Super Rk-vertex-connectedness

Applied Mathematics and Computation 339 (2018) 812–819

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Super Rk -vertex-connectednessR Xiaomin Hu, Yingzhi Tian, Jixiang Meng∗ College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China

a r t i c l e

i n f o

Keywords: Conditional connectivity Rk -vertex-connectivity Cayley graphs Wheel graphs Super Rk -vertex-connectedness

a b s t r a c t For a graph G = (V, E ), a subset F ⊆ V(G) is called an Rk -vertex-cut of G if G − F is disconnected and each vertex u ∈ V (G ) − F has at least k neighbours in G − F . The Rk -vertexconnectivity of G, denoted by κ k (G), is the cardinality of a minimum Rk -vertex-cut of G. In this paper, we further study the Rk -vertex-connectivity by introducing the concept, called super Rk -vertex-connectedness. The graph G is called super Rk -vertex-connectedness if, for every minimum Rk -vertex-cut S, G − S contains a component which is isomorphic to a certain graph H, where H is related to the graph G and integer k. For the Cayley graphs generated by wheel graphs, H is isomorphic to K2 when k = 1 and H is isomorphic to C4 when k = 2. In this paper, we show that the Cayley graphs generated by wheel graphs are super R1 -vertex-connectedness and super R2 -vertex-connectedness. Our studies generalize the main result in [8]. © 2018 Elsevier Inc. All rights reserved.

1. Introduction We follow [1] for graph-theoretical terminology and notation not defined here. Let G = (V, E ) be a simple and connected graph. It is known that the underlying topology of an interconnection network can be modeled by a graph G. The connectivity is an important indicator of the reliability and fault tolerability of a network. A vertex cut of a graph G is a set S ⊆ V(G) such that G − S has more than one component. The connect ivit y of a graph G, denoted by κ (G), is the cardinality of a minimum vertex cut of G. It is known that κ (G) ≤ δ (G), where δ (G) is the minimum degree of G. A graph G is said to be maximally connected, if κ (G ) = δ (G ). A graph G is said to be super connected if every minimum vertex cut of G isolates a vertex. However, this parameter has a deficiency. That is, it tacitly assume that all vertices adjacent to the same vertices of G could fail at the same time, which is highly unlikely for large-scale systems. To compensate this shortcoming, Harary [5] introduced the concept of conditional connectivity. The Rk -vertex-connectivity follows this trend. For a simple connected graph G = (V, E ), a subset F ⊆ V is called an Rk − vertex − set of G if each vertex u ∈ V − F has at least k neighbours in G − F , in other words, if the minimum degree δ of the survival graph satisfies δ (G − F ) ≥ k. An Rk − vertex − cut of a connected graph G is an Rk -vertex-set F of G such that G − F is disconnected. The Rk − vertex − connect ivit y of G, denoted by κ k (G), is the cardinality of a minimum Rk -vertex-cut of G. Cayley graphs have a lot of properties which are desirable in an interconnection network [6,7]: vertex symmetry makes it possible to use the same routing protocols and communication schemes at all nodes; hierarchical structure facilitates recursive constructions; high fault tolerance implies robustness, among others. In recent years, the problem of studying Rk vertex-connectivity for some special classes of Cayley graphs has received a lot of attention (see [2–4,8–12]). On the other R ∗

The research is supported by NSFC (No.11531011). Corresponding author. E-mail address: [email protected] (J. Meng).

https://doi.org/10.1016/j.amc.2018.07.012 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.

