Strong Menger connectivity with conditional faults on the class
of hypercube-like networks
✩Lun-Min Shih
a, Chieh-Feng Chiang
a, Lih-Hsing Hsu
b, Jimmy J.M. Tan
a,∗ aDepartment of Computer Science, National Chiao Tung University, Hsinchu, Taiwan 30050, ROC bDepartment of Computer Science and Information Engineering, Providence University, Taichung, Taiwan 43301, ROCReceived 18 April 2007; received in revised form 15 October 2007; accepted 16 October 2007 Available online 26 October 2007
Communicated by L. Boasson
Abstract
In this paper, we study the Menger property on a class of like networks. We show that in all n-dimensional hypercube-like networks with n− 2 vertices removed, every pair of unremoved vertices u and v are connected by min{deg(u), deg(v)} vertex-disjoint paths, where deg(u) and deg(v) are the remaining degree of vertices u and v, respectively. Furthermore, under the restricted condition that each vertex has at least two fault-free adjacent vertices, all hypercube-like networks still have the strong Menger property, even if there are up to 2n− 5 vertex faults.
©2007 Elsevier B.V. All rights reserved.
Keywords: Strong Menger connectivity; Conditional faults; Hypercube-like network; Interconnection networks
1. Introduction
Interconnection networks have been widely studied recently. The architecture of an interconnection network is usually denoted as an undirected graph G. Among all fundamental properties for interconnection networks, the (vertex) connectivity is a major parameter widely discussed for the connection status of networks. A ba-sic definition of the connectivity κ(G) of a graph G is defined as the minimum number of vertices whose removal from G produces a disconnected graph. In
con-✩ This work was supported in part by the National Science Council
of the Republic of China under Contract NSC 96-2221-E-009-137-MY3.
* Corresponding author at: Department of Computer Science, Na-tional Chiao Tung University, Hsinchu City, Taiwan 30050, ROC.
E-mail address: [email protected] (J.J.M. Tan).
trast to this concept, Menger [5] provided a local point of view, and define the connectivity of any two vertices as the minimum number of internally vertex-disjoint paths between them.
In this paper, we study the Menger property on a class of hypercube-like networks [9], which is a vari-ation of the classical hypercube network by twisting some pairs of links in it. We show that in all n-di-mensional hypercube-like networks with some vertices removed, every pair of unremoved vertices u and v are connected by min{deg(u), deg(v)} vertex-disjoint paths, where deg(u) and deg(v) are the remaining degree of vertices u and v, respectively. This concept is firstly ap-plied on hypercubes and stars by Oh and Chen [6–8]. In this paper, we give a simpler proof of this result. Furthermore, if we restrict a condition such that each vertex has at least two fault-free adjacent vertices, all hypercube-like networks still have this strong Menger 0020-0190/$ – see front matter © 2007 Elsevier B.V. All rights reserved.
property, even if there are up to 2n− 5 vertex faults. The bound of 2n− 5 is sharp.
2. Preliminary
The topology of a multiprocessor system can be modeled as an undirected graph G= (V, E), where
V (G) represents the set of all processors and E(G) represents the set of all connecting links between the processors. For a subset of vertices F⊂ V (G), the in-duced graph obtained by deleting the vertices of F from
G is denoted by G− F . Let u be a vertex, we use
N (u)to denote the set of vertices adjacent to u, and use deg(u) to denote the cardinality of N (u). For a set of vertices V, the neighborhood of Vis defined as the set
N (V)= {v∈VN (v)} − V. Let G be a graph with a
set F of faulty vertices, the number of fault-free neigh-bors of u in G− F is denoted by degG−F(u).
Let G0= (V0, E0)and G1= (V1, E1) be two dis-joint graphs with the same number of vertices. A one-to-one connection between V (G0)and V (G1)is defined as an edge set M= {(v, φ(v)) | v ∈ V0, φ(v)∈ V1and
φ: V0→ V1is a bijection}. We use G0⊕M G1 to de-note the graph G= (V0∪ V1, E0∪ E1∪ M). Different bijection functions φ lead to different operations ⊕M
and generate different graphs.
