DET L for network M, and let N be a

subnetwork of M.

Ž .i Let N be a child subnetwork of N such that L1 Gw xN1 is a trail co¨er in T . Then L must begin or end at an edge in N .F 1

Ž .ii N has at most two child subnetworks N such that each L_{i G}w xN is a_i trail co¨er in T ._F

Proof. Each nondistinguished trail contains at least one internal vertex as a terminal, and therefore one of its termini cannot be concatenated with a trail cover of another subnetwork. Trail covers in T contain at least oneF

nondistinguished trail. Consequently, these nondistinguished trails must be Ž . the beginning section or the ending section of L. Therefore, statement i follows. If N contains more than two child subnetworks N such that eachi

w x

L_G N is a trail cover in T , N contains more than two nondistinguished_{i F}

trails. Since trail covers in TC contain at most two nondistinguished trails,

N cannot have any trail cover in TC. Furthermore, M cannot be DET,

which contradicts the assumption. Hence the lemma follows. w x

LEMMA3.4. Let M be a DET network, G M realize a DET L, and N be

a type S subnetwork of M. Then the following statements hold:

Ž .i N can contain at most two child subnetworks C such that each_i

w x

L_M C is a trail co_i ¨er in T ._C

Ž .ii N can contain at most one child subnetwork D such that L_Mw xD is a trail co_¨er in T ._D

Žiii. N cannot contain both child subnetworks C and D, where C andi i

D are as defined abo¨e.

Ž .

Proof. To prove statement i , we assume without loss of generality that there are exactly three child subnetworks C , C , and C of N such that1 2 3

w x w x

each L C is a trail cover in T . Then G_{G i C M} N can be written as

w x

w x

w x

w x

w x

G_M N s G N_{M 1} s G C s G N s G C_{M 1 M 2 M 2}

1

w x

w x

w x

s G N s G C s G N ,M 3 M 3 M 4

where N can be vacuous for is 1, 2, 3, 4. Since cs1_cs1_{c: cs cs c s B,} i

w x w x w x

it follows that GM C , G1 M C , and G2 M C3 cannot all be placed consecu-w x

tively in G M , i.e., N and N cannot both be vacuous. Next, we consider2 3

w x w x

the case where only two G_M C_i are placed consecutively in G_M N .

w x w x

w x Ž possible trail covers of G_M N are contained in the set t s TC y₁  4.c s c s c s TC y c s c s TC y c . It follows from Table 11 4 1 4 1Ž  4. 1 4 1Ž  4. that

 4

 4

 4

 4

 4

 4

Ž

Ž

Ž

Ž

Ž

Ž

.

.

.

Ž

.

.

.

Ž

.

.



4

 4

 4

 4

 4

Ž

Ž

Ž

Ž

.

Ž

.

.

.

Ž

.

.



4

 4

 4

 4

Ž

Ž

Ž

Ž

.

.

.

Ž

.

.



4

 4

 4

Ž

Ž

.

.

Thus there is no trail cover in G_M N for G M to form a DET, i.e., N

does not have a trail cover in TC. Finally, we consider the case where N2

w x and N are not vacuous. In this case, the possible trail covers of G_{3 M} N

Ž  4. 1 4 1Ž  4. 1 4 1Ž are contained in the sett s TC y c s c s TC y c s c s TC y₂  4.c s c s TC y c . As in the first case, we can show that the set t is1 4 1Ž  4. ₂

Ž .

empty. Therefore statement i follows. Ž . Ž .

Statements ii and iii can be proved by arguments similar to those Ž .

given for statement i .

w x

LEMMA 3.5. Let M be a DET network, G M realize a DET L, N be a

type S subnetwork of M, and A , . . . , A be the only child subnetworks of N₁ n

w x w x

such that each LG Ai is in T . Then in GA M N , A , . . . , A are represented1 n

by at most two connected subgraphs.

w x

Proof. Suppose that in GM N , the subnetworks A , A , . . . , A1 2 n are

w x w x

represented by three connected components, say, GM K , G1 M K , and2

w x w x

GM K . It follows that G3 M N can be written as

w x

w x

w x

w x

w x

GM N s G NM 1 s G K s G N s G KM 1 M 2 M 2

1

w x

w x

w x

s G N s G K s G N ,M 3 M 3 M 4

where N and N can be vacuous. As in the deduction in Example 2.1, we_{1 4}

1 w x

have T_As T s T . It follows that each L K is in T for every i. By_{A A G i A} Table 1, T can be obtained only by T_{A A}s T . It follows that the induced_A

w x  4

trail cover in each G_M N is contained in Y_i s TC y a, b . Moreover, N₂

and N cannot be vacuous, since otherwise the A ’s can be represented by₃ i

less than three connected components. Therefore the induced trail cover

w x 1_{ 4} 1 1_{ 4} 1 1_{ 4} 1

in LG N is contained in Ys a, b s Ys a, b s Ys a, b s Y. It is

obvi-1_{ 4} 1 _{ 4} 1 1_{ 4} 1_{ 4}

ous that Ys a, b s Y : Ys a, b s Y and Ys Ys a, b : Ys a, b :