X. Hu et al. / Applied Mathematics and Computation 339 (2018) 812–819

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hand, as far as we know, there are no results on characterizing all minimum Rk -vertex-cuts of a graph. This motivates the authors to introduce a new concept, called super Rk -vertex-connectedness. The graph G is called super Rk − vertex − connectedness if, for every minimum Rk -vertex-cut S, G − S contains a component which is isomorphic to a certain graph H, where H is related to the graph G and integer k. The concept of super Rk -vertex-connectedness is a more enhanced definition than the Rk -vertex-connectivity. Since if every minimum Rk -vertexcut of G is classified, then Rk -vertex-connectivity of G follows easily. In [8], Tu et al. showed that κ 1 (W Gn ) = 4n − 6 and κ 2 (W Gn ) = 8n − 18, where WGn is the Cayley graphs generated by wheel graphs and n ≥ 5. In this paper, we characterize all the minimum R1 -vertex-cuts and R2 -vertex-cuts of WGn . The rest of this paper is organized as follows. In Section 2, we introduce some definitions and results that will be useful in our arguments. Section 3 presents the main work of the paper. 2. Preliminaries Let  be a group and S be a subset of  \{1 }, where 1 is the identity of  . Cayley digraph Cay( , S) is the digraph with vertex set  and arc set {(g, g · s): g ∈  , s ∈ S}. We say that arc (g, g · s) has label s. In particular, if S−1 = S, then Cay( , S) is an undirected graph, called Cayley graph. In this paper, we consider Cayley graph Cay(Sym(n ), T ), where Sym(n) is the symmetric group on {1, 2, . . . , n} and T is a set of transpositions on Sym(n). Let G(T ) be the graph on n vertices {1, 2, . . . , n} such that there is an edge (i, j) in G(T ) if and only if transposition (i j ) ∈ T . The graph G(T ) is called the transposition generating graph of Cay(Sym(n ), T ). If G(T ) is a tree, Cay(Sym(n ), T ) is denoted by  n . If G(T ) is a path,  n is the bubble sort graph, denoted by Bn . If G(T ) is a star,  n is the star graph Sn . If G(T ) is a complete graph, Cay(Sym(n ), T ) is denoted by C n . If G(T ) is a unicyclic graph, Cay(Sym(n ), T ) is denoted by UGn . If G(T ) is a cycle, UGn is the modified bubble sort graph MBn . If G(T ) is a wheel Wn , which is a graph with n vertices, formed by connecting a single vertex to all vertices of an (n − 1 )-cycle, then we use nation WGn to denote the graph Cay(Sym(n ), T ). For a graph G, NG (u) is the neighbour set of u in G, and dG (u ) = |NG (u )| is the degree of vertex u in G. For a vertex set S ⊆ V(G), NG (S ) = {∪u∈S NG (u )} − S, G[S] is the subgraph of G whose vertex set is S and whose edge set consists of all edges of G which have both ends in S. We say that G[S] is a graph induced by S. When graph G is obvious in the context, we omit the subscript G and use N(u), d(u) and N(S) to denote NG (u), dG (u) and NG (S), respectively. Sometimes, we use a graph itself to represent its vertex set, for instance, N(G) means N(V(G)). For a given permutation p = p1 p2 . . . pn and a transposition (ij) in Sym(n), p(ij) is obtained from p by swapping pi and pj , that is p(i j ) = p1 p2 . . . pi−1 p j pi+1 . . . p j−1 pi p j+1 . . . pn . The next result is due to Cheng and Lipták. Theorem 2.1. [2] Let G be the Cayley graph obtained from a generating graph G(T ) on {1, 2, . . . , n} with m edges, where n ≥ 3. Suppose T is a subset of V(G). (1) Graph G does not have the subgraph K2,4 . (2) If G(T ) does not contain a triangle, then G does not have the subgraph K2,3 . (3) The edges of a 4-cycle C4 in G correspond to either the following transpositions in the given order: (ab), (cd), (ab), (cd) (say C4 is a Type A 4-cycle) or (ab), (bc), (ab), (ac) (say C4 is a Type B 4-cycle), where a, b, c, d are distinct. (4) If n ≥ 4, |T | ≤ 2m − 2 and G(T ) is not K4 − e, then G − T is either connected or it is disconnected with at most six vertices in total in the components excluding the largest one. (5) G is m-regular, bipartite (with the two parts of bipartition containing even and odd permutations, respectively). (6) κ (G ) = m. The next lemma characterizes the properties of modified bubble sort graph MBn . Lemma 2.2. Let n be an integer greater than 3 and T be a subset of V(MBn ) satisfies |T | ≤ 2n − 2. (1) (2) (3) (4) (5) (6) (7) (8)

MBn does not have subgraph K2,3 . Every 4-cycle of MBn must be Type A 4-cycle. If MBn − T is disconnected, then the order of each component H of MBn − T can not be 3, 4, 5 and 6. MBn − T has at most two isolated vertices. If MBn − T has one or two isolated vertices, then every non-singleton component of MBn − T has at least seven vertices. If MBn − T has exactly two isolated vertices, then |T | = 2n − 2. If MBn − T has a component H ∼ = K2 , then |T | = 2n − 2 and every component H = H of MBn − T has at least seven vertices. If MBn − T is disconnected, then one of the following is true: (i) MBn − T has exactly two components, one of which is a singleton. (ii) MBn − T has exactly two components, one of which is K2 , and |T | = 2n − 2. (iii) MBn − T has exactly three components, two of which are singletons, and |T | = 2n − 2.

Proof. (1) The result is a consequence of Theorem 2.1(2). (2 ) If MBn has a Type B 4-cycle, then G(T ) has a triangle.