The hypercube network is one of the popular topolo-gies in interconnection networks. Several variants of hy-percubes are proposed by twisting some pairs of links in hypercubes, including twisted cubes [1,4], Möbius cubes [2], and crossed cubes [3], to name a few. To make a unified study on these variants, Vaidya et al. [9] pro-posed a class of graphs, called a class of hypercube-like
networks. We now give a recursive definition of the
n-dimensional hypercube-like networks HLn as follows:
(1) HL0= K1, where K1is a trivial graph in the sense that it has only one vertex; and (2) G∈ HLnif and only
if G= G0⊕M G1 for some G0, G1∈ HLn−1. By the
definitions above if G is a graph in HLn, then G is
a composition of G0⊕MG1 with both G0 and G1 in
HLn−1, n 1. Each vertex in G0has exactly one
neigh-bor in G1.
A graph G is r-regular if the degree of every vertex in G is r. We say that a graph G is connected if there is a path between every pair of two distinct vertices. A sub-set S of V (G) is a cut sub-set if G− S is disconnected. The connectivity of G, written as κ(G), is defined as the minimum size of a vertex cut if G is not a com-plete graph, and κ(G)= |V (G)| − 1 if otherwise. We say that a graph G is k-connected if k κ(G). In addi-tion, a graph has connectivity k if it is k-connected but not (k+ 1)-connected.
A classical theorem about connectivity was provided by Menger as follows.
Theorem 1. (See [5].) Let x and y be two distinct
ver-tices of a graph G and (x, y) /∈ E(G). The minimum size of an x, y-cut equals the maximum number of pair-wise internally disjoint x, y-paths.
Following this theorem, Oh and Chen [7] gave a de-finition to extend the Menger’s theorem.
Definition 1. (See [7].) A k-regular graph G is strongly
Menger-connected if for any subgraph G− F of G with
at most k− 2 vertices removed, each pair of vertices
uand v in G− F are connected by min{degG−F(u),
degG−F(v)} vertex-disjoint fault-free paths in G − F ,
where degG−F(u) and degG−F(v)are the degree of u
and v in G− F , respectively.
By Definition 1, Oh and Chen [6–8] showed that an
n-dimensional star graph Sn (respectively, an
n-dimen-sional hypercube Qn) with at most n− 3 (respectively, n− 2) vertices removed is strongly Menger-connected.
In order to be consistent with Definition 1, we say that a graph G possess the strongly Menger-connected prop-erty with respect to a vertex set F if, after deleting F from G, there are min{degG−F(u),degG−F(v)}
vertex-disjoint fault-free paths connecting u and v, for each pair of vertices u and v in G−F . Throughout this paper, we shall call a graph “strongly Menger-connected”, and omit the description of the remaining structure G− F of the graph, if there is no ambiguous.
It is known that the connectivity of an n-dimensional hypercube-like network HLnis n [9]. To extend the
con-nectivity result of HLn further, we study the strongly
Menger-connected property of HLnwith at most n− 2
vertices deleted. Moreover, if we restrict a condition such that each vertex has at least two fault-free adja-cent vertices, HLnstill have the strong Menger property,
even if there are up to 2n− 5 vertex faults. 3. Strong Menger connectivity
In this section, we will prove that all graphs in the class of n-dimensional hypercube-like networks are strongly Menger-connected if there are at most n− 2 vertex faults. Before proving this main result, we need the following lemma, essentially it says that every n-di-mensional hypercube-like network with no more than 2n− 3 vertex faults, still contains a large connected component.
Lemma 1. Let G∈ HLn be an n-dimensional hyper-cube-like network, and S be a set of vertices with|S|
2n− 3, for n 2. There exists a connected component
C in G− S such that |V (C)| 2n− |S| − 1.
