Optimal 1-edge fault-tolerant designs for ladders

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Optimal 1-edge fault-tolerant designs for ladders

✩

Yen-Chu Chuang

a

_{, Lih-Hsing Hsu}

a,∗

_{, Chung-Haw Chang}

b

a_{Department of Computer and Information Science, National Chiao Tung University, Hsinchu, Taiwan 30050, ROC} b_{Ming-Hsin Institute of Technology, Hsinchu, Taiwan, ROC}

Received 14 September 2001; received in revised form 14 January 2002 Communicated by K. Iwama

Abstract

A graph G∗is 1-edge fault-tolerant with respect to a graph G, denoted by 1-EFT(G), if every graph obtained by removing any edge from G∗contains G. A 1-EFT(G) graph is optimal if it contains the minimum number of edges among all 1-EFT(G) graphs. The kth ladder graph, Lk, is defined to be the cartesian product of the Pkand P2where Pnis the n-vertex path graph.

In this paper, we present several 1-edge fault-tolerant graphs with respect to ladders. Some of these graphs are proven to be optimal.

Keywords: Cartesian product; Edge fault tolerance; Meshes; Ladders; Fault tolerance

1. Introduction and notations

In this paper, any graph means an undirected graph in which multiple edges are allowed. Let G= (V, E) be a graph where V (= V (G)) is the vertex x of V , deg_G(x) denotes its degree in G. Let E be a subset of E. We use G− Eto denote the spanning subgraph of G with its edge set to be E− E. For convenience,

G− e denotes G − {e}. Let G1= (V1, E1) and G2=

(V2, E2) be two graphs. The cartesian product of G1

and G2, denoted by G1× G2, is the smallest graph

with the vertex set V1× V2 such that the subgraph

induced by V1× {v2} is isomorphic to G1 for every

✩

This work was supported in part by the National Science Council of the Republic of China under Contract NSC 90-2213-E-009-148.

* _{Corresponding author.}

E-mail address: [email protected] (L.-H. Hsu).

v2∈ V2, and the subgraph induced by {v1} × V2 is

isomorphic to G2for every v1∈ V1.

Motivated by the study of computers and commu-nication networks that tolerate failure of their compo-nents, Harary and Hayes [6] have formulated the con-cept of edge fault tolerance in graphs. Given a

tar-get graph G= (V, E), let G∗= (V, E∗) be a

span-ning supergraph of G. G∗ is said to be k-EFT(G), if

G∗−F contains a subgraph isomorphic to G, which is

called a reconfiguration for k-edge fault F (or simply

reconfiguration), for any F ⊂ E∗ and|F | = k. A re-configuration can be viewed as a relabeling of ver-tices of G∗ such that G∗− F contains G. We some-times write “G∗ is a k-EFT(G) graph” as “G∗ is a

k-EFT(G)”, for short. The graph G∗is said to be op-timal if G∗ contains the smallest number of edges among all k-EFT(G) graphs. We use eftk(G) to denote

the difference between the number of edges in an

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mal k-EFT(G) graph and that in G. Families of k-EFT graphs with respect to some graphs have been studied in literature [1,2,4,6–8,10–17].

The n-dimensional mesh M(m1, m2, . . . , mn) is

defined to be the cartesian product Pm1 × Pm2 ×

· · · × Pmn of n paths. Mesh is a widely used graph

model for computer networks [9]. Farrag [4] has pre-sented families of 1-EFT graphs with respect to the

n-dimensional meshes. In [6], the graph C(m1, m2,

. . . , mn)= Cm1× Cm2× · · · × Cmn was proposed as

a 1-EFT graphs with respect to the n-dimensional mesh M(m1, m2, . . . , mn). We call such graphs mul-tidimensional torus graphs because their construction

is similar to that of the torus for n= 2 [5]. Harary and Hayes [6] conjectured that these multidimensional torus graphs are optimal if mi 3 for every i. There is

another 1-EFT graph for the n-dimensional meshes. We assume the vertices of M(m1, m2, . . . , mn) are

labeled canonically. Thus, xi1,i2,...,in is a vertex of

M(m1, m2, . . . , mn) if and only if 1 ij mj for

1 j n. Moreover, xi1,i2,...,in is adjacent to another

vertex xj1,j2,...,jn if there exist a index k such that |ik− jk| = 1 and it = jt for all indices t= k. Then,

