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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 424023, 6 pages
http://dx.doi.org/10.1155/2012/424023
Research Article

On Arc Connectivity of Direct-Product Digraphs

Department of Mathematics, Wuyi University, Jiangmen 529020, China

Received 20 March 2012; Accepted 8 June 2012

Academic Editor: Luca Formaggia

Copyright © 2012 Tiedan Zhu and Jianping Ou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Four natural orientations of the direct product of two digraphs are introduced in this paper. Sufficient and necessary conditions for these orientations to be strongly connected are presented, as well as an explicit expression of the arc connectivity of a class of direct-product digraphs.

1. Introduction

Various product operations are employed for constructing larger networks from smaller ones, among which direct-product operation is the most frequently employed one. The direct product of two graphs 𝐺1 and 𝐺2, denoted by 𝐺1×𝐺2, is defined on vertex set 𝑉(𝐺1)×𝑉(𝐺2), where two vertices (𝑥1,𝑥2) and (𝑦1,𝑦2) are adjacent to each other in 𝐺1×𝐺2 if and only if 𝑥1𝑦1𝐸(𝐺1) and 𝑥2𝑦2𝐸(𝐺2). Other names for direct product are tensor product, categorical product, Kronecker product, cardinal product, relational product, and weak direct product [1]. Some basic connectivity properties of direct-product graphs are presented in [2, 3] and elsewhere. Specially, the authors characterize connected product graphs by presenting the following Theorem 1.1 in [1]; an explicit expression of the connectivity of a direct-product graph is presented in [4].

Theorem 1.1 (see [1, Theorem  5.29]). Let 𝐺 and 𝐻 be connected nonempty graphs. Then 𝐺×𝐻 is connected if and only if at least one of them is nonbipartite. Furthermore, if both 𝐺 and 𝐻 are bipartite, then 𝐺×𝐻 has exactly two components.

Let 𝐷1=(𝑉1,𝐴1) and 𝐷2=(𝑉2,𝐴2) be two digraphs. It is naturally to define the vertex set and adjacency relationship between vertices of direct product of digraph as those of undirected graphs. But the orientation is not unique, for every pair of arcs (𝑥1,𝑦1)𝐴1 and (𝑥2,𝑦2)𝐴2, there are four natural orientations of the two edges (𝑥1,𝑥2)(𝑦1,𝑦2) and (𝑦1,𝑥2)(𝑥1,𝑦2). For clarity and comparison, we depicture the Cartesian product 𝐾2𝐾2 and the four orientations of direct-product digraph 𝐾2×𝐾2 in Figure 1.

424023.fig.001
Figure 1: Orientations of direct-product digraph 𝐾2×𝐾2.

A digraph is strongly connected or disconnected if any vertex is reachable from any other vertex, where a vertex 𝑢 is said to be reachable form another vertex 𝑣 if there is a directed path from 𝑣 to 𝑢. The minimum number 𝜆(𝐷) of arcs needed to be removed for destroying the strong connectivity of a digraph is called its arc connectivity. This work characterizes strongly connected direct-product digraphs of the above four orientations. We follow [5] for symbols and terminology not specified in this work.

2. Strongly Connected Direct-Product Digraphs

As will be shown in next section, the above four orientations can be transformed by one another to some extent. So we assume in this section that 𝐷1×𝐷2 contains an arc from vertex (𝑥1,𝑥2) to (𝑦1,𝑦2) if and only if arc (𝑥2,𝑦2)𝐴(𝐷2), and (𝑥1,𝑦1)𝐴(𝐷1) or (𝑦1,𝑥1)𝐴(𝐷1).

Lemma 2.1. Let 𝐷 be a digraph. Then 𝐾2×𝐷 is strongly connected if and only if 𝐷 is nonbipartite and strongly connected.