Proof. We prove this statement by induction on n. For
n= 2, HL2 is a cycle of length four, the result is triv-ially true. Assume this lemma holds for n− 1, for some
n 3, we will prove that it is true for n.
Let G be an n-dimensional hypercube-like network,
G= G0⊕MG1, and G0, G1∈ HLn−1. Let S be a set
of vertices with|S| 2n − 3, for n 3, and let S0and
S1be subsets of set S in G0and G1, respectively. Then |S0| + |S1| = |S| 2n − 3. Without loss of generality, we assume |S0| |S1|. The proof is divided into two major cases:
Case 1: 0 |S0| 1.
Since G0is (n−1)-connected, G0−S0is connected, for n 3. All the vertices in G0− S0 are connected and form a connected component C0 with|V (C0)| =
2n−1− S0. By definition, all the vertices in G1− S1are adjacent to the vertices in G0= C0∪ S0. Thus, G− S contains a connected component C such that the number of vertices in C is greater than|V (G0)−S0|+|V (G1)− S1| − |S0| = |V (G)| − |S| − |S0| 2n− |S| − 1. (See
Fig. 1.)
Case 2:|S0| 2 and consequently |S1| 2n − 5.
Since 2 |S0| |S1| 2n − 5, so |S0| n − 2 and
n 4. By induction hypothesis, there exists a connected
component C1in G1−S1, and|V (C1)| 2n−1−|S1|−
1. Since the connectivity of G0is n−1 and |S0| n−2,
G0− S0is connected. Then G− S contains a connected component C such that the number of vertices in C is greater than |V (G0)− S0| + (|V (G1)− S1| − 1) =
|V (G)| − |S| − 1 = 2n− |S| − 1. 2
By Lemma 1, we have the following corollary. Corollary 1. Let G be an n-dimensional hypercube-like
network, n 2, and let Vbe a set of vertices in G with
|V| = 2. Then |N(V)| 2n − 2.
Fig. 1. The illustration of the proof of Case 1 in Lemma 1.
In the following, we show that with up to n− 2 ver-tex faults, an n-dimensional hypercube-like network has strongly Menger-connected property. Referring to the relative study proposed by Oh [6], the strong Menger connectivity of regular hypercube networks has been proved. Here we provide a significantly simpler proof for the general hypercube-like networks.
Theorem 2. Consider an n-dimensional
hypercube-like network G∈ HLn, for n 2. Let F be a set of faulty vertices with |F | n − 2. Then each pair of vertices u and v in G − F are connected by
min{degG−F(u),degG−F(v)} vertex-disjoint fault-free paths, where degG−F(u) and degG−F(v) are the re-maining degree of u and v in G− F , respectively.
Proof. Let G be an n-dimensional hypercube-like network, and u and v be two fault-free vertices in
G − F . We first assume, without loss of
general-ity, that degG−F(u) degG−F(v), so min{degG−F(u),
degG−F(v)} = degG−F(u). We now show that u is
con-nected to v if the number of vertices deleted is smaller than degG−F(u)− 1 in G − F . By Theorem 1, this
implies that each pair of vertices u and v in G− F are connected by degG−F(u) vertex-disjoint fault-free
paths, where|F | n − 2.
For the sake of contradiction, suppose that u and v are separated by deleting a set of vertices Vf, where
|Vf| degG−F(u)− 1. As a consequence, |Vf| n − 1
because of degG−F(u) deg(u) n. Then, the
sum-mation of the cardinality of these two sets F and Vf is
|F | + |Vf| 2n − 3. Let S = F ∪ Vf. By Lemma 1,
there exists a connected component C in G− S such that|V (C)| 2n−|S|−1. It means that (i) either G−S is connected, or (ii) G− S has two components, one of which contains only one vertex. If G− S is connected, it contradicts to the assumption that u and v are discon-nected. Otherwise, if G− S has two component and one of which contains only one vertex x. Since we assume that u and v are separated, one of u and v is the vertex x, say u= x. Thus, the set Vf must be the neighborhood of uand|Vf| = degG−F(u), which is also a contradiction.