Vp= {xi1,i2,...,in| ik= 1 or mk for some 1 k n} is

the set of peripheral vertices. Let xi1,i2,...,inbe a vertex

in Vp. The antipodal vertex of xi1,i2,...,inis xj1,j2,...,jn,

with jk= mk−ik+1, which is another vertex in Vp. It

is easy to check that every vertex in Vphas exactly one

antipodal. In M(m1, m2, . . . , mn), we add the edges

joining each vertex in Vp to its antipodal

counter-part to form a new graph P (m1, m2, . . . , mn). We

call these P (m1, m2, . . . , mn) projective-plane graphs

because their construction is similar to that of the projective plane when n = 2 [5]. It is proven in [3] that P (m1, m2, . . . , mn) is also 1-EFT(M(m1, m2,

. . . , mn)) and it contains fewer edges than that of

C(m1, m2, . . . , mn). Thus, the conjecture posed in [6]

is disproved with these projective-plane graphs. The projective-plane graphs are optimal for some cases but not for all. Note that every n-dimensional hypercube can be viewed as the mesh M(2, 2, . . . , 2). Our P (2, 2, . . . , 2) is actually the same 1-EFT graph as that proposed in [1,6,7,13,16]. Thus, P (2, 2, . . . , 2) is an optimal 1-EFT graph. It is proved in [3] that the graph in Fig. 1(a) is a 1-EFT(M(3, 2)) and the graph in Fig. 1(b) is a 1-EFT(M(4, 2)). With these two examples, we know that the projective-plane graphs may not be optimal for some cases. Furthermore,

Fig. 1. (a) A 1-EFT(M(3, 2)), L∗₃; (b) a 1-EFT(M(4, 2)), L∗₄.

the problem of finding the optimal 1-EFT for all n-dimensional meshes remains unsolved.

In this paper, we only aim at the 1-EFT graphs for M(k, 2) with k 2. For simplicity, the kth

lad-der graph Lk is defined to be M(k, 2). Since the

projective-plane graph P (k, 2) is a 1-EFT(Lk) graph,

we know that eft1(Lk) k. In this paper, we will prove

by constructing a 1-EFT(Lk) graph L∗kthat eft1(Lk)

k− 1 if k is odd and k 7, and eft1(Lk) k − 2 if k

is even and k 4. Moreover, we prove that eft1(L2)=

eft1(L3)= eft1(L4)= 2, and eft1(L5)= 3.

2. Some 1-EFT designs for ladders

The vertices of Lk can be labeled by xi,j with

1 i k and 1 j 2 canonically. The vertices x1,1,

xk,1, x1,2, and xk,2are called the corner vertices of Lk.

We have the following theorem:

Theorem 1. Let L∗_k be a 1-EFT(Lk) graph. Then we have

(i) deg_L∗

k(x) 3 for any vertex x of L ∗ k, and

(ii) eft1(Lk) 2.

Proof. Suppose some vertex x with deg_L∗

k(x)= 2. Let

e be any edge incident with x. Obviously, deg_L∗_k−e(x)

= 1. Since degLk(x) 2 for any vertex x of Lk, Lkis

not a subgraph of L∗_k− e. We obtain a contradiction that L∗_k is a 1-EFT(Lk) graph. Hence, degL∗_k(x) 3.

Since there are exactly four corner vertices in every

Lk, we have eft1(Lk) 2. ✷

Corollary 1. eft1(Lk) > 2 if k > 4.

Proof. It is observed that there are exactly three

differ-ent ways of joining the four corner vertices in Lk with

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Fig. 2. A 1-EFT(M(5, 2)), L∗₅.

xk,1), (x1,2, xk,2)}, and {(x1,1, xk,2), (x1,2, xk,1)}. It is

observed that none of the graphs obtained by joining two edges to the corner vertices of Lk with k > 4 is

1-EFT(Lk). Hence eft1(Lk) > 2 if k > 4. ✷ 2.1. Optimal 1-EFT(L2), 1-EFT(L3), 1-EFT(L4)

graphs

Let L∗₂ (L∗₃, and L∗₄, respectively) be the graph

P (2, 2) (the graph in Figs. 1(a) and 1(b), respectively).