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X. Hu et al. / Applied Mathematics and Computation 339 (2018) 812–819

Fig. 1. Illustration for Lemma 2.2(3).

v

u

(23) (41)

(23) (41)(12) (34)

x

y

u

v



z

t

w Fig. 2. Illustration for Lemma 2.2(4).

(3 ) Each connected subgraph H of MBn with |V(H)| ∈ {3, 4, 5} is depicted in Fig. 1. If |V (H )| = 6, since H is bipartite and it does not contain K2,3 as a subgraph, it is easy to check that there are 12 graphs by discussing the maximum degree of H. According to Theorem 2.1(5) and Lemma 2.2(2), we have |N (H )| > 2n − 2. (4 ) Suppose u, v, w are isolated vertices of MBn − T . Since MBn is n-regular and it does not have subgraph K2, 3 , we have |T | = 2n − 2. We can assume T = N (u ) ∪ N (v ) and N (w ) ⊆ T . By Lemma 2.2(1), |N (w ) ∩ N (u )| ≤ 2 and |N (w ) ∩ N (v )| ≤ 2. So |N (w )| ≤ 4 and then n = 4. Recall that MBn has only Type A 4-cycle, we can assume u = u(12 ), v = u(34 )(see Fig. 2). By Lemma 2.2(1), |N (w ) ∩ {u , v }| ≤ 1. Hence, |N (w ) ∩ {x, y}| ≥ 1 and |N (w ) ∩ {z, t }| ≥ 1. Again by MBn has only Type A 4-cycle, |N (w ) ∩ {u , v }| = 0 and then N (w ) = {x, y, z, t }. We can obtain that y = w(23 ) and t = w(23 ), a contradiction. (5 ) Let {u} and H be singleton and non-singleton component of MBn − T , respectively. By Lemma 2.2(3), |V(H)| = 3, 4, 5, 6. If |V (H )| = 2, assume H = K2 = (v, w ). Since MBn is n-regular and |T | ≤ 2n − 2, we have T = N (H ) and N(u)⊆T. Because d (u ) = n, we see that N (u ) ∩ {N (v )\{w}} = ∅ and N (u ) ∩ {N (w )\{v}} = ∅. It follows that MBn has a cycle C5 , a contradiction. (6 ) Combining Theorem 2.1(5) with Lemma 2.2(1), the result follows. (7 ) By Lemma 2.2(3) and Lemma 2.2(5), |V(H )| = 1, 3, 4, 5, 6. If |V (H )| = 2, assume H = (u, v ) and H = (w, z ). Clearly, T = N (H ) = N (H ) and |T | = 2n − 2. By Lemma 2.2(1), N (w )\{z} = N (u )\{v}. So {N (w )\{z}} ∩ {N (v )\{u}} = ∅. Similarly, {N (w )\{z}} ∩ {N (u )\{v}} = ∅. Hence MBn has a cycle C5 , a contradiction. (8) Applying Theorem 2.1(4), Lemma 2.2(3),(4),(5),(6),(7), the result follows.  Now we introduce the hierarchical structure and some interesting results about WGn . Recall that the transposition generating graph G(T ) of WGn is a wheel Wn . Without loss of generality, assume V (Wn ) = {1, 2, . . . , n} and E (Wn ) = {(i, n ) : 1 ≤ i ≤ n − 1} ∪ {(i, i + 1 ) : 1 ≤ i ≤ n − 2} ∪ {(n − 1, 1 )}. If deleting the center vertex of Wn , then the resulting graph G(T ) = G(T ) − n is a cycle on vertex set {1, 2, . . . , n − 1}. Hence, the wheel-transposition graph WGn can be decomposed into n copies of MBn−1 as follows: For each i ∈ {1, 2, . . . , n}, let MBin−1 be the subgraph of WGn ∼ induced by vertex set {x1 x2 . . . xn−1 i|x1 x2 . . . xn−1 ranges over all permutations of {1, 2, . . . , n}\{i}}. Hence MBi = MBn−1 . Let n−1

[i, j] = {l : i ≤ l ≤ j} for i < j. We use MB[i, j] to denote the subgraph of WGn induced by vertices {u : u ∈ V (MBln−1 ), l ∈ [i, j]}. It can be seen that for any i ∈ [1, n] and j ∈ [1, n − 1], each vertex u ∈ V (MBin−1 ) has a unique neighbour u outside of

MBin−1 such that the edge (u, u ) has label (jn). We call u is the outside neighbour of u. Let N out (u ) = NWGn (u )\NMBi For a vertex set S ⊆

V (MBin−1 ),

let

N out (S )

=

{Nout (u )|u

n−1

∈ S}. It is easy to prove the following Lemma.