Then, u is connected to v when the number of vertices deleted is smaller than degG−F(u)− 1 in G − F .
The proof is complete. 2
4. Strong Menger connectivity with conditional faults
As proved in the last section, an n-dimensional hypercube-like network with at most n− 2 faulty ver-tices is strongly Menger-connected. But the result
can-not be guaranteed, if there are n− 1 faulty vertices and all these faulty vertices are adjacent to the same vertex. In most circumstances, the possibility of all the neigh-bors of a vertex being faulty simultaneously is very small. Motivated by the deficiency of traditional fault tolerance, we consider a measure of conditional faults by restricting that every vertex has at least two fault-free neighboring vertices.
Under this condition, we claim that for every n-di-mensional hypercube-like network with at most 2n− 5 faulty vertices and n 5, the resulting network is still strongly Menger-connected. We have an example to show that this result does not hold for n= 4. Consider a 4-dimensional HL4, this network may not be strongly Menger-connected, if the number of conditional faults is 3. (See Fig. 2. The remaining degrees of nodes u and
vare both four, with three vertices deleted as indicated in the graph. But the number of vertex-disjoint paths be-tween u and v is three.) So we can only expect the result holds for n 5.
To prove this result, we need some preliminary lemma. In the following, we show that an n-dimensional hypercube-like network with at most 3n− 6 vertex faults S has a connected component having at least 2n− |S| − 2 vertices.
The proof is by induction, and the case for n= 5 is proved in the following two lemmas.
Lemma 2. Let V be a set of vertices in a 4-dimen-sional hypercube-like network with |V| = 3. Then,
|N(V)| 7.
Proof. Let G be a 4-dimensional hypercube-like net-work. G is a composition of two 3-dimensional hyper-cube-like networks G0 and G1, G= G0⊕M G1, for
a matching operation ⊕M. Without loss of
general-ity, let V be a subset of V (G) containing three ver-tices {x, y, z}. If x, y, z are all in G0, by Lemma 1, {x, y, z} has at least 4 neighboring vertices in G0. Be-sides,{x, y, z} has 3 neighboring vertices in G1. Then, |N({x, y, z})| 4 + 3 = 7. If x, y are in G0, and z is in
G1, by Lemma 1,{x, y} has at least 4 neighboring
ver-Fig. 2. An example showing that an HL4 is not strongly Menger-connected.
tices in G0. In addition,{z} has 3 neighboring vertices in G1. Then,|N({x, y, z})| 4 + 3 = 7. 2
Lemma 3. Let G be a 5-dimensional hypercube-like
network and S be a set of vertices with|S| 9. (3n −
6= 9, for n = 5.) There exists a connected component
C in G− S such that |V (C)| 25− |S| − 2.
Proof. Let G be a 5-dimensional hypercube-like net-work, G0, G1∈ HL4, and G= G0⊕MG1, for a
match-ing operation⊕M. Let S be a set of vertices with|S|
3n− 6 = 9, for n = 5, and let S0and S1be subsets of S in G0and G1, respectively. Without loss of generality, we assume|S0| |S1|. (Note that |S| 9, so |S0| 4.) We then consider three cases:
Case 1: 0 |S0| 2.
Since G0 is (n− 1)-connected, G0− S0 is con-nected, for n= 4. So G0− S0has only one connected component C0with|V (C0)| = 24− S0. By definitions,
all vertices in G1− S1 are adjacent to the vertices of
G0= C ∪ S0. Let C be the connected component of
G− S containing C0. Then the number of vertices in Cis greater than|V (G0)− S0| + |V (G1)− S1| − |S0| =
|V (G)| − |S| − |S0| 25− |S| − 2.
Case 2:|S0| = 3 and therefore |S1| 6.
