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Conditional connectivity of Cayley graphs generated by transposition trees
Information Processing Letters 110 (2010) 1027–1030
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Information Processing Letters www.elsevier.com/locate/ipl
Conditional connectivity of Cayley graphs generated by transposition trees ✩ Weihua Yang a,b , Hengzhe Li a , Jixiang Meng a,∗ a b
College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang, 830046, PR China School of Mathematical Science, Xiamen University, Xiamen Fujian 361005, PR China
a r t i c l e
i n f o
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Article history: Available online 18 September 2010 Communicated by A.A. Bertossi Keywords: Interconnection networks Cayley graphs Conditional connectivity Transposition tree
Given a graph G and a non-negative integer h, the R h -(edge)connectivity of G is the minimum cardinality of a set of (edges)vertices of G, if any, whose deletion disconnects G, and every remaining component has minimum degree at least h. Similarly, given a nonnegative integer g, the g-(edge)extraconnectivity of G is the minimum cardinality of a set of (edges)vertices of G, if any, whose deletion disconnects G, and every remaining component has more than g vertices. In this paper, we determine R 2 -(edge)connectivity and 2-extra(edge)connectivity of Cayley graphs generated by transposition trees. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The traditional connectivity, is an important measure for the networks, which can correctly reflects the fault tolerance of systems with few processor, but it always underestimates the resilience of large networks. The discrepancy incurred is because events whose occurrence would disrupt a large network after a few processor/link failures are highly unlikely, therefore, the disruption envisaged occurs in a worst case scenario. Motivated by the shortcomings of the traditional connectivity, Harary [7] introduced the concept of conditional connectivity. Let G be a connected undirected graph, and P be a graph-theoretic property, Harary [7] defined the conditional connectivity κ (G ; P ) as the minimum cardinality of a set of vertices, if any, whose deletion disconnects G and every remaining component has property P . Some kinds of conditional connectivity introduced in [7] have received much attention in the literature, such as the R h -(edge)connectivity and the extra(edge)connectivity.
✩ The research is supported by NSFXJ (No. 2010211A06) and NSFC (No. 10671165). Corresponding author. E-mail addresses: [email protected] (W. Yang), [email protected] (J. Meng).
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0020-0190/$ – see front matter doi:10.1016/j.ipl.2010.09.001
© 2010
Elsevier B.V. All rights reserved.
Let h be a non-negative integer, P h be the property of having at last h neighbors for any vertex, Latifi [10] called this κ (G ; P h ) (resp. λ(G ; P h )) the R h -(edge)connectivity, written κ h (G ) (resp. λh (G )). That is, the R h -(edge)connectivity of G is the minimum cardinality of a set of (edges)vertices of G, if any, whose deletion disconnects G, and every remaining component has minimum degree at least h. Similarly, let g be a non-negative integer, and P g be the property of having more than g vertices. Fàbrega and Fiol [6] defined the g-extra(edge)connectivity κ g (G ) (resp. λ g (G )) of G as κ (G ; P g ) (resp. λ(G ; P g )). That is, the g-extra(edge)connectivity of G is the minimum cardinality of a set of (edges)vertices of G, if any, whose deletion disconnects G, and every remaining component has more than g vertices. A vertex cut is called a κ g -vertex cut (resp. λ g -vertex cut) of G if G − S having property P g , and an R g -vertex cut (resp. λ g -vertex cut) if G − S having property P g . Let X be a group, S be a subset of X and e be the identity element of X . The Cayley digraph Cay( X , S ) is a digraph with vertex set X and arc set {( g , gs) | g ∈ X , s ∈ S }. Clearly, if S = S −1 , where S −1 = {s−1 | s ∈ S }, then Cay( X , S ) is an undirected graph. Denote by Sym(n) the group of all permutations on {1, . . . , n}. For convenience, we use ( p 1 p 2 · · · pn ) to denote the permutation
1 2 ···n p 1 p 2 ··· pn
,
and (i j ), which is called a transposition, denotes the per-
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mutation that swaps the objects at position i and j, that is, ( p 1 · · · p i · · · p j · · · pn )(i j ) = ( p 1 · · · p j · · · p i · · · pn ). Let T be a set of transpositions. We call G (T ) the transposition generating graph, where the vertex set of G (T ) is {1, 2, . . . , n} and edge set E (G (T )) = {(i j ) | (i j ) ∈ T }. It is well known that the Cayley graph Cay(Sym(n), T ) is connected if and only if the transposition generating graph G (T ) is connected (see [8]). We call G (T ) a transposition tree if G (T ) is a tree. We denote Cay(Sym(n), T ) by Γn if G (T ) is a tree. It is well known that if G (T ) ∼ = K 1,n−1 , then Cay(Sym(n), T ) is called star graph, which is a well known network model; and Cay(Sym(n), T ) is called bubble-sort graph if G (T ) ∼ = P n . In this paper, we always use S n to denote the star graph and G n to denote any graph whose transposition generating graph is a tree and not isomorphic to K 1,n−1 . Clearly, the Cayley graphs Γn obtained by these transposition trees are (n − 1)-regular bipartite graph and have n! vertices, see [9] for the details. The networks based on Cayley graphs generated by transpositions have many advantageous properties, see, for example, [2–4,9]. In particular, the star graphs as a network model received much attention, see [2–4,8,9,11] for the details. Assume that A is a subgraph of graph G, we use N G ( A ) to denote the neighborhood of V ( A ) in G, and use ω( A ) to denote the set of edges with one end in V ( A ) and the other end in V ( A ). We follow Bondy [1] for graph terminologies not given here. It is known that κ (Γn ) = n − 1, κ1 (Γn ) = 2n − 4 for n 3, κ 2 ( S n ) = 6(n − 3) for n 4, see [2,3,11] for the detail. In this paper, we shall determine κ2 (Γn ), λ2 (Γn ) and κ 2 (Γn ), λ2 (Γn ). 2. Preliminaries We give some useful properties of Γn , which can be found in [2–5]. Property 1. If n is a leaf of G (T ), then Γn can be decomposed into n disjoint copies of Γn−1 , say Γn1−1 , Γn2−1 , . . . ,
Γnn−1 , where Γni−1 is an induced subgraph by vertex set {( p 1 p 2 · · · pn−1 i ) | ( p 1 p 2 · · · pn−1 ) ranges over all permutations of {1, . . . , i − 1, i + 1, . . . , n}}. We denote this decomposition by Γn = Γn1−1 ⊕ Γn2−1 ⊕ · · · ⊕ Γnn−1 (that is, G n = G n1−1 ⊕ G n2−1 ⊕ · · · ⊕ G nn−1 and S n = S n1−1 ⊕ S n2−1 ⊕ · · · ⊕ S nn−1 , respectively).