Proof. Necessity. If 𝐷 is bipartite, then by Theorem 1.1 the underlying graph of 𝐾2×𝐷 is disconnected. This contradiction shows that 𝐷 is nonbipartite. Let {𝑎,𝑏} be the vertex set of 𝐾2. Since 𝐾2×𝐷 is strongly connected, it follows that for any two vertices 𝑢,𝑣𝑉(𝐷), 𝐾2×𝐷 contains a directed walk 𝑊1 from (𝑎,𝑢) to (𝑎,𝑣). Let 𝑊1=(𝑎,𝑢)(𝑏,𝑣1)(𝑎,𝑣2)(𝑏,𝑣𝑘)(𝑎,𝑣). Then it yields a directed walk 𝑢𝑣1𝑣2𝑣𝑘𝑣 of 𝐷. The necessity follows from this observation.
Sufficiency. Since 𝐷 is strongly connected, it contains a spanning closed directed walk 𝑊=𝑢0𝑣1𝑣2𝑣𝑘𝑢0, which corresponds two directed walks 𝑊=(𝑎,𝑢0)(𝑏,𝑣1)(𝑎,𝑣2)(𝑥,𝑢0) and 𝑊=(𝑏,𝑢0)(𝑎,𝑣1)(𝑏,𝑣2)(𝑦,𝑢0) of 𝐾2×𝐷, where 𝑥,𝑦{𝑎,𝑏}. It is obvious that 𝑊𝑊 is a spanning subgraph of 𝑉(𝐾2×𝐷).
Case  1. The length of 𝑊=𝑢0𝑣1𝑣2𝑣𝑘𝑢0 is odd, say, 𝑘=2𝑛.
In this case, 𝑊=(𝑎,𝑢0)(𝑏,𝑣1)(𝑎,𝑣2)(𝑎,𝑣2𝑛)(𝑏,𝑢0) and 𝑊=(𝑏,𝑢0)(𝑎,𝑣1)(𝑏,𝑣2)(𝑏,𝑣2𝑛)(𝑎,𝑢0). And so 𝑊𝑊 is a spanning closed directed walk of 𝐾2×𝐷. Hence, 𝐾2×𝐷 is strongly connected.
Case  2. The length of 𝑊=𝑢0𝑣1𝑣2𝑣𝑘𝑢0 is even, say, 𝑘=2𝑛+1.
In this case, both 𝑊=(𝑎,𝑢0)(𝑏,𝑣1)(𝑎,𝑣2)(𝑎,𝑣2𝑛)(𝑎,𝑢0) and 𝑊=(𝑏,𝑢0)(𝑎,𝑣1)(𝑏,𝑣2)(𝑏,𝑣2𝑛)(𝑏,𝑢0) are closed directed walks. Since 𝐷 is nonbipartite, it contains an arc joining two vertices of 𝑊 whose suffixes have same parity. Without loss of generality, let (𝑢0,𝑣2𝑙)𝐴(𝐷). This arc corresponds to two arcs ((𝑎,𝑢0),(𝑏,𝑣2𝑙)), ((𝑏,𝑢0),(𝑎,𝑣2𝑙)) of 𝐾2×𝐷. These two arcs make 𝑊 and 𝑊 reachable from each other. Therefore, 𝐾2×𝐷 is strongly connected.

Theorem 2.2. Let 𝐷1 and 𝐷2 be two digraphs. Then 𝐷1×𝐷2 is strongly connected if and only if 𝐷2 is strongly connected and 𝐷1 or 𝐷2 is nonbipartite.

