Faithful 1-edge fault tolerant
graphs
Shih-Yih Wang
Lih-Hsing Hsu
Ting-Ti Sung
Information Processing Letters 61 (1997) 173-181
• F. Harary and J. P. Hayes, Edge fault
tolerance in graphs, Networks 23 (1993)
135-142.
• Let G be a graph of order n and k be a
positive integer. An n-vertex graph G* is
said to be k-edge fault tolerant, or k-EFT,
with respect to G, if every graph obtained
by removing any k-edges from G* contains
G.
• For brevity, we refer to G* as k-EFT(G)
graph.
• Obviously, k+1G is an k-EFT(G) for any
graph G.
• Moreover, Cn is an optimal 1-EFT(Pn).
• It is easy to check that the torus graph
C(m1,m2,…, mn) is an 1-edge fault tolerant
graph for M(m1,m2,…, mn) where mi  2
with 1  i  n.
• A k-EFT(G) graph G* is optimal if it
contains the least number of edges among
all k-EFT(G) graphs.
• We use eftk(G) to denote the difference
between the number of edges in an
optimal k-EFT(G) graph and that in G.
(C6*,C6)
(C4*,C4)
C6 C4
(C6*-E(C6))  (C4*-E(C4) )
Tensor product
(C6*,C6)  (C4*,C4)
(C6*,C6)  (C4*,C4) = ((C6*,C6)  (C4*,C4) , C6*  C4)
Optimal!
• Generalization.
• More theoretic approach!
Commutative and Associative
Theorem 1. (G1*,G1)  (G2*,G2) = (G2*,G2)  (G1*,G1) .
and
((G1*,G1)  (G2*,G2))  (G3*,G3) = (G1*,G1)  ((G2*,G2)  (G3*,G3)) .
Proof. Cartesian product and tensor product are commutative
and associative.
Basic mathematical observation!
• Recursively define
(G1*,G1)  (G2*,G2)  …  (Gn*,Gn) as
((G1*,G1)  (G2*,G2)  …  (Gn-1*,Gn-1))  (Gn*,Gn).
• Recursively define
(G1*,G1)  (G2*,G2)  …  (Gn*,Gn) as
((G1*,G1)  (G2*,G2)  …  (Gn-1*,Gn-1))  (Gn*,Gn).
Corollary 2. For any permutation  on the set {1,2,…,n},
we have
• (G1*,G1)  (G2*,G2)  …  (Gn*,Gn) =
(G(1)*,G(1))  (G(2)*,G(2))  …  (G(n)*,G(n)).
•
(G1*,G1)  (G2*,G2)  …  (Gn*,Gn) =
(G(1)*,G(1))  (G(2)*,G(2))  …  (G(n)*,G(n)).
• For 1  i  n, we define the ith projection of
V1V2…Vn as the function
• pi: V1V2…Vn Vi given by
• pi((x1,x2,…,xn))=xi where xjVj for 1  j  n
C2 = 2K2
Faithful edge fault tolerant graphs
More definition
(P4*,P4)
((P4*,P4)(C2,K2))
Edge set X
Faithful edge fault tolerant graphs
e
Edge set Ye
• A graph G* is said to be faithful or a faithful graph with
respect to G, denoted by FG(G), if it satisfies the following
two conditions:
(1) There exists a function : V(G)V(G) such that the
function h:V(GK2) h:V(GK2) given by h((xi,z1))=(xi,z1)
and h((xi,z2))=( (xi),z2) induces an isomorphism from G K2
into a subgraph of ((G*,G)(C2,K2))-X.

(2) For any edge e=(xi,xj) in G, there exists an isomorphism
fe from GK2 into a subgraph of ((G*,G)(C2,K2))-Ye such
that p1(fe((xi,z1)) = p1(fe((xi,z2)), where p1 is the 1st
projection.
e
Lemma 3. Pn* is FG(Pn).
x0
x1
x2
x3
x4
(x0,z2)
(x1,z2)
(x2,z2)
(x3,z2)
(x4,z2)
(x0,z1)
(x1,z1)
(x2,z1)
(x3,z1)
(x4,z1)
(x4,z2)
(x3,z2)
(x2,z2)
(x1,z2)
(x0,z2)
(x0,z1)
(x1,z1)
(x2,z1)
(x3,z1)
(x4,z1)
(x3,z2)
(x4,z2)
(x0,z1)
(x1,z1)
(x2,z1)
(x3,z1)
(x4,z1)
(x0,z4)
(x1,z2)
(x2,z2)
Lemma 4. Any faithful graph G* with respect
to G is 1-EFT(G).
Moreover, |E(G*) – E(G)|  |V(G)/2 for any
faithful graph G* with respect to G.
fe
X
Lemma 5. Any graph has a faithful
supergraph.
E(G' )  E ( G) {( x , x ) | x V (G)}
2
i
i
i
Harary and Hayes: 1-EFT(Cn)
Not faithful
Lemma 6. Let n be a positive integer,
Cn* is FG(Cn) if and only if n  4.
(x2,z2)
(x3,z2)
(x0,z2)
(x1,z2)
(x0,z1)
(x1,z1)
(x2,z1)
(x3,z1)
(x0,z2)
(x0,z1)
(x1,z2)
(x3,z1)
(x2,z1)
(x1,z1)
(x3,z2)
(x2,z2)
1
2
Theorem 7. Let W = {G*,G) | G* is FG(G)} .
Then W is closed under .
Try this!
Corollary 8. Let Gi* be faithful graph of Gi for i=1,2.
The graph (G1*,G1)(G2*,G2) is 1-EFT(G1G2) .
Furthermore,
eft1(G1G2)2(|E(G1*)|-E(G1)|)(|E(G2*)-E(G2)|).
Tensor product
Lemma 10. Let mi4 be a positive even
integer for all i. C*(m1,m2,…,mn) is an
optimal 1-EFT graph with respect to
C(m1,m2,…,mn). Furthermore,
eft1(C(m1,m2,…,mn))=(i=1n mi)/2.
Conjecture 11.
m 
eft1(C(m1,m2 ,...,mn ))  2   i 
i 1  2 
if each mi is an odd integer wi th mi  5 and n  2.
n-1
n