Property 2. Let (tn) ∈ T be a pendant edge of G (T ). For any vertex u of Γni−1 , u (tn) is the unique neighbor of u
outside Γni−1 , is called the out neighbor of u, written u , and any two distinct vertices of Γni−1 have different out neighbors. Hence, there are exactly (n − 2)! independent j edges between Γni−1 and Γn−1 if i = j, that is, | N (Γni−1 ) ∩ j
V (Γn−1 )| = (n − 2)! if i = j. Property 3. G n has girth 4 and S n has girth 6. The next useful result can be found in [8]. Lemma 2.1. Let T be a set of transpositions from S ym(n) and let g and h be elements of T . If the transposition generating
graph of T contains no triangles, then g and h have exactly one common neighbor in Cay(Sym(n), T ) if gh = hg, and exactly two common neighbors otherwise. Clearly, for any two distinct transpositions (i j ) and (kl), (i j )(kl) = (kl)(i j ) if and only if {i , j } ∩ {k, l} = ∅. It is not difficult to see that if C 4 = x1 x2 x3 x4 is a 4-cycle of Γn , then there must exist two transpositions (i j ) and (kl) such that {i , j } ∩ {k, l} = ∅ and x2 = x1 (i j ), x3 = x2 (kl), x4 = x3 (i j ) = x1 (kl). In fact, there exists an automorphism of Γn , say ψ ∈ Aut(Γn ), such that ψ(x1 ) = e since Γn is vertex transitive, furthermore, C 4 = ψ(x1 x2 x3 x4 ) is also a 4-cycle of Γn . Let g = ψ(x2 ), h = ψ(x4 ), clearly, g and h are two transpositions of T . By Lemma 2.1, we have that gh = hg since they have two common neighbors. That is, (i j ) = ψ −1 ( g ), (kl) = ψ −1 (h). By the above argument, we have the following corollary. Corollary 2.2. For any two distinct vertices of Γn , then have at most two common neighbors if they have any. Lemma 2.3. (See [2,3,11].) κ (Γn ) = n − 1, κ1 (Γn ) = 2n − 4 for n 3, κ 2 ( S n ) = 6(n − 3) for n 4. Lemma 2.4. Let n 4, G n = G n1−1 ⊕ G n2−1 ⊕ · · · ⊕ G nn−1 . If C 4
is a 4-cycle of G ni −1 , then N G n (C 4 ) is an R 2 -vertex cut of G n .
Proof. If n = 4, then G n = Cay(Sym(4), {(12), (23), (34)}). Without loss of generality, we assume C 4 = x1 x2 x3 x4 x1 , where x1 = (1234), x2 = x1 (12) = (2134), x3 = x2 (34) = (2143), x4 = x3 (12) = x1 (34) = (1243). Clearly, C 4 is 2-regular. Next we show that G 4 − V (C 4 ) ∪ N G n (C 4 ) has minimum degree at least 2. Since G 4 is 3-regular, it is sufficient to show that for any two vertices of N G n (C 4 ) have no common neighbor in G 4 − V (C 4 ) ∪ N G n (C 4 ). Note that G 4 is a bipartite graph. Then the neighbors of any two adjacent vertices of C 4 have no common neighbor in G 4 − V (C 4 ) ∪ N G n (C 4 ). Now we show that the neighbors of x2 and x4 have no common neighbor in G 4 − V (C 4 ) ∪ N G n (C 4 ). Suppose there exists a common neighbor. So there are two transpositions (i j ) and (kl) of {(12), (23), (34)} such that x2 (23)(i j ) = x4 (23)(kl), which is impossible. So is x1 and x3 since G 4 is vertex-transitive. That is N G n (C 4 ) is a R 2 -vertex cut of G 4 . Let n 5. Without loss of generality, we assume that C 4 = x1 x2 x3 x4 x1 ⊂ G nn−1 , where x1 is a vertex of G nn−1 and x2 = x1 (i j ), x3 = x2 (kl), x4 = x3 (i j ) = x1 (kl), and i < j < k < l (note that {i , j } ∩ {k, l} = ∅). Let x1 = ( p 1 · · · p i · · · p j · · · pk · · · pl · · · n). Thus we have x2 = ( p 1 · · · p j · · · p i · · · pk · · · pl · · · n), x3 = ( p 1 · · · p j · · · p i · · · pl · · · pk · · · n), x4 = ( p 1 · · · p i · · · p j · · · pl · · · pk · · · n). Since n is a leaf of G (T ), there has exactly one transposition contains n in T , say (tn). Clearly, if (tn) ∈ / ∈{(in), ( jn), (kn), (ln)}, then all the out vertices x1 = x1 (tn), x2 = x2 (tn), x3 = x3 (tn), x4 = x4 (tn) are in G nt −1 which clearly induce a 4-cycle of G nt −1 . Otherwise, without loss of generality, we assume t = i. We have that {x1 , x4 } ⊂ V (G ni −1 ) and
{x2 , x3 } ⊂ V (G n−1 ). Clearly, v ∈ V (G nt −1 ) or V (G n−1 ), we j
j
W. Yang et al. / Information Processing Letters 110 (2010) 1027–1030
have | N G t
n −1
( v ) ∩ {x1 , x2 , x3 , x4 }| 1 since G nt −1 contains
no triangle and for any two distinct vertices have at most two common neighbors (Corollary 2.2). That is, every vern tex of s=2 G ns −1 has degree at least n − 3. Combining the above arguments, we have that N G n (C 4 ) is an R 2 -vertex cut of G n for n 4. 2 Remark 1. Let P 3 be a path with order 3. If P 3 ⊂ C 4 ⊂ G ni −1 , by an arguments similar to that of Lemma 2.4, we can see that N ( P 3 ) is a κ2 -vertex cut of G n for n 4. Furthermore, ω(C 4 ) and ω( P 3 ) are the R 2 -edge cut and λ2 -edge cut, respectively. 3. R 2 -(edge)connectivity of Γn In this section we shall derive
κ 2 (Γn ) and λ2 (Γn ).