Proof. Sufficiency. If 𝐷2 is nonbipartite and strongly connected, by Lemma 2.1, 𝐷1×𝐷2 is strongly connected.
If 𝐷2 is strongly connected and bipartite, but 𝐷1 is nonbipartite, then 𝐷2 contains a spanning closed directed walk 𝑊2=𝑢0𝑢1𝑢2𝑢𝑘𝑢0. Since 𝐷2 is bipartite, it follows that 𝑘 is odd. Let 𝑊1=𝑣0𝑣1𝑣2𝑣𝑖𝑣2𝑚+𝑖𝑣𝑖𝑣𝑠𝑣0 be a spanning closed walk of 𝐷1, where 𝐶=𝑣𝑖𝑣2𝑚+𝑖𝑣𝑖 is an odd cycle. For any vertex 𝑣𝑗𝑉(𝑊1) and any one of its neighbor 𝑣𝑙 in 𝑊1, 𝑊𝑗=(𝑣𝑗,𝑢0)(𝑣𝑙,𝑢1)(𝑣𝑗,𝑢2)(𝑣𝑙,𝑢𝑘)(𝑣𝑗,𝑢0)𝑊2 and 𝑉(𝐷)=𝑠𝑖=0𝑉(𝑊𝑖). Since 𝑣𝑗 and 𝑣𝑙 are adjacent in 𝑊1, it follows that 𝑙=𝑗+1 or 2𝑚+𝑗. Let 𝑊=𝑣𝑗,𝑢0𝑣𝑗+1,𝑢1𝑣𝑗,𝑢2𝑣𝑗+1,𝑢𝑘𝑣𝑗,𝑢0,𝑊=𝑣𝑗,𝑢0𝑣2𝑚+𝑗,𝑢1𝑣𝑗,𝑢2𝑣2𝑚+𝑗,𝑢𝑘𝑣𝑗,𝑢0.(2.1)
The sum of the suffixes of each vertex in 𝑊 has the same parity as the integer 𝑗. Since 𝑣𝑗 may be any vertex of 𝑊1, it follows that the vertices of 𝑉(𝐷1)×𝑉(𝐷2) whose suffix sum has same parity induce a strong component (a strongly connected vertex-induced subgraph with as many as possible vertices). Since 𝑊 contains vertices with odd suffix sum as well vertices with even suffix sum, it follows that 𝐷1×𝐷2 is strongly connected.
Necessity. Since 𝐷1×𝐷2 is strongly connected, every two vertices (𝑢𝑖,𝑣𝑗) and (𝑢𝑘,𝑣𝑙) are reachable from each other in 𝐷1×𝐷2. It follows that 𝑣𝑗 and 𝑣𝑙 are reachable from each other in 𝐷2. Hence, 𝐷2 is strongly connected. If 𝐷2 is strongly connected, but both 𝐷1 and 𝐷2 are bipartite, then 𝐷1×𝐷2 is not strongly connected by Lemma 2.1. The necessity follows from this contradiction.

3. Relationship of the Four Orientations

The last three orientations of Figure 1 can be defined as follows, respectively.

Orientation 2. Let 𝐷1=(𝑉1,𝐴1) and 𝐷2=(𝑉2,𝐴2) be two digraphs. 𝐷1×𝐷2 contains an arc from (𝑥1,𝑥2) to (𝑦1,𝑦2) if and only if (𝑦2,𝑥2)𝐴2, and (𝑦1,𝑥1) or (𝑥1,𝑦1)𝐴1.

Orientation 3. Let 𝐷1=(𝑉1,𝐴1) and 𝐷2=(𝑉2,𝐴2) be two digraphs. 𝐷1×𝐷2 contains an arc from (𝑥1,𝑥2) to (𝑦1,𝑦2) if and only if (𝑥1,𝑦1)𝐴1, and (𝑥2,𝑦2) or (𝑦2,𝑥2)𝐴2.

Orientation 4. Let 𝐷1=(𝑉1,𝐴1) and 𝐷2=(𝑉2,𝐴2) be two digraphs. 𝐷1×𝐷2 has an arc from (𝑥1,𝑥2) to (𝑦1,𝑦2) if and only if (𝑦1,𝑥1)𝐴1, and (𝑥2,𝑦2) or (𝑦2,𝑥2)𝐴2.

The readers are suggested to refer to (2), (3), and (4) of Figure 2 for Orientations 2, 3, and 4 respectively. It is not difficult to see that Orientation 2 is the converse of the first orientation in Figure 1. And so, the following Corollary 3.1 follows directly from Theorem 2.2. Similarly, if 𝐷2×𝐷1 has Orientation  1 (refer to (2) of Figure 1) and 𝐷1×𝐷2 has Orientation 3 then 𝐷2×𝐷1𝐷1×𝐷2. From this observation and Corollary 3.1, Corollary 3.2 follows directly.