G0− S0is connected by the fact that G0is (n− 1)-connected, for n 4. Thus, G0− S0has only one con-nected component C0 with|V (C0)| = 24− S0. Then,
all vertices in G1 are connected to component C0, ex-cept for the three vertices in G1 adjacent to the ver-tices in S0. Since |S1| 6 and by Lemma 2, at least one of these three vertices is connected to component
G1− S1. So at least 24− |S1| − 2 vertices are connected to component C0. Let C be the connected component of G− S containing C0. Then, the number of vertices in C is|V (C)| |V (G0)− S0| + |V (G1)− S1− 2| =
|V (G)| − |S| − 2 = 25− |S| − 2.
Case 3:|S0| = 4 and consequently 4 |S1| 5.
Since 5 2n − 3, for n 4. By Lemma 1, there ex-ists a connected components C0 (respectively, C1) in
G0− S0 (respectively, G1− S1) such that|V (C0)| 24− |S0| − 1 (respectively, |V (C1)| 24− |S1| − 1).
Thus, there exists a connected component C in G− S such that|V (C)| |V (G0)− S0− 1| + |V (G1)− S1−
1| = |V (G)| − |S| − 2 = 25− |S| − 2. 2
Based on Lemma 3, the general case for n 5 is stated as follows.
Lemma 4. Let G be an n-dimensional hypercube-like
n 5. There exists a connected component C in G − S such that|V (C)| 2n− |S| − 2.
Proof. We prove this statement by induction on n. By Lemma 3, the result holds for n= 5. Assume the lemma holds for n− 1, for some n 6. We now show that it is true for n.
Let G be an n-dimensional hypercube-like network,
G0, G1∈ HLn−1, and G= G0⊕MG1, for some
match-ing operation⊕M. Let S be a set of vertices with|S|
3n− 6, for n 6, and let S0 and S1 be subsets of S in G0 and G1, respectively. Therefore, |S0| + |S1| = |S| 3n − 6. Without loss of generality, we assume |S0| |S1|. The proof is divided into two major cases:
Case 1: 0 |S0| 2.
Since G0 is (n− 1)-connected, G0− S0 is con-nected, for n 6. Let C0= G0− S0, C0is a connected component with |V (C0)| 2n−1− S0. By definitions,
all vertices in G1− S1 are adjacent to the vertices in
G0= C0∪ S0. Let C be the connected component of
G− S containing C0. The number of vertices in C is
greater than |V (G0)− S0| + |V (G1)− S1| − |S0| =
|V (G)| − |S| − |S0| 2n− |S| − 2.
Case 2:|S0| 3 and consequently |S1| 3n − 9.
By induction hypothesis, there are two connected components C0 and C1 in G0− S0 and G1− S1, and |V (C0)| 2n−1−|S0|−2 and |V (C1)| 2n−1−|S1|−
2, respectively. Without loss of generality, we assume that |V (C0)| |V (C1)|. Now we focus on the
num-ber of vertices in the component C1, and discuss two situations. First, suppose |V (C1)| = 2n−1− |S1| − 2.
By Corollary 1, |S1| 2(n − 1) − 2 = 2n − 4. So |S0| = |S|−|S1| n−2. Since G0is (n−1)-connected,
G0− S0is connected. G0− S0has only one connected component C0and|V (C0)| = 2n−1− |S0|. Let C be the
connected component containing C0. Then |V (C)| = |V (C0)| + |V (C1)| 2n−1− |S0| + 2n−1− |S1| − 2
2n− |S| − 2. Second, suppose that |V (C1)| 2n−1−
|S1| − 1. Since |V (C0)| |V (C1)| 2n−1− |S1| − 1,
there exists a connected component C containing C0 such that|V (C)| = |V (C0)| + |V (C1)| 2n−1− |S0| −
1+ 2n−1− |S1| − 1 2n− |S| − 2. 2
Corollary 2. Let G be an n-dimensional hypercube-like
network, n 5, and let Vbe a set of vertices in G with
|V| = 3. Then |N(V)| 3n − 5.