 4  4

Ys a, b . The trail covers in Ys a, b are concatenated with trail covers in

 4

other subnetworks by means of a trail cover in a, b , while the trail covers

1_{ 4} 1

in Ys a, b s Y are concatenated by means of both trail covers in Y. Since  4

ŽYs a, b s Y l Ys a, b s B. Therefore, Ys a, b s Y : Ys a,1 4 1 . Ž  4. 1 4 1 Ž 

4 . Ž  4.  4  4 1 1 4

bs Y y Ys a, b s j, m, n, o, p s W. Moreover, a, b s Ws a, b

Ž 4 .  4 w x

: a, bs W s a, b s B, i.e., L N s B. Therefore, there exists noG

w x w x

trail cover in GM N for G M to form a DET, i.e., N has no trail cover in TC. This leads to a contradiction, and hence the lemma follows.

w x

LEMMA 3.6. Let M be a DET network, G M realize a DET L, N be a

type S subnetwork of M, and E , . . . , E be the only child subnetworks of N1 n

w x w x

such that each LG E is in T . Then in Gi E M N , E , . . . , E are represented by1 n

at most two connected subgraphs.

w x

Proof. Suppose that in GM N , the subnetworks E , E , . . . , E1 2 n are

w x w x

represented by three connected components, say, G_M K , G_{1 M} K , and₂

w x w x

G_M K . It follows that G_{3 M} N can be written as

w x

w x

w x

w x

w x

G_M N s G N_{M 1} s G K s G N s G K_{M 1 M 2 M 2}

1

w x

w x

w x

s G N s G K s G N ,M 3 M 3 M 4

where N and N can be vacuous. Similar to the deduction in Example 2.1,1 4

1 w x

we have TEs T s T . It follows that each L K is in T for every i. ByE E G i E

Table 1, T can be obtained only by T_{E E}s T . It follows that the induced_E

w x  4

trail covers in each G_M N are contained in Y_i s TC y e . Moreover, N₂

and N cannot be vacuous, since otherwise the E ’s can be represented by_{3 i} less than three connected components. Therefore the induced trail cover

w x 1_{ 4} 1 1_{ 4} 1 1_{ 4} 1

in L_G N is contained in Ys e s Ys e s Ys e s Y. It is obvious that

1_{ 4} 1 _{ 4} 1 1_{ 4} 1_{ 4  4}

Ys e s Y : Ys e s Y and Ys Ys e : Ys e : Ys e . The trail covers

 4

in Ys e are concatenated with trail covers in other subnetworks by means

 4 1 4 1

of a trail cover in e , while the trail covers in Ys e s Y are concatenated  4

by means of both trail covers in Y. Since Yl e s B and e is a primary Ž 1 4 1 . Ž  4.

trail cover class, it follows that Ys e s Y l Ys e s B. Therefore,

1_{ 4} 1 _{Ž  4 . Ž  4.  4}

Ys e s Y : Ys e s Y y Ys e s f, k, l, p s W. Similarly, we have

 4es Ws e : e s W s e s B. Therefore, there exists no trail cover in1 1 4 Ž 4 .  4

w x w x

G_M N for G M to form a DET, i.e., N has no trail cover in TC. This

leads to a contradiction, and hence the lemma follows.

w x LEMMA 3.7. Let M be a DET network that realizes a DET L in G M . Then the following statements hold:

Ž .i All child subnetworks A of M such that each L_{i G}w xA_i is in T are_A w x

represented by only one connected component in G M .

Ž .ii M does not contain a child subnetwork N such that LGw xN g T .D

Ž .

Proof. To prove statement i , suppose that all of the A ’s are repre-i

w x

sented by two connected components in G M . We use arguments similar w x to those in the proof of Lemma 3.5 and obtain the result that G M can be

written as follows:

w x

w x

w x

w x

w x

w x

G M s G NM 1 s G K s G N s G K s G N ,M 1 M 2 M 2 M 3

w x

where N and N can be vacuous, N cannot be vacuous, L1 3 2 G Ki is in TA

w x

for is 1, 2, and trail covers in each G N are contained in Y s TC yM i

a, b . It follows that L is contained in Y4 s a, b s Ys a, b s Y. It1 4 1 1 4 1  4 1 4 1 4 follows from repeated applications of Table 1 that a, bs c, d, e s a, b

Ž 4  4  4. Ž 4  4.  4  4 1Ž

: a, bs c, d, e s a, b y a, b s c, d, e s k, l , a, bs Y y

c, d, e4.s a, b : a, b s Y y c, d, e s a, b y a, b s Y y c, d, e1 4 Ž 4 Ž  4.  4. Ž 4 Ž  4..  4 1 1 4 1  4

s B. Therefore, we have a, bs Ys a, b s Y : k, l s Y s B. This con-Ž .

tradicts the existence of L, and hence statement i follows. Ž .

To prove statement ii , we suppose that M has a child subnetwork N

w x  4

such that L_G N g T s d . It follows from Table 1 that we have_D

 4  4  4

ds TC s d, h, n , ds TCs TC : d, h, n s TC s d, h, k, l, n , and

 4  4

ds TCs TCs TC : d, h, k, l, n s TC s d, h, k, l, n . Therefore, the

possi-ble trail covers in M are contained in d, h, k, l, n and, as a result, M is not DET. This leads to a contradiction. Hence the lemma follows.

Ž .

Remark 3.1. Statement i of Lemmas 3.4 and 3.6 also holds for

Ns M. In other words, let M be a DET network that realizes a DET L in

w x

G M . Then M contains at most two child subnetworks C such that eachi

w x

L CG i is in T . All of the child subnetworks E of M such that eachC i

w x

LG E is in T are represented by at most two connected components ini E

w x

G M .

4. RULES FOR TRAIL COVER CLASS COMPOSITION AND REFINEMENT

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 trail_{1 1 2 1 2}

covers in TA are represented by at most two connected components in w x