( u ).

j

Lemma 2.3. [8] (1) For any i = j, each vertex in MBin−1 has a unique outside neighbour in MBn−1 .

(2) Let [MBin−1 , MBn−1 ](|[MBin−1 , MBn−1 ]| ) be (the number of) edges with one end-vertex in MBin−1 and the other one in j

j

j MBn−1

j for any i = j. Then [MBin−1 , MBn−1 ] is exactly a perfect matching of the graph induced by vertices j i i and hence |[MBn−1 , MBn−1 ]| = |V (MBn−1 )| = (n − 1 )!. (3) Any k( ≥ 3) vertices from different copies MBin−1 (1 ≤ i ≤ k) cannot constitute a cycle of WGn with

j

V (MBin−1 ) ∪ V (MBn−1 ),

length k.

Lemma 2.4. Assume Ht is a component of MBtn−1 − Ft for each t ∈ [i, j − 1], MBsn−1 − Fs is connected for each s ∈ [j, l]. If there is an edge between Ht and MB[ j, l] − F for each t ∈ [i, j − 1], and an edge between MBsn−1 − Fs and MBsn+1 − Fs+1 for each s ∈ −1 [ j, l − 1], then the graph induced by vertices {∪t=i V (Ht )} ∪ V (MB[ j, l] − F ) is connected. j−1

X. Hu et al. / Applied Mathematics and Computation 339 (2018) 812–819

i M Bn−1

u

j

j i M Bn−1 M Bn−1 (pn) x u

l M Bn−1

M Bn−1 (pn) x

(qn) v

v

y

(pn)

(pq)

(pq)

y

(qn)

815

j

i M Bn−1 M Bn−1 (pn) x u

(st) v

(2)

(1)

(st) y

(pn) (3)

Fig. 3. In (1), v = u( pq ), but p and q may be not adjacent in G(T ).

j i l t M Bn−1 M Bn−1 M Bn−1 MB

n−1

s · · · M Bn−1

j i l t M Bn−1 M Bn−1 M Bn−1 MB

n−1

(tn) (qn) u (pq)

v

(qn)

(sn) u

(pn) (pq)

(pq)

(pq)

(qn)

v (sn)

(pn)

(pq)

(qn) (pn)

s · · · M Bn−1

(pn)

(qn) (pn)

(tn)

(2)

(1) Fig. 4. Illustration for Lemma 2.5(2).

Lemma 2.5. Let u, v and w be three arbitrary vertices in MBin−1 , then (1) any path u, x, y, v is isomorphic to one of graphs in Fig. 3, where x ∈ Nout (u) and y ∈ N out (v ). (2) If the graph induced by N out (u ) ∪ N out (v ) has at least one edge, then it is isomorphic to one of graphs in Fig. 4. (3) If there is at least one edge with end-vertex in Nout (u), N out (v ) and N out (w ), respectively, then the graph induced by N out (u ) ∪ N out (v ) ∪ N out (w ) is isomorphic to one of graphs in Fig. 5. Proof. (1) We can assume x = u( pn ). Clearly, either y = x(qn ), or y = x( pq ), or y = x(st ), where p, q, s, t, n are distinct integers. Then by analysing the location of the entry i in the permutation y, the result follows. (2) Since WGn is bipartite, Nout (u) and N out (v ) are independent sets. Because the graph induced by N out (u ) ∪ N out (v ) has at least one edge, we can assume v = u( pq ). By discussing whether p and q are adjacent in G(T ) or not, the result follows. (3) We can assume either v = u( pq ) and w = v(qs ), or v = u( pq ) and w = v(st ). If |E (W Gn [u, v, w] )| = 0, then p and q, q and s, s and t are not adjacent in G(T ). Hence the graph induced by N out (u ) ∪ N out (v ) ∪ N out (w ) is isomorphic to one of graphs in Fig. 5(1) and Fig. 5(2). If |E (W Gn [u, v, w] )| = 1, we can assume ( p, q ) ∈ E (G(T )), q and s, s and t are not adjacent in G(T ). Hence the graph induced by N out (u ) ∪ N out (v ) ∪ N out (w ) is isomorphic to one of graphs in Fig. 5(3) and Fig. 5(4). If |E (W Gn [u, v, w] )| = 2, then {( p, q ), (q, s ), (s, t )} ⊆ E (G(T )) and the graph induced by N out (u ) ∪ N out (v ) ∪ N out (w ) is isomorphic to one of graphs in Fig. 5(5) and Fig. 5(6).  3. Main results In this section, we will prove that WGn is super R1 -vertex-connectedness and super R2 -vertex-connectedness. Lemma 3.1. For any integer n ≥ 4, κ 1 (W Gn ) ≤ 4n − 6. Proof. By Theorem 2.1(5), WGn is (2n − 2 )-regular. Let T = N ({u, v} ) and w ∈ V (W Gn ) − T ∪ {u, v}, where (u, v ) is an arbitrary edge of WGn . Since WGn is bipartite, |N (w ) ∩ {N (u )\{v}}| = 0 if |N (w ) ∩ {N (v )\{u}}| ≥ 1. Similarly, if |N (w ) ∩ {N (u )\{v}}| ≥ 1, then |N (w ) ∩ {N (v )\{u}}| = 0. Therefore, |N (w ) ∩ T | ≤ 2n − 3 < d (w ) and then w is not an isolated vertex of W Gn − T . It follows that T is an R1 -vertex-cut and then κ 1 (W Gn ) ≤ |T | = 4n − 6.  Theorem 3.2. For any integer n ≥ 4, WGn is super R1 -vertex-connectedness.