j
G n−1 − F j if n = 4 or 5. Combining the above arguments, we have that G n − F is connected. But F is an R 2 -vertex cut, a contradiction. Case 2. There exists an integer i such that | F i | 2n − 6. Since | F | 4n − 13, there is exactly one integer i such that | F i | 2n − 6. Without loss of generality, we assume | F 1 | 2n − 6. That is, | F j | 2n − 7 for j = 1. By an argun ment similar to that of Case 1, we have that l=2 (G ln−1 −
¯ F l ) is a connected subgraph of G n − F , denoted by G. Since G n − F is disconnected, there must be a component of G n − F in G n1−1 , say C . Note that g (G n ) = 4, there must exist a 4-vertex path in C , say P 4 = u 0 u 1 u 2 u 3 . By Corollary 2.2, we have | N G 1 ( P 4 )| 4(n − 2) − 6 − 2. Let x ∈ NG1
N −1
Theorem 3.1. For n 4, girth of Γn .
κ 2 (Γn ) = g (n − 3), where g is the
Proof. Wan and Zhang in [11] have shown that κ 2 ( S n ) = g (n − 3) = 6(n − 3). Next we shall show that κ 2 (G n ) = g (n − 3) = 4(n − 3). Note that if G (T ) is not a star, then there exists two non-adjacent edges in G (T ), say (i j ) and (kl). Let C 4 = e 0 e 1 e 2 e 3 be a cycle of G ni −1 for an integer i, where e 1 = e 0 (i j ), e 2 = e 1 (kl), e 3 = e 2 (i j ). By Corollary 2.2, it is easy to see that | N G n (C 4 )| = 4(n − 3). By Lemma 2.4, we have that κ 2 (G n ) 4(n − 3). Next we shall show that κ 2 (G n ) 4(n − 3). Suppose by way of contradiction that κ 2 (G n ) 4(n − 3) − 1. Let F be a minimum R 2 -vertex cut, then | F | 4(n − 3) − 1. Denote by F i the vertex set F ∩ V (G iN −1 ) in the following arguments. Case 1. | F i | 2n − 7 for all i. We first show that G ni −1 − F i is connected for all i. If G ni −1 − F i is disconnected for an integer i, then by
Property 2, F i is an R 1 -vertex cut of G ni −1 , so we have | F i | κ 1 (G n−1 ) = κ1 (G n−1 ) = 2n − 6. But | F i | 2n − 7. That is, G ni −1 − F i is connected for all integers i. We claim that any two distinct integers i and j, then j G ni −1 − F i is connected with G n−1 − F j . Otherwise, suppose
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N −1
( P 4 ), then at least one of x and x in F . Clearly,
{u 0 , u 1 , u 2 , u 3 } ⊂ F . By Property 2, | F | | N G 1 |{u 0 , u 1 , u 2 , u 3 }| 4n − 12, a contradiction. 2
N −1
( P 4 )| +
Theorem 3.2. λ2 (Γn ) = g (n − 3) for n 4. Proof. By an argument similar to that of Theorem 3.1. We have that λ2 (Γn ) = g (n − 3). 2 4. 2-extra(edge)connectivity Γn In this section, we discuss the 2-extra(edge)connectivity of Γn . Theorem 4.1. κ2 (G n ) = 3n − 8, κ2 ( S n ) = 3n − 7 for n 4. Proof. We first show that κ2 (G n ) = 3n − 8. Note that if G (T ) is not a star, there exists two disjoint edges in G (T ), say (i j ) and (kl). Let P = x0 x1 x2 where x1 = x0 (i j ), x2 = x1 (kl). By Corollary 2.2, we have | N ( P )| = 3n − 8. Clearly, Remark 1 implies that κ2 (G n ) 3n − 8. Next we shall show that κ2 (G n ) 3n − 8. Suppose by the way of contradiction that κ2 (G n ) 3n − 9. Let F be a minimum κ2 -vertex cut of Γn , then | F | 3n − 9. Let F i = F ∩ V (G ni −1 ). Since | F | 3n − 9, there are at most two distinct integers k and l such that F k n − 2 and F l n − 2.
j
that G ni −1 − F i is disconnected from G n−1 − F j for two disj tinct integers i and j, that is, N (G ni −1 − F i ) ∩ V (G n−1 ) ⊂ j
F j . Note that | N (G ni −1 ) ∩ V (G n−1 )| = (n − 2)!. We have that (n − 2)! − (2n − 7) (n − 2)! − | F i | | F j | 2n − 7, which is impossible for n 6. Thus G ni −1 − F i is conj
nected to G n−1 − F j for n 6. On the other hand, if n = 4 or 5, the above inequalities become equalities. That is, | F i | = | F j | = 2n − 7. Since | F | 4n − 13 and n 4, there must exist an integer k such that F k = ∅. Note that | N (G nk −1 ) ∩ V (G ni −1 − F i )| (n − 2)! − (2n − 7) = 1 when
n = 4 or 5, we have that G nk −1 − F k (emptyset) is connected to G ni −1 − F i . Similarly, we have that G nk −1 − F k (emptyset) j
is connected to G n−1 − F j . Thus G ni −1 − F i is connected to
Case 1. | F i | n − 3 for all i. Since
κ (G n−1 ) = n − 2, we have that all G ni −1 − F i
are connected. Let G nk −1 − F k and G ln−1 − F l be two distinct subgraphs of G n − F . Since | N (G nk −1 ) ∩ V (G ln−1 )| =
(n − 2)!, we have that | N (G nk −1 − F k ) ∩ V (G ln−1 − F l )| (n − 2)! − 2(n − 3) > 0 for n 5. That is, for any two distinct subgraphs G nk −1 − F k and G ln−1 − F l of G n − F are connected if n 5. Thus G n − F is connected for n 5, a contradiction. In particular, if n = 4, then | F | 3. But κ1 (G n ) = 2n − 4 = 4 > 3 = 3n − 9, a contradiction. Case 2. There exists an integer i such that | F i | n − 2.