From the above discussion, L∗_k is 1-EFT(Lk) for

k = 2, 3, and 4. Since there are exactly 2 edges

added to Lk with k = 2, 3, and 4, by Theorem 1

these graphs are optimal. It can be verified that the optimal 1-EFT(Lk) is unique for k = 2, 3, and 4

by checking all the three cases joining two edges to the corner vertices of Lk. We obtain the following

theorem:

Theorem 2. eft1(Lk)= 2 for k = 2, 3, and 4. 2.2. An optimal 1-EFT(L5) graph

Consider the spanning supergraph L∗₅ of L5 given

by E(L∗₅)= E(L5)∪ {(x1,1, x5,2), (x1,2, x4,2), (x2,1,

x5,1)} as shown in Fig. 2.

Edges of L5 can be divided into the following 7

classes: namely, A=(x1,1, x1,2) , B=(xi,1, xi,2)| 2 i 4 , C=(x5,1, x5,2) , D=(x1,1, x2,1), (x1,2, x2,2) , E=(x2,1, x3,1), (x2,2, x3,2) , F =(x3,1, x4,1), (x3,2, x4,2) , G=(x4,1, x5,1), (x4,2, x5,2) .

We can reconfigure L5 in L∗₅ for any faulty edge e

in A (B, C, D, E, F , and G, respectively) as shown in Figs. 3(a), (3(b), 3(c), 3(d), 3(e), 3(f), and 3(g), respectively). Hence L∗₅is 1-EFT(L5). The following

theorem follows from Corollary 1.

Theorem 3. eft1(L5)= 3.

2.3. 1-EFT(Lk) for graphs where k 4 and even

In this subsection, we are going to construct 1-EFT(Lk) graphs where k is an even integer with k 4.

Let the spanning supergraph L∗_kof Lkbe the graph that

adds E= {(xi,j, xk−i+1,j)| 1 i < k/2, j = 1, 2} to

E(Lk) as shown in Fig. 4(a). The graph in Fig. 4(a)

is actually isomorphic to M(k/2, 2, 2) as shown in Fig. 4(b). We can reconfigure Lk in L∗_k as shown in

Fig. 4(c) for any faulty edge of the form (xi,1, xi,2) or

as shown in Fig. 4(d) for any faulty edge of the form

(xi,1, xi+1,1) or (xi,2, xi+1,2). Hence, M(k/2, 2, 2) is

a 1-EFT(Lk). We obtain the following theorem:

Theorem 4. eft1(Lk) k − 2 where k is an even integer with k 4.

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Fig. 4. (a) L∗_k, a 1-EFT(Lk) where k is even and k 4; (b) the 3-dimensional mesh M(k/2, 2, 2); (c) reconfigure Lkfor any faulty edge of the

form (xi,1, xi,2); and (d) reconfigure Lkfor any faulty edge of the form (xi,1, xi_+1,1), or (xi,2, xi_+1,2).

Fig. 5. A 1-EFT(Lk) where k is odd and k 7.

2.4. 1-EFT(Lk) graphs for k 7 and odd

Assume k is an odd integer with k 7. Construct the spanning supergraph L∗_k of Lk by adding E=

{(x1,2, x4,2), (x3,2, x6,2), (x2,1, x5,1), (x4,1, x7,1), (x1,1,

x5,2), (x3,1, x7,2)} ∪ {(x2i,j, x2i+3,j)| 3 i (k −

3)/2, j= 1, 2} as shown in Fig. 5.

Edges of Lk can be divided into the following 7

We can reconfigure Lk in L∗_k for any faulty edge e

in A, B, C, D, E, F , and G respectively as shown in Figs. 6(a), 6(b), 6(c), 6(d), 6(e), 6(f), and 6(g), respectively. Hence L∗_k is 1-EFT(Lk). We obtain the

following theorem:

Theorem 5. eft1(Lk) k −1 where k is an odd integer with k 7.

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Fig. 6. Reconfigures of Lkin L∗kwhere k is odd and k 7 for any faulty edge in A, B, C, D, E, F , and G, respectively.

Acknowledgements

The authors are very grateful to the anonymous referees for their thorough review of the paper and many concrete and helpful suggestions.

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