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X. Hu et al. / Applied Mathematics and Computation 339 (2018) 812–819

j i l t M Bn−1 M Bn−1 M Bn−1 MB

n−1

s ··· M Bn−1

k M Bn−1

j i l t M Bn−1 M Bn−1 M Bn−1 MB

s ··· M Bn−1

k M Bn−1

j i l t s M Bn−1 M Bn−1 M Bn−1 ··· M Bn−1 M Bn−1

k M Bn−1

n−1

u

u

v

v

w

w (2)

(1)

j i l t s M Bn−1 M Bn−1 M Bn−1 ··· M Bn−1 M Bn−1

k M Bn−1

u

u

v

v

w

w (4)

(3)

j i l t s M Bn−1 M Bn−1 M Bn−1 ··· M Bn−1 M Bn−1

k M Bn−1

j i l t s M Bn−1 M Bn−1 M Bn−1 ··· M Bn−1 M Bn−1

u

u

v

v

w

k M Bn−1

w (6)

(5) Fig. 5. Illustration for Lemma 2.5(3).

Proof. For n = 4, we defer the discussion to Appendix. We assume that n ≥ 5 in the following. Let F be a minimum R1 vertex-cut of WGn and Fi = F ∩ V (MBin−1 ). By Lemma 3.1, |F | ≤ 4n − 6. We will show that F is the vertex set N ({u, v} ), where u and v are two end-vertices of an edge. Claim 1. |Fi | ≥ 2 for each i ∈ [1, n]. If |Fi | = 0 for some i ∈ [1, n], assume i = 1. Since MB1n−1 is connected and every vertex in MBtn−1 − Ft (t ∈ [2, n]) is adjacent to one vertex in MB1n−1 , we see that W Gn − F is connected. Hence |Fi | ≥ 1 for each i ∈ [1, n]. If |Fi | = 1 for some i ∈ [1, n], assume i = 1 and F1 = {u1 }. By a similar discuss as above, we see that the graph induced by V (W Gn ) − F ∪ N out (u1 ) is connected. Recall that F is an R1 -vertex-cut and Nout (u1 ) is an independent set. So every vertex in N out (u1 ) − F is adjacent to some vertex in V (W Gn ) − F ∪ N out (u1 ). Hence W Gn − F is connected, a contradiction. Claim 2. 1 ≤ |J| ≤ 2, where J = {i|MBin−1 − Fi is disconnected}.