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Let I = {i || F i | n − 2}. Clearly, | I | 2. Note that | F j | n − 3 if j ∈ / I . By an argument similar to that of Case 1, we j know that G¯ = j ∈/ I (G n−1 − F j ) is connected. Subcase 2.1. | I | = 1. Without loss of generality, we assume | F 1 | n − 2. If G n1−1 is disconnected, then there exists a component of
G n − F contained in G n1−1 − F 1 , say C , and | V (C )| 3. Let P = u 0 u 1 u 2 be a 3-vertex connected subgraph of C . By Corollary 2.2, we have that | N G 1 ( P )| 3(n − 2) − 4 − 1 = n −1
3n − 11. Note that for any u ∈ N G 1 ( P ), at least one n −1
of {u , u } in F . By Property 2, {u 0 , u 1 , u 2 } ∩ N G 1 ( P ) =
∅. Therefore, | F | | N G 1 ( P )| + n−1
3n − 9, a contradiction. If
G n1−1
|{u 0 , u 1 , u 2 }|
n −1
3n − 8 >
is connected, then note that
(n − 1)! > 3n − 9 for n 4, that is, N (G n1−1 − F 1 )∩ V (G¯ ) = ∅. Thus G n − F is connected, a contradiction. Subcase 2.2. | I | = 2. Assume I = {k, l}. We consider the following three cases. If G nk −1 − F k and G ln−1 − F l are connected. Note that
k k l k ¯ |N (G n−1 − F k )∩ V (G )| | N (G n−1 )|−| V (G n−1 )∩ N (G n−1 )|− | j =l F j | (n − 1)! − (n − 2)! − (2n − 7) > 0 for n 5, that is, G nk −1 − F k is connected to G¯ for n 5, and so is G ln−1 − F l . Thus, G n − F is connected for n 5. Clearly, if n = 4, then | F | 3. But κ1 (G n ) = 2n − 4 = 4 > 3 = 3n − 9 κ2 (G n ), a contradiction. Thus, G n − F is connected for n 4, a contradiction. If exactly one of G nk −1 − F k and G ln−1 − F l is connected. Assume G nk −1 − F k is connected and G ln−1 − F l is discon-
nected. By an argument similar to that of above para¯ That is, graph, we have that G nk −1 − F k is connected to G.
¯ (i) If y = x, then P = xy is nected and is connected to G. connected to G¯ (since F is a κ2 -cutset). For any component C of G nk −1 − F k with order at least 3. By an argument simi-
¯ lar to that of Subcase 2.1, we have that C is connected to G, thus G n − F is connected. (ii) Suppose that y is contained in V (G nk −1 − F k − {x}) or V (G¯ ). Firstly, if y is in V (G¯ ), it
is easy to see that every component of G nk −1 − F k − {x}
¯ If y is in G k − F k − {x}, assume is connected to G. n−1
y ∈ V (C 1 ), where C 1 is component of G nk −1 − F k − {x}. Let P = x0 x1 x2 be a path of C 1 . Since G n contains no odd cycle, at least one of {x0 , x1 , x2 } is not contained in F l (since F l = N G l ( y )). We have | F | | F k | + | F l | + 1 = 3n − 8 > n−1
¯ There3n − 9, a contradiction. That is, y is connected to G. fore, G nk −1 − F k − {x} contains a component of G n − F . By an argument similar to that of Subcase 2.1, we have G n − F is connected. Combining Case 1 with Case 2, we have that κ2 (G n ) = 3n − 8. Note that the girth of S n is 6. By an argument similar to that of G n , we have κ2 ( S n ) = 3n − 7. 2 Theorem 4.2. λ2 (Γn ) = 3n − 7 for n 4. Proof. Theorem 4.2 follows immediately from Remark 1 and an argument similar to that of Theorem 4.1. 2 Remark 2. In [5], Cheng and Lipták determined the R 2 connectivity of Γn independently. Acknowledgements The authors wish to thank Prof. Eddie Cheng and the referees for their very useful suggestions and their kind help.
j
j =l G n−1 − F j is connected. Finally, by an argument similar to that of Subcase 2.1, we have that G n − F is connected, a contradiction. If G nk −1 − F k and G ln−1 − F l are disconnected, then note that κ1 (G n−1 ) = 2n − 6 and κ (G n−1 ) = n − 2, so it is not difficult to see that each of G nk −1 − F k and G ln−1 − F l contains exactly one isolated vertex, say x and y, such that x ∈ V (G nk −1 − F k ), y ∈ V (G ln−1 − F l ).
We now assume that G nk −1 − F k − {x} and G ln−1 − F l − { y } are connected, then since | F k | n − 2, | F l | n − 2 and | F | 3n − 9, there exists an integer j such that F j = ∅. Note that (n − 2)! > 2n − 6 for n 5, it is not difficult j to see that | N (G nk −1 − F k − {x}) ∩ V (G n−1 )| > 0. That is, j
G nk −1 − F k − {x} is connected to G n−1 , so is G ln−1 − F l − { y }. We have that G nk −1 − F k − {x} and G ln−1 − F l − { y } con-
¯ Thus we get a component (G n [{x, y }]) of G n − F nect to G. having order at most 2, but F is a κ2 -vertex cut, a contradiction. In particular, if n = 4, then 3n − 9 = 3 < κ1 (G n ), a contradiction. If G nk −1 − F k − {x} (or G ln−1 − F l − { y }) is disconnected, then | F k ∪ {x}| κ1 (G n−1 ) = 2n − 6. Clearly, | F k | = 2n − 7, | F l | = n − 2. It is easy to see that G ln−1 − F l − { y } is con-
References [1] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, Macmillan, London, 1976. [2] E. Cheng, L. Lipták, N. Shawash, Orienting Cayley graphs generated by transposition trees, Computers & Mathematics with Applications 55 (2008) 2662–2672. [3] E. Cheng, M. Lipman, L. Liptáka, Strong structural properties of unidirectional star graphs, Discrete Applied Mathematics 156 (2008) 2939–2949. [4] E. Cheng, L. Lipták, Linearly many faults in Cayley graphs generated by transposition trees, Information Sciences 177 (2007) 4877–4882. [5] E. Cheng, L. Lipták, A kind of conditional vertex connectivity of Cayley graphs generated by transposition trees, Congressus Numerantium, in press. [6] J. Fàbrega, M.A. Fiol, On the extraconnectivity of graphs, Discrete Mathematics 155 (1996) 49–57. [7] F. Harary, Conditional connectivity, Networks 13 (1983) 347–357. [8] C. Godsil, G. Royle, Algebraic Graph Theory, Springer, pp. 52–54. [9] S. Lakshmivarahan, J. Jwo, S.K. Dhall, Symmetry in interconnection networks based on Cayley graphs of permutation groups: a survey, Parallel Computing 19 (1993) 361–407. [10] S. Latifi, M. Hegde, M. Naraghi-Pour, Conditional connectivity measures for large multiprocessor systems, IEEE Transactions on Computers 43 (2) (2002) 218–222. [11] M. Wan, Z. Zhang, A kind of conditional vertex connectivity of star graphs, Applied Mathematics Letters 22 (2) (2009) 264–267.