424023.fig.002
Figure 2: Partition of  𝑉(𝐾2×𝐷𝑆).

Corollary 3.1. Let 𝐷1×𝐷2 be oriented as Orientation 2. Then it is strongly connected if and only if 𝐷2 is strongly connected and 𝐷1 or 𝐷2 is nonbipartite.

Corollary 3.2. Let 𝐷1×𝐷2 be oriented as in Orientations 3 or 4. Then it is strongly connected if and only if 𝐷1 is strongly connected and 𝐷1 or 𝐷2 is nonbipartite.

4. Strong Arc Connectivity

Let 𝐷 be a strongly arc connected digraph and 𝑋𝑉(𝐷) be a nonempty vertex set. Denote by (𝑋,𝑉(𝐷)𝑋) the set of arcs with tail in 𝑋 and head in 𝑉(𝐷)𝑋, which is called a directed cut of 𝐷. If 𝑇 is a directed cut of strongly connected digraph 𝐷, then 𝐷𝑇 is connected but is not strongly connected. The size 𝜆(𝐷) of minimum directed cuts of digraph 𝐷 is called its strong arc connectivity. Let 𝛽(𝐷)=min{|𝑆|𝑆𝐴(𝐷)suchthat𝐷𝑆isabipartitegraph}, 𝛿(𝐷)=min{𝛿+(𝐷),𝛿(𝐷)} be the minimum degree of 𝐷. For every integer 𝛿(𝐷)𝑗𝜆(𝐷), let 𝛽𝑗=min{𝛽(𝐶)𝐶 be any strongly connected component of 𝐷𝑇 and 𝑇 is an arbitrary directed cut of 𝐷 of size 𝑗}.

Let 𝐷1×𝐷2 be oriented according to the first orientation of Figure 1. For every arc ((𝑥1,𝑥2),(𝑦1,𝑦2)) of 𝐷1×𝐷2, the arc (𝑥2,𝑦2)𝐴(𝐷2) is called its projection on 𝐷2. The following Lemma 4.1 is immediate, so we omit its proof herein.

Lemma 4.1. If directed graph 𝐷 is not strongly connected, then it contains two strongly connected components, one of which has no outer neighbors and the other has no inner neighbors.

Theorem 4.2. If 𝐾2×𝐷 is strongly connected, then 𝜆(𝐾2×𝐷)=min{2𝜆(𝐷),𝛽(𝐷),min{𝑗+𝛽𝑗𝜆(𝐷)𝑗𝛿}}.