Since (n − 1 )! − |F | ≥ 8(n − 2 ) − (4n − 6 ) ≥ 10, there are at least ten edges between the pair MBin−1 − Fi and MBn−1 − Fj for any distinct i and j. If |J| = 0, then W Gn − F = MB[1, n] − F is connected by Lemma 2.4. If |J| ≥ 3, by Theorem 2.1(6) and Claim 1, we have in=1 |Fi | ≥ 3(n − 1 ) + 2(n − 3 ) > |F |. Hence 1 ≤ |J| ≤ 2. We consider two cases. Case 1. |J| = 1. Assume MB1n−1 − F1 is disconnected. Clearly, |F1 | = |F | − (in=2 |Fi | ) ≤ 2n − 4. By Lemma 2.2(8), we can assume all components of MB1n−1 − F1 are {u1 } and H1 , or {u1 }, {v1 } and H1 , or (u1 , v1 ) and H1 . Recall that (n − 1 )! − (4n − 6 ) ≥ 10, there is an edge between H1 and MB2n−1 − F2 . By Lemma 2.4, the graph induced by vertices V (H1 ) ∪ V (MB[2, n] − F ) is connected. Since W Gn − F is disconnected and it has no isolated vertices, the vertex set of some nonsingleton component H in W Gn − F belongs to V (MB1n−1 − F1 − H1 ). Therefore, (u1 , v1 ) and H1 are all components of MB1n−1 − F1 , and F = N ({u1 , v1 } ). Case 2. |J| = 2. Assume MB1n−1 − F1 and MB2n−1 − F2 are disconnected. By Theorem 2.1(6) and Claim 1, |F1 | = |F2 | = n − 1 and |Ft | = 2 for each t ∈ [3, n]. Applying Lemma 2.2(8), we can assume {xi } and Hi are all components of MBin−1 − Fi for each i ∈ [1, 2]. According to Lemma 2.4, the graph induced by vertices ∪2i=1V (Hi ) ∪ V (MB[3, n] − F ) is connected. Since W Gn − F is disconnected and it has no isolated vertices, the vertex set of some non singleton component H of W Gn − F belongs to {x1 , x2 }. Hence H = (x1 , x2 ) and then F = N (H ).  j

X. Hu et al. / Applied Mathematics and Computation 339 (2018) 812–819

(ac) (ab) u1

u2

(ac)

(ab)

(bc) (ac)

(bc) u4

817

u3

(ab)

Fig. 6. Illustration for Lemma 3.4.

Fig. 7. Illustration for Theorem 3.5.

Corollary 3.3. [8] κ 1 (W Gn ) = 4n − 6 for n ≥ 5. Proof. By Theorem 3.2, every minimum R1 -vertex-cut of WGn is the vertex set N ({u, v} ), where (u, v ) is an edge of WGn . Hence κ 1 (W Gn ) = 4n − 6.  Next, we will characterize the minimum R2 -vertex-cut of WGn . In [10], Wang et al. showed that κ 2 (W G4 ) = 16. Let F be a minimum R2 -vertex-cut of WG4 . As |V (W G4 )| = 24, W G4 − F has exactly two components, and every component is isomorphic to C4 . Hence F = N (C4 ) and then WG4 is super R2 -vertex-connectedness. In the following, we focus our attention on the case n ≥ 5. Lemma 3.4. For any integer n ≥ 5, κ 2 (W Gn ) ≤ 8n − 18. Proof. Let C4 be a Type B 4-cycle(see Fig. 6). By Theorem 2.1(1), |N (C4 )| = 8n − 18 and |N (w ) ∩ N (ui )| ≤ 3 for each w ∈ V (W Gn ) − C4 ∪ N (C4 ) and each i ∈ [1, 4]. Since WGn is bipartite, |N (w ) ∩ N (C4 )| ≤ 6 ≤ d (w ) − 2. Hence N(C4 ) is an R2 -vertexcut and then κ 2 (W Gn ) ≤ 8n − 18.  Theorem 3.5. For any integer n ≥ 5, WGn is super R2 -vertex-connectedness. Proof. Let F be a minimum R2 -vertex-cut of WGn , and Fi = F ∩ V (MBin−1 ) for each i ∈ [1, n]. By Lemma 3.4, |F | ≤ 8n − 18. We will prove that F = N (C4 ), where C4 is a Type B 4-cycle. Claim. |Fi | ≥ 2 for each i ∈ [1, n]. Proof of this Claim is similar to Theorem 3.2(Claim 1). We consider two cases. Case 1. |Fi | = 2 for some i ∈ [1, n]. Assume |F1 | = 2 and F1 = {u, v}. By a similar discuss as Theorem 3.2(Claim 1), the graph induced by vertices V (W Gn ) − N out (u ) ∪ N out (v ) ∪ F is connected. Since W Gn − F is disconnected, we see that the vertex set of some component H in W Gn − F belongs to N out (u ) ∪ N out (v ) − F . By δ (H) ≥ 2 and Lemma 2.5(2), H is a Type B 4-cycle(see Fig. 4(1)). Hence F = N (C4 ). Case 2. |Fi | ≥ 3 for each i ∈ [1, n]. Case 2.1. |Fi | = 3 for some i ∈ [1, n]. Assume |F1 | = 3 and F1 = {u, v, w}. By a similar discuss as Case 1, we see that the vertex set of some component H in W Gn − F belongs to N out (u ) ∪ N out (v ) ∪ N out (w ) − F . If V (H ) ⊆ N out (u ) ∪ N out (v ) − F , by δ (H) ≥ 2 and Lemma 2.5(2), then H is a Type B 4-cycle. Recall that any non-adjacent two vertices in WGn have at most three common neighbours. Since {u, v} ⊆ N (H ), we have NMB[2,n] (H ) = 4(n − 2 ) + 4(n − 3 ) > |F | − |F1 |, a contradiction. Hence, we can assume V(H) ∩ Nout (x) = ∅ for any x ∈ {u, v, w}. As δ (H) ≥ 2, H is isomorphic to the graph induced by those edges labeled by heavy in Fig. 5(5). However, |N (H )| > 8n − 18, a contradiction. Case 2.2. |Fi | ≥ 4 for each i ∈ [1, n]. j Since (n − 1 )! − |F | ≥ 8(n − 2 ) − (8n − 18 ) ≥ 2, there are at least two edges between the pair MBin−1 − Fi and MBn−1 − Fj