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Information Processing Letters www.elsevier.com/locate/ipl
Conditional connectivity of Cayley graphs generated by transposition trees ✩ Weihua Yang a,b , Hengzhe Li a , Jixiang Meng a,∗ a b
College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang, 830046, PR China School of Mathematical Science, Xiamen University, Xiamen Fujian 361005, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history: Available online 18 September 2010 Communicated by A.A. Bertossi Keywords: Interconnection networks Cayley graphs Conditional connectivity Transposition tree
Given a graph G and a non-negative integer h, the R h -(edge)connectivity of G is the minimum cardinality of a set of (edges)vertices of G, if any, whose deletion disconnects G, and every remaining component has minimum degree at least h. Similarly, given a nonnegative integer g, the g-(edge)extraconnectivity of G is the minimum cardinality of a set of (edges)vertices of G, if any, whose deletion disconnects G, and every remaining component has more than g vertices. In this paper, we determine R 2 -(edge)connectivity and 2-extra(edge)connectivity of Cayley graphs generated by transposition trees. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The traditional connectivity, is an important measure for the networks, which can correctly reflects the fault tolerance of systems with few processor, but it always underestimates the resilience of large networks. The discrepancy incurred is because events whose occurrence would disrupt a large network after a few processor/link failures are highly unlikely, therefore, the disruption envisaged occurs in a worst case scenario. Motivated by the shortcomings of the traditional connectivity, Harary [7] introduced the concept of conditional connectivity. Let G be a connected undirected graph, and P be a graph-theoretic property, Harary [7] defined the conditional connectivity κ (G ; P ) as the minimum cardinality of a set of vertices, if any, whose deletion disconnects G and every remaining component has property P . Some kinds of conditional connectivity introduced in [7] have received much attention in the literature, such as the R h -(edge)connectivity and the extra(edge)connectivity.
✩ The research is supported by NSFXJ (No. 2010211A06) and NSFC (No. 10671165). Corresponding author. E-mail addresses: [email protected] (W. Yang), [email protected] (J. Meng).
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0020-0190/$ – see front matter doi:10.1016/j.ipl.2010.09.001
© 2010
Elsevier B.V. All rights reserved.
Let h be a non-negative integer, P h be the property of having at last h neighbors for any vertex, Latifi [10] called this κ (G ; P h ) (resp. λ(G ; P h )) the R h -(edge)connectivity, written κ h (G ) (resp. λh (G )). That is, the R h -(edge)connectivity of G is the minimum cardinality of a set of (edges)vertices of G, if any, whose deletion disconnects G, and every remaining component has minimum degree at least h. Similarly, let g be a non-negative integer, and P g be the property of having more than g vertices. Fàbrega and Fiol [6] defined the g-extra(edge)connectivity κ g (G ) (resp. λ g (G )) of G as κ (G ; P g ) (resp. λ(G ; P g )). That is, the g-extra(edge)connectivity of G is the minimum cardinality of a set of (edges)vertices of G, if any, whose deletion disconnects G, and every remaining component has more than g vertices. A vertex cut is called a κ g -vertex cut (resp. λ g -vertex cut) of G if G − S having property P g , and an R g -vertex cut (resp. λ g -vertex cut) if G − S having property P g . Let X be a group, S be a subset of X and e be the identity element of X . The Cayley digraph Cay( X , S ) is a digraph with vertex set X and arc set {( g , gs) | g ∈ X , s ∈ S }. Clearly, if S = S −1 , where S −1 = {s−1 | s ∈ S }, then Cay( X , S ) is an undirected graph. Denote by Sym(n) the group of all permutations on {1, . . . , n}. For convenience, we use ( p 1 p 2 · · · pn ) to denote the permutation
1 2 ···n p 1 p 2 ··· pn
,
and (i j ), which is called a transposition, denotes the per-
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mutation that swaps the objects at position i and j, that is, ( p 1 · · · p i · · · p j · · · pn )(i j ) = ( p 1 · · · p j · · · p i · · · pn ). Let T be a set of transpositions. We call G (T ) the transposition generating graph, where the vertex set of G (T ) is {1, 2, . . . , n} and edge set E (G (T )) = {(i j ) | (i j ) ∈ T }. It is well known that the Cayley graph Cay(Sym(n), T ) is connected if and only if the transposition generating graph G (T ) is connected (see [8]). We call G (T ) a transposition tree if G (T ) is a tree. We denote Cay(Sym(n), T ) by Γn if G (T ) is a tree. It is well known that if G (T ) ∼ = K 1,n−1 , then Cay(Sym(n), T ) is called star graph, which is a well known network model; and Cay(Sym(n), T ) is called bubble-sort graph if G (T ) ∼ = P n . In this paper, we always use S n to denote the star graph and G n to denote any graph whose transposition generating graph is a tree and not isomorphic to K 1,n−1 . Clearly, the Cayley graphs Γn obtained by these transposition trees are (n − 1)-regular bipartite graph and have n! vertices, see [9] for the details. The networks based on Cayley graphs generated by transpositions have many advantageous properties, see, for example, [2–4,9]. In particular, the star graphs as a network model received much attention, see [2–4,8,9,11] for the details. Assume that A is a subgraph of graph G, we use N G ( A ) to denote the neighborhood of V ( A ) in G, and use ω( A ) to denote the set of edges with one end in V ( A ) and the other end in V ( A ). We follow Bondy [1] for graph terminologies not given here. It is known that κ (Γn ) = n − 1, κ1 (Γn ) = 2n − 4 for n 3, κ 2 ( S n ) = 6(n − 3) for n 4, see [2,3,11] for the detail. In this paper, we shall determine κ2 (Γn ), λ2 (Γn ) and κ 2 (Γn ), λ2 (Γn ). 2. Preliminaries We give some useful properties of Γn , which can be found in [2–5]. Property 1. If n is a leaf of G (T ), then Γn can be decomposed into n disjoint copies of Γn−1 , say Γn1−1 , Γn2−1 , . . . ,
Γnn−1 , where Γni−1 is an induced subgraph by vertex set {( p 1 p 2 · · · pn−1 i ) | ( p 1 p 2 · · · pn−1 ) ranges over all permutations of {1, . . . , i − 1, i + 1, . . . , n}}. We denote this decomposition by Γn = Γn1−1 ⊕ Γn2−1 ⊕ · · · ⊕ Γnn−1 (that is, G n = G n1−1 ⊕ G n2−1 ⊕ · · · ⊕ G nn−1 and S n = S n1−1 ⊕ S n2−1 ⊕ · · · ⊕ S nn−1 , respectively).