Proof . By Lemma 2.1, Theorem 4.2 is clearly true in the case when 𝐷 is bipartite, and so we assume in what follows that 𝛽(𝐷)1 and 𝐷 is nonbipartite. Let 𝑇 be a minimum direct cut of 𝐷 and let 𝐹 be a minimum arc-set of 𝐷 such that 𝐷𝐹=(𝑋,𝑌;𝐸) is a bipartite graph with bipartition 𝑋 and 𝑌. Then 𝛽(𝐷)=|𝐹| and, 𝜆(𝐷)=|𝑇|1 by Lemma 2.1.
Claim  1. 𝜆(𝐾2×𝐷)min{2𝜆(𝐷),𝛽(𝐷),min{𝑗+𝛽𝑗𝜆(𝐷)𝑗𝛿}.
Let 𝐴(𝐾2)={(𝑎,𝑏)}. By Theorem 1.1, 𝐾2×(𝐷𝐹) consists of two components. The vertex sets of these two components are {(𝑎,𝑥)𝑥𝑋}{(𝑏,𝑦)𝑦𝑌} and {(𝑏,𝑥)𝑥𝑋}{(𝑎,𝑦)𝑦𝑌}. It is not difficult to see that {((𝑎,𝑥),(𝑏,𝑦))(𝑥,𝑦)𝐹𝐴(𝐷[𝑋])}{((𝑏,𝑥),(𝑎,𝑦))(𝑥,𝑦)𝐹𝐴(𝐷[𝑌])} is a directed cut of 𝐾2×𝐷, since its removal makes {(𝑏,𝑥)𝑥𝑋}{(𝑎,𝑦)𝑦𝑌} not reachable from {(𝑎,𝑥)𝑥𝑋}{(𝑏,𝑦)𝑦𝑌}. And so, 𝜆(𝐾2×𝐷)|𝐹|=𝛽(𝐷).
By Lemma 4.1, 𝐷𝑇 contains a strongly connected component 𝐷1 that has no outer neighbors. By the minimality of 𝑇, we have (𝐷1,𝐷1)=𝑇. Obviously, {((𝑎,𝑥),(𝑏,𝑦))(𝑥,𝑦)𝑇}{((𝑎,𝑦),(𝑏,𝑥))(𝑥,𝑦)𝑇} is a directed cut of 𝐾2×𝐷. Hence 𝜆(𝐾2×𝐷)2|𝑇|=2𝜆(𝐷).
Let 𝑇𝑗 be a directed cut of 𝐷 that has size 𝑗, 𝐶𝑗 be a strongly connected component of 𝐷𝑇𝑗 with (𝐶𝑗,𝐷𝑇𝑗𝐶𝑗)= (by Lemma 4.1 such components exist), 𝐹𝑗 be an arc set of 𝐶𝑗 such that 𝐶𝑗𝐹𝑗=(𝑋𝑗,𝑌𝑗;𝐸𝑗) is a bipartite subgraph with bipartition (𝑋𝑗,𝑌𝑗). Then 𝐾2×(𝐶𝑗𝐹𝑗) consists of two components, whose vertex sets are {(𝑎,𝑥)𝑥𝑋𝑗}{(𝑏,𝑦)𝑦𝑌𝑗} and {(𝑏,𝑥)𝑥𝑋𝑗}{(𝑎,𝑦)𝑦𝑌𝑗}. It’s not difficult to see that the union of {((𝑎,𝑥),(𝑏,𝑦))(𝑥,𝑦)𝐹𝑗𝐴(𝐷[𝑋𝑗])}, {((𝑏,𝑥),(𝑎,𝑦))(𝑥,𝑦)𝐹𝑗𝐴(𝐷[𝑌𝑗])}, {((𝑎,𝑥),(𝑏,𝑦))(𝑥,𝑦)(𝑋𝑗,𝐷𝐶𝑗)} and {((𝑏,𝑥),(𝑎,𝑦))(𝑥,𝑦)(𝑌𝑗,𝐷𝐶𝑗)} is a directed cut of 𝐾2×𝐷, since its removal makes {(𝑎,𝑦)𝑦𝑌𝑗}{(𝑏,𝑥)𝑥𝑋𝑗} not reachable from {(𝑎,𝑥)𝑥𝑋𝑗}{(𝑏,𝑦)𝑦𝑌𝑗}. Noticing that |{((𝑎,𝑥),(𝑏,𝑦))(𝑥,𝑦)𝐹𝑗𝐴(𝐷[𝑋𝑗])}{((𝑏,𝑥),(𝑎,𝑦))(𝑥,𝑦)𝐹𝑗𝐴(𝐷[𝑌𝑗])}|=|𝐹𝑗|, |{((𝑎,𝑥),(𝑏,𝑦))(𝑥,𝑦)(𝑋𝑗,𝐷𝐶𝑗)}{((𝑏,𝑥),(𝑎,𝑦))(𝑥,𝑦)(𝑌𝑗,𝐷𝐶𝑗)}|=|𝑇𝑗| and |𝑇𝑗𝐹𝑗|=|𝑇𝑗|+|𝐹𝑗|=𝑗+𝛽𝑗, we deduce that 𝜆(𝐾2×𝐷)min{𝑗+𝛽𝑗𝜆(𝐷)𝑗𝛿}. And so, Claim  1 follows.