for any distinct i and j. Let J = {i|MBin−1 − Fi is disconnected} If |J| = 0, then W Gn − F = MB[1, n] − F is connected by Lemma 2.4. If |J| = 1, assume MB1n−1 − F1 is disconnected. By Lemma 2.4, MB[2, n] − F is connected. Hence the vertex set of some component H in W Gn − F belongs to V (MB1n−1 ) − F1 . Since MB1n−1 is bipartite, MB1n−1 does not have the subgraph K2, 3 and δ (H) ≥ 2, we see that |V(H)| ≥ 4 and |V(H)| = 5. If |V (H )| = 4, then H∼ =C4 and |N (H )| = |NMB1 (H )| + |N out (H )| = n−1

4(n − 3 ) + 4(n − 1 ) > |F |, a contradiction. If |V (H )| = 6, then H is isomorphic one of the graphs in Fig. 7. Hence |N (H )| =

818

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123

132

213

312

231

321 Fig. 8. MB3 .

|NMB1 (H )| + |Nout (H )| ≥ 2(n − 3 ) + 6(n − 1 ) > |F |, a contradiction. If |V(H)| ≥ 7, then |N (H )| ≥ κ (MB1n−1 ) + |Nout (H )| ≥ (n − n−1 1 ) + 7(n − 1 ) > |F |, a contradiction. If |J| = 2, assume MB1n−1 − F1 and MB2n−1 − F2 are disconnected, and |F1 | ≥ |F2 |. Since |F1 | + |F2 | = |F | − (in=3 |Fi | ) ≤ (8n − 18 ) − 4(n − 2 ) = 4n − 10, we have |F2 | ≤ 2n − 5. By Lemma 2.2(8), we can assume {x2 } and H2 are all components of MB2n−1 − F2 . Since dWGn −F (x2 ) ≥ 2, we have dWGn −MB1 (x2 ) ≥ 1. Hence x2 has at least one neighbour in MB[3, n] − F . By n−1 Lemma 2.4, the graph induced by vertices V (H2 ) ∪ V (MB[3, n] − F ) is connected. It follows that MB[2, n] − F is connected. So the vertex set of some component H in W Gn − F belongs to V (MB1n−1 ) − F1 . Then by a similar argument as the above

paragraph, we can obtain a contradiction. If |J| = 3, assume MB1n−1 − F1 , MB2n−1 − F2 and MB3n−1 − F3 are disconnected, and |F1 | ≥ |F2 | ≥ |F3 |. Clearly, |F1 | = |F | − (|F2 | + |F3 | ) − (in=4 |Fi | ) ≤ (8n − 18 ) − 2(n − 1 ) − 4(n − 3 ) = 2n − 4 and |F2 | < 2n − 4, |F3 | < 2n − 4. By Lemma 2.2(8), we can assume {xi } and Hi are all components of MBin−1 − Fi for each i ∈ [2, 3], and either {x1 } and H1 , or {x1 }, {y1 } and H1 , or K2 = (x1 , y1 ) and H1 are all components of MB1n−1 − F1 . By Lemma 2.4, graph induced by vertices ∪3i=1V (Hi ) ∪ V (MB[4, n] − F ) is connected. Hence the vertex set of some component H in W Gn − F belongs to V (W Gn ) − ∪3i=1V (Hi ) ∪ V (MB[4, n] − F ). Clearly, |V(H)| ≥ 4. By Lemma 2.3(1), all components of MB1n−1 − F1 are K2 = (x1 , y1 ) and H1 , and then H is a Type B 4-cycle. It follows that F = N (C4 ). |J | If |J| ≥ 4, assume MB1n−1 − F1 , . . . , MBn−1 − F|J| are disconnected, and |F1 | = max{|Fi | : 1 ≤ i ≤ |J|}. When n ≥ 6, |F1 | = |F | − (i4=2 |Fi | ) − (in=5 |Fi | ) ≤ (8n − 18 ) − 3(n − 1 ) − 4(n − 4 ) < 2n − 4. Similarly, |Ft | < 2n − 4 for each t ∈ [1, |J|]. By Lemma 2.2(8), |J |