Property 2. Let (tn) ∈ T be a pendant edge of G (T ). For any vertex u of Γni−1 , u (tn) is the unique neighbor of u
outside Γni−1 , is called the out neighbor of u, written u , and any two distinct vertices of Γni−1 have different out neighbors. Hence, there are exactly (n − 2)! independent j edges between Γni−1 and Γn−1 if i = j, that is, | N (Γni−1 ) ∩ j
V (Γn−1 )| = (n − 2)! if i = j. Property 3. G n has girth 4 and S n has girth 6. The next useful result can be found in [8]. Lemma 2.1. Let T be a set of transpositions from S ym(n) and let g and h be elements of T . If the transposition generating
graph of T contains no triangles, then g and h have exactly one common neighbor in Cay(Sym(n), T ) if gh = hg, and exactly two common neighbors otherwise. Clearly, for any two distinct transpositions (i j ) and (kl), (i j )(kl) = (kl)(i j ) if and only if {i , j } ∩ {k, l} = ∅. It is not difficult to see that if C 4 = x1 x2 x3 x4 is a 4-cycle of Γn , then there must exist two transpositions (i j ) and (kl) such that {i , j } ∩ {k, l} = ∅ and x2 = x1 (i j ), x3 = x2 (kl), x4 = x3 (i j ) = x1 (kl). In fact, there exists an automorphism of Γn , say ψ ∈ Aut(Γn ), such that ψ(x1 ) = e since Γn is vertex transitive, furthermore, C 4 = ψ(x1 x2 x3 x4 ) is also a 4-cycle of Γn . Let g = ψ(x2 ), h = ψ(x4 ), clearly, g and h are two transpositions of T . By Lemma 2.1, we have that gh = hg since they have two common neighbors. That is, (i j ) = ψ −1 ( g ), (kl) = ψ −1 (h). By the above argument, we have the following corollary. Corollary 2.2. For any two distinct vertices of Γn , then have at most two common neighbors if they have any. Lemma 2.3. (See [2,3,11].) κ (Γn ) = n − 1, κ1 (Γn ) = 2n − 4 for n 3, κ 2 ( S n ) = 6(n − 3) for n 4. Lemma 2.4. Let n 4, G n = G n1−1 ⊕ G n2−1 ⊕ · · · ⊕ G nn−1 . If C 4
is a 4-cycle of G ni −1 , then N G n (C 4 ) is an R 2 -vertex cut of G n .
Proof. If n = 4, then G n = Cay(Sym(4), {(12), (23), (34)}). Without loss of generality, we assume C 4 = x1 x2 x3 x4 x1 , where x1 = (1234), x2 = x1 (12) = (2134), x3 = x2 (34) = (2143), x4 = x3 (12) = x1 (34) = (1243). Clearly, C 4 is 2-regular. Next we show that G 4 − V (C 4 ) ∪ N G n (C 4 ) has minimum degree at least 2. Since G 4 is 3-regular, it is sufficient to show that for any two vertices of N G n (C 4 ) have no common neighbor in G 4 − V (C 4 ) ∪ N G n (C 4 ). Note that G 4 is a bipartite graph. Then the neighbors of any two adjacent vertices of C 4 have no common neighbor in G 4 − V (C 4 ) ∪ N G n (C 4 ). Now we show that the neighbors of x2 and x4 have no common neighbor in G 4 − V (C 4 ) ∪ N G n (C 4 ). Suppose there exists a common neighbor. So there are two transpositions (i j ) and (kl) of {(12), (23), (34)} such that x2 (23)(i j ) = x4 (23)(kl), which is impossible. So is x1 and x3 since G 4 is vertex-transitive. That is N G n (C 4 ) is a R 2 -vertex cut of G 4 . Let n 5. Without loss of generality, we assume that C 4 = x1 x2 x3 x4 x1 ⊂ G nn−1 , where x1 is a vertex of G nn−1 and x2 = x1 (i j ), x3 = x2 (kl), x4 = x3 (i j ) = x1 (kl), and i < j < k < l (note that {i , j } ∩ {k, l} = ∅). Let x1 = ( p 1 · · · p i · · · p j · · · pk · · · pl · · · n). Thus we have x2 = ( p 1 · · · p j · · · p i · · · pk · · · pl · · · n), x3 = ( p 1 · · · p j · · · p i · · · pl · · · pk · · · n), x4 = ( p 1 · · · p i · · · p j · · · pl · · · pk · · · n). Since n is a leaf of G (T ), there has exactly one transposition contains n in T , say (tn). Clearly, if (tn) ∈ / ∈{(in), ( jn), (kn), (ln)}, then all the out vertices x1 = x1 (tn), x2 = x2 (tn), x3 = x3 (tn), x4 = x4 (tn) are in G nt −1 which clearly induce a 4-cycle of G nt −1 . Otherwise, without loss of generality, we assume t = i. We have that {x1 , x4 } ⊂ V (G ni −1 ) and
{x2 , x3 } ⊂ V (G n−1 ). Clearly, v ∈ V (G nt −1 ) or V (G n−1 ), we j
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have | N G t
n −1
( v ) ∩ {x1 , x2 , x3 , x4 }| 1 since G nt −1 contains
no triangle and for any two distinct vertices have at most two common neighbors (Corollary 2.2). That is, every vern tex of s=2 G ns −1 has degree at least n − 3. Combining the above arguments, we have that N G n (C 4 ) is an R 2 -vertex cut of G n for n 4. 2 Remark 1. Let P 3 be a path with order 3. If P 3 ⊂ C 4 ⊂ G ni −1 , by an arguments similar to that of Lemma 2.4, we can see that N ( P 3 ) is a κ2 -vertex cut of G n for n 4. Furthermore, ω(C 4 ) and ω( P 3 ) are the R 2 -edge cut and λ2 -edge cut, respectively. 3. R 2 -(edge)connectivity of Γn In this section we shall derive
κ 2 (Γn ) and λ2 (Γn ).