As stated in the last section, we showed that every
n-dimensional hypercube-like network with at most
n− 2 vertex faults is strongly Menger-connected. In
the following, we will show another main result that, by restricting every vertex having at least two fault-free
neighboring vertices, every n-dimensional hypercube-like network with up to 2n− 5 vertex faults is still strongly Menger-connected.
For the next theorem, we define a set of vertices Fc
in graph G to be a conditional faulty vertex set if, in the induced subgraph G− Fc, every vertex has at least two
fault-free neighboring vertices. We also call the sub-graph G− Fca conditional faulty graph.
Theorem 3. Consider an n-dimensional hypercube-like
network G∈ HLn, for n 5. Let Fc be a set of con-ditional faulty vertices with |Fc| 2n − 5. Then each pair of vertices u and v in G− Fc are connected by
min{degG−Fc(u),degG−Fc(v)} vertex-disjoint fault-free
paths, where degG−Fc(u) and degG−Fc(v) are the
de-gree of u and v in G− Fc, respectively.
Proof. Without loss of generality, we assume degG−Fc(u) degG−F
c(v), and therefore mindegG−Fc(u),degG−Fc(v)
= degG−Fc(u).
We want to prove that each pair of vertices u and v in
G− Fc are connected by degG−Fc(u) vertex-disjoint fault-free paths, for |Fc| 2n − 5. We are going to
show that u is connected to v if the number of vertices deleted is smaller than degG−Fc(u)−1 in G−Fc, where |Fc| 2n − 5.
Suppose on the contrary that u and v are sepa-rated by deleting a set of vertices Vfc, where |Vfc| degG−Fc(u) − 1. By degG−Fc(u) deg(u) n, we have|Vfc| n − 1. We sum up the cardinality of these two sets Fc and Vfc. Since|Fc| 2n − 5 and |Vfc|
n− 1, then |Fc| + |Vfc| 3n − 6. Let S = Fc∪ Vfc. By Lemma 4, there exits a connected component C in
G−S such that |V (C)| 2n−|S|−2 and |S| 3n−6.
It means that there are at most two vertices in G− S not belonging to C. We then consider three cases:
Case 1:|V (C)| = 2n− |S|. It means that all vertices
in G− S are connected, which contradicts to the as-sumption that u and v are disconnected.
Case 2:|V (C)| = 2n− |S| − 1. Only one vertex is
disconnected to G−S. Since |Vfc| degG−Fc(u)−1 degG−Fc(v)− 1, neither u nor v can be the only one disconnected vertex, a contradiction.
Case 3:|V (C)| = 2n−|S|−2. Let a and b be the two
vertices in G− S not belonging to C. We consider two situations. (i) Suppose first that u∈ C. If v ∈ C, then
u and v are connected, a contradiction. If v∈ {a, b}, since|Vfc| degG−Fc(v)− 1, v is connected to at least one vertex in component C, a contradiction. (ii) Sup-pose u∈ {a, b}. We without loss of generality let u = a, and consider the adjacency between a and b.
Subcase 1: Suppose that a is not adjacent to b. By the
assumption that u and v are separated by deleting a set of vertices Vfcwith|Vfc| = degG−Fc(u)− 1. Let Vfc be a subset of the neighborhood of u, that is, Vfc⊂ N(u). Since |Vfc| < |N(u)|, vertex u and component C are connected, which is a contradiction.
Subcase 2: Suppose that a is adjacent to b. Let Vfc=
N (u)− {b}. Since G − Fcis a conditional faulty graph,
one of the neighbors of b is in C. Then, b is connected to C, which is a contradiction.
Therefore, vertex u and v are still connected with up to degG−Fc(u)− 1 vertex faults. By Theorem 1, this
implies that each pair of vertices u and v in G− Fc
are connected by min{degG−Fc(u),degG−Fc(v)} vertex-disjoint fault-free paths, where|Fc| 2n − 5. The proof
is complete. 2
Acknowledgements
The authors are grateful to the anonymous referees for a number of comments and suggestions that improve the quality of this paper.
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