we can assume {xt } and Ht are all components of MBtn−1 − Ft for each t ∈ [1, |J|]. Graph induced by vertices ∪i=1V (Hi ) ∪ V (MB[|J| + 1, n] − F ) is connected. Hence the vertex set of some component H in W Gn − F belongs to {x1 , x2 , . . . , x|J| }. Since δ (H) ≥ 2, H contains a cycle. This contradicts to Lemma 2.3(3). When n = 5, discuss whether |F1 | = 2n − 4 or |F1 | < 2n − 4. Note that any edge (u, v ) has the label (jn) for some j ∈ [1, n − 1], if u and v are not lie in same copy MBin−1 . Then by a similar discuss as the above paragraph, we see that F = N (C4 ), where C4 is a Type B 4-cycle.  Corollary 3.6. [8] κ 2 (W Gn ) = 8n − 18 for n ≥ 5. Proof. By Theorem 3.5, every minimum R2 -vertex-cut of WGn is the vertex set N(C4 ), where C4 is a Type B 4-cycle. Hence κ 2 (W Gn ) = |N (C4 )| = 8n − 18.  Appendix A

Proof. Let F be a minimum R1 -vertex-cut of WG4 . By Lemma 3.1, |F| ≤ 10. Claim: |Fi | ≥ 2 for each i ∈ [1, 4]. Proof of this Claim is similar to Theorem 3.2(Claim 1). Let J = {i|MBi3 − Fi is disconnected}. We consider two cases. Case 1: |J| = 0. If there is an edge between MBi3 − Fi and MBi3+1 − Fi+1 for each i ∈ [1, 3], then W G4 − F is connected by Lemma 2.4. So we can assume there is no edge between MB13 − F1 and MB23 − F2 . Hence |F1 | + |F2 | ≥ |V (MB3 )| = 6. Since |F| ≤ 10 and |Fi | ≥ 2 for each i ∈ [1, 4], we have |F1 | + |F2 | = 6 and |F3 | = |F4 | = 2. Suppose |F1 | ≥ |F2 |, then |F1 | ≥ 3 and |F2 | ≤ 3. By Lemma 2.4, MB[2, 4] − F is connected. Since W G4 − F is disconnected and it has no isolated vertices, the vertex set of some non singleton component H in W G4 − F belongs to V (MB13 ) − F1 . Clearly, 2 ≤ |V (H )| ≤ |V (MB13 ) − F1 | ≤ 3. If |V (H )| = 2, then H is an edge and F = N (H ). If |V (H )| = 3, then |F1 | = 3 and H = MB13 − F1 . Since |F1 | + |F4 | < 6, there is an edge between MB13 − F1 and MB43 − F4 . Recall that MB[2, 4] − F is connected, hence MB[1, 4] − F is connected. Case 2: |J| ≥ 1. Since κ (MB3 ) = 3 and |Fi | ≥ 2 for each i ∈ [1, 4], we have |J| ≤ 2.

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If |J| = 1, assume MB13 − F1 is disconnected and then |F1 | ≥ 3. Assume |F2 | ≥ |F3 | ≥ |F4 |. Since |F| ≤ 10 and |Fi | ≥ 2 for each i ∈ [1, 4], we have |F2 | ≤ 3 and |F3 | = |F4 | = 2. By Lemma 2.4, MB[2, 4] − F is connected. Hence the vertex set of some component H in W G4 − F belongs to V (MB13 ) − F1 . Since |V(H)| ≥ 2, |V (MB13 ) − F1 | ≤ 3 and MB13 − F1 is disconnected, we see that |F1 | = 3 and MB13 − F1 has exactly two components that isomorphic to K2 and K1 , respectively. It is impossible(see Fig. 8). If |J| = 2, assume MB13 − F1 and MB23 − F2 are disconnected. Clearly, |F1 | = |F2 | = 3 and |F3 | = |F4 | = 2. By Lemma 2.4, MB[3, 4] − F is connected. Hence the vertex set of some non singleton component H in W G4 − F belongs to V (MB[1, 2] − F ). We can easily obtain that MBi3 − Fi is an independent set for each i ∈ [1, 2]. By Lemma 2.3(1) and |V(H)| ≥ 2, H is an edge with one end-vertex in MB13 and the other in MB23 . Hence F = N (H ).  References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

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