j
G n−1 − F j if n = 4 or 5. Combining the above arguments, we have that G n − F is connected. But F is an R 2 -vertex cut, a contradiction. Case 2. There exists an integer i such that | F i | 2n − 6. Since | F | 4n − 13, there is exactly one integer i such that | F i | 2n − 6. Without loss of generality, we assume | F 1 | 2n − 6. That is, | F j | 2n − 7 for j = 1. By an argun ment similar to that of Case 1, we have that l=2 (G ln−1 −
¯ F l ) is a connected subgraph of G n − F , denoted by G. Since G n − F is disconnected, there must be a component of G n − F in G n1−1 , say C . Note that g (G n ) = 4, there must exist a 4-vertex path in C , say P 4 = u 0 u 1 u 2 u 3 . By Corollary 2.2, we have | N G 1 ( P 4 )| 4(n − 2) − 6 − 2. Let x ∈ NG1
N −1
Theorem 3.1. For n 4, girth of Γn .
κ 2 (Γn ) = g (n − 3), where g is the
Proof. Wan and Zhang in [11] have shown that κ 2 ( S n ) = g (n − 3) = 6(n − 3). Next we shall show that κ 2 (G n ) = g (n − 3) = 4(n − 3). Note that if G (T ) is not a star, then there exists two non-adjacent edges in G (T ), say (i j ) and (kl). Let C 4 = e 0 e 1 e 2 e 3 be a cycle of G ni −1 for an integer i, where e 1 = e 0 (i j ), e 2 = e 1 (kl), e 3 = e 2 (i j ). By Corollary 2.2, it is easy to see that | N G n (C 4 )| = 4(n − 3). By Lemma 2.4, we have that κ 2 (G n ) 4(n − 3). Next we shall show that κ 2 (G n ) 4(n − 3). Suppose by way of contradiction that κ 2 (G n ) 4(n − 3) − 1. Let F be a minimum R 2 -vertex cut, then | F | 4(n − 3) − 1. Denote by F i the vertex set F ∩ V (G iN −1 ) in the following arguments. Case 1. | F i | 2n − 7 for all i. We first show that G ni −1 − F i is connected for all i. If G ni −1 − F i is disconnected for an integer i, then by
Property 2, F i is an R 1 -vertex cut of G ni −1 , so we have | F i | κ 1 (G n−1 ) = κ1 (G n−1 ) = 2n − 6. But | F i | 2n − 7. That is, G ni −1 − F i is connected for all integers i. We claim that any two distinct integers i and j, then j G ni −1 − F i is connected with G n−1 − F j . Otherwise, suppose
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N −1
( P 4 ), then at least one of x and x in F . Clearly,
{u 0 , u 1 , u 2 , u 3 } ⊂ F . By Property 2, | F | | N G 1 |{u 0 , u 1 , u 2 , u 3 }| 4n − 12, a contradiction. 2
N −1
( P 4 )| +
Theorem 3.2. λ2 (Γn ) = g (n − 3) for n 4. Proof. By an argument similar to that of Theorem 3.1. We have that λ2 (Γn ) = g (n − 3). 2 4. 2-extra(edge)connectivity Γn In this section, we discuss the 2-extra(edge)connectivity of Γn . Theorem 4.1. κ2 (G n ) = 3n − 8, κ2 ( S n ) = 3n − 7 for n 4. Proof. We first show that κ2 (G n ) = 3n − 8. Note that if G (T ) is not a star, there exists two disjoint edges in G (T ), say (i j ) and (kl). Let P = x0 x1 x2 where x1 = x0 (i j ), x2 = x1 (kl). By Corollary 2.2, we have | N ( P )| = 3n − 8. Clearly, Remark 1 implies that κ2 (G n ) 3n − 8. Next we shall show that κ2 (G n ) 3n − 8. Suppose by the way of contradiction that κ2 (G n ) 3n − 9. Let F be a minimum κ2 -vertex cut of Γn , then | F | 3n − 9. Let F i = F ∩ V (G ni −1 ). Since | F | 3n − 9, there are at most two distinct integers k and l such that F k n − 2 and F l n − 2.
j
that G ni −1 − F i is disconnected from G n−1 − F j for two disj tinct integers i and j, that is, N (G ni −1 − F i ) ∩ V (G n−1 ) ⊂ j
F j . Note that | N (G ni −1 ) ∩ V (G n−1 )| = (n − 2)!. We have that (n − 2)! − (2n − 7) (n − 2)! − | F i | | F j | 2n − 7, which is impossible for n 6. Thus G ni −1 − F i is conj
nected to G n−1 − F j for n 6. On the other hand, if n = 4 or 5, the above inequalities become equalities. That is, | F i | = | F j | = 2n − 7. Since | F | 4n − 13 and n 4, there must exist an integer k such that F k = ∅. Note that | N (G nk −1 ) ∩ V (G ni −1 − F i )| (n − 2)! − (2n − 7) = 1 when
n = 4 or 5, we have that G nk −1 − F k (emptyset) is connected to G ni −1 − F i . Similarly, we have that G nk −1 − F k (emptyset) j
is connected to G n−1 − F j . Thus G ni −1 − F i is connected to
Case 1. | F i | n − 3 for all i. Since
κ (G n−1 ) = n − 2, we have that all G ni −1 − F i
are connected. Let G nk −1 − F k and G ln−1 − F l be two distinct subgraphs of G n − F . Since | N (G nk −1 ) ∩ V (G ln−1 )| =
(n − 2)!, we have that | N (G nk −1 − F k ) ∩ V (G ln−1 − F l )| (n − 2)! − 2(n − 3) > 0 for n 5. That is, for any two distinct subgraphs G nk −1 − F k and G ln−1 − F l of G n − F are connected if n 5. Thus G n − F is connected for n 5, a contradiction. In particular, if n = 4, then | F | 3. But κ1 (G n ) = 2n − 4 = 4 > 3 = 3n − 9, a contradiction. Case 2. There exists an integer i such that | F i | n − 2.
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Let I = {i || F i | n − 2}. Clearly, | I | 2. Note that | F j | n − 3 if j ∈ / I . By an argument similar to that of Case 1, we j know that G¯ = j ∈/ I (G n−1 − F j ) is connected. Subcase 2.1. | I | = 1. Without loss of generality, we assume | F 1 | n − 2. If G n1−1 is disconnected, then there exists a component of
G n − F contained in G n1−1 − F 1 , say C , and | V (C )| 3. Let P = u 0 u 1 u 2 be a 3-vertex connected subgraph of C . By Corollary 2.2, we have that | N G 1 ( P )| 3(n − 2) − 4 − 1 = n −1
3n − 11. Note that for any u ∈ N G 1 ( P ), at least one n −1
of {u , u } in F . By Property 2, {u 0 , u 1 , u 2 } ∩ N G 1 ( P ) =
∅. Therefore, | F | | N G 1 ( P )| + n−1
3n − 9, a contradiction. If
G n1−1
|{u 0 , u 1 , u 2 }|
n −1
3n − 8 >
is connected, then note that
(n − 1)! > 3n − 9 for n 4, that is, N (G n1−1 − F 1 )∩ V (G¯ ) = ∅. Thus G n − F is connected, a contradiction. Subcase 2.2. | I | = 2. Assume I = {k, l}. We consider the following three cases. If G nk −1 − F k and G ln−1 − F l are connected. Note that
k k l k ¯ |N (G n−1 − F k )∩ V (G )| | N (G n−1 )|−| V (G n−1 )∩ N (G n−1 )|− | j =l F j | (n − 1)! − (n − 2)! − (2n − 7) > 0 for n 5, that is, G nk −1 − F k is connected to G¯ for n 5, and so is G ln−1 − F l . Thus, G n − F is connected for n 5. Clearly, if n = 4, then | F | 3. But κ1 (G n ) = 2n − 4 = 4 > 3 = 3n − 9 κ2 (G n ), a contradiction. Thus, G n − F is connected for n 4, a contradiction. If exactly one of G nk −1 − F k and G ln−1 − F l is connected. Assume G nk −1 − F k is connected and G ln−1 − F l is discon-
nected. By an argument similar to that of above para¯ That is, graph, we have that G nk −1 − F k is connected to G.
¯ (i) If y = x, then P = xy is nected and is connected to G. connected to G¯ (since F is a κ2 -cutset). For any component C of G nk −1 − F k with order at least 3. By an argument simi-
¯ lar to that of Subcase 2.1, we have that C is connected to G, thus G n − F is connected. (ii) Suppose that y is contained in V (G nk −1 − F k − {x}) or V (G¯ ). Firstly, if y is in V (G¯ ), it
is easy to see that every component of G nk −1 − F k − {x}
¯ If y is in G k − F k − {x}, assume is connected to G. n−1
y ∈ V (C 1 ), where C 1 is component of G nk −1 − F k − {x}. Let P = x0 x1 x2 be a path of C 1 . Since G n contains no odd cycle, at least one of {x0 , x1 , x2 } is not contained in F l (since F l = N G l ( y )). We have | F | | F k | + | F l | + 1 = 3n − 8 > n−1
¯ There3n − 9, a contradiction. That is, y is connected to G. fore, G nk −1 − F k − {x} contains a component of G n − F . By an argument similar to that of Subcase 2.1, we have G n − F is connected. Combining Case 1 with Case 2, we have that κ2 (G n ) = 3n − 8. Note that the girth of S n is 6. By an argument similar to that of G n , we have κ2 ( S n ) = 3n − 7. 2 Theorem 4.2. λ2 (Γn ) = 3n − 7 for n 4. Proof. Theorem 4.2 follows immediately from Remark 1 and an argument similar to that of Theorem 4.1. 2 Remark 2. In [5], Cheng and Lipták determined the R 2 connectivity of Γn independently. Acknowledgements The authors wish to thank Prof. Eddie Cheng and the referees for their very useful suggestions and their kind help.
j
j =l G n−1 − F j is connected. Finally, by an argument similar to that of Subcase 2.1, we have that G n − F is connected, a contradiction. If G nk −1 − F k and G ln−1 − F l are disconnected, then note that κ1 (G n−1 ) = 2n − 6 and κ (G n−1 ) = n − 2, so it is not difficult to see that each of G nk −1 − F k and G ln−1 − F l contains exactly one isolated vertex, say x and y, such that x ∈ V (G nk −1 − F k ), y ∈ V (G ln−1 − F l ).
We now assume that G nk −1 − F k − {x} and G ln−1 − F l − { y } are connected, then since | F k | n − 2, | F l | n − 2 and | F | 3n − 9, there exists an integer j such that F j = ∅. Note that (n − 2)! > 2n − 6 for n 5, it is not difficult j to see that | N (G nk −1 − F k − {x}) ∩ V (G n−1 )| > 0. That is, j
G nk −1 − F k − {x} is connected to G n−1 , so is G ln−1 − F l − { y }. We have that G nk −1 − F k − {x} and G ln−1 − F l − { y } con-
¯ Thus we get a component (G n [{x, y }]) of G n − F nect to G. having order at most 2, but F is a κ2 -vertex cut, a contradiction. In particular, if n = 4, then 3n − 9 = 3 < κ1 (G n ), a contradiction. If G nk −1 − F k − {x} (or G ln−1 − F l − { y }) is disconnected, then | F k ∪ {x}| κ1 (G n−1 ) = 2n − 6. Clearly, | F k | = 2n − 7, | F l | = n − 2. It is easy to see that G ln−1 − F l − { y } is con-
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