Claim  2. 𝜆(𝐾2×𝐷)min{2𝜆(𝐷),𝛽(𝐷),min{𝑗+𝛽𝑗𝜆(𝐷)𝑗𝛿}.
Let 𝑆 be a minimum direct cut of 𝐾2×𝐷, 𝑉1={𝑥𝑉(𝐷): vertices (𝑎,𝑥) and (𝑏,𝑥) lie in common strongly connected component of 𝐾2×𝐷𝑆}, 𝑉2={𝑥𝑉(𝐷): vertices (𝑎,𝑥) and (𝑏,𝑥) lie in different components of 𝐾2×(𝐷𝑆)}. Then (𝑉1,𝑉2) is a partition of 𝑉(𝐷). From the minimality of 𝑆 it follows that 𝐾2×𝐷𝑆 consists of two strongly connected components, say 𝐶1 and 𝐶2. Noticing that (𝐶1,𝐶2)𝑆 or (𝐶2,𝐶1)𝑆, we assume without loss of generality that (𝐶1,𝐶2)𝑆. Now three different cases occur: 𝑉1=𝑉2; 𝑉1=𝑉2; 𝑉1𝑉2.
Consider at first the case when 𝑉1=𝑉2. By Lemma 2.1, we deduce that {𝑥𝑉(𝐷)(𝑎,𝑥)𝑉(𝐶1)} induces a strongly connected component of 𝐷 as well as {𝑦𝑉(𝐷)(𝑎,𝑦)𝑉(𝐶2)} in this case. Furthermore, ({𝑥𝑉(𝐷)(𝑎,𝑥)𝑉(𝐶1)},{𝑦𝑉(𝐷)(𝑎,𝑦)𝑉(𝐶2)}) is a directed cut of 𝐷 and ({(𝑎,𝑥)(𝑎,𝑥)𝑉(𝐶1)},{(𝑏,𝑦)(𝑏,𝑦)𝑉(𝐶2)})({(𝑏,𝑥)(𝑏,𝑥)𝑉(𝐶1)},{(𝑎,𝑦)(𝑎,𝑦)𝑉(𝐶2)})𝑆. It follows from these observations that ||𝑆||||𝐶(𝑎,𝑥)𝑉1,𝐶(𝑏,𝑦)𝑉2||+||(𝐶𝑏,𝑥)𝑉1,(𝐶𝑎,𝑦)𝑉2||2𝜆(𝐷).(4.1)
Consider secondly the case when 𝑉1=𝑉2. Let 𝑀={𝑥𝑉(𝐷)(𝑎,𝑥)𝑉(𝐶1)} and 𝑁={𝑦𝑉(𝐷)(𝑏,𝑦)𝑉(𝐶1)}. Then (𝑀,𝑁) is a partition of 𝑉(𝐷) and (𝐶1,𝐶2)={((𝑎,𝑥),(𝑏,𝑦))(𝑥,𝑦)𝐴(𝐷[𝑀])}{((𝑏,𝑥),(𝑎,𝑦))(𝑥,𝑦)𝐴(𝐷[𝑁])}. Since 𝐷𝐴(𝐷[𝑀])𝐴(𝐷[𝑁]) is a bipartite subgraph of 𝐷, it follows that |(𝐶1,𝐶2)|=|𝐴(𝐷[𝑀])|+|𝐴(𝐷[𝑁])|𝛽(𝐷). Recalling that (𝐶1,𝐶2)𝑆, we have |𝑆||(𝐶1,𝐶2)|𝛽(𝐷).
Consider finally the case when 𝑉1𝑉2. Let 𝐶𝐻=𝑥𝑉(𝐷)(𝑎,𝑥)𝑉1,𝑥𝑉1,𝐶𝑄=𝑥𝑉(𝐷)(𝑎,𝑥)𝑉2,𝑥𝑉1,𝐶𝑊=𝑥𝑉(𝐷)(𝑎,𝑥)𝑉1,𝑥𝑉2,𝐶𝑍=𝑥𝑉(𝐷)(𝑎,𝑥)𝑉2,𝑥𝑉2.(4.2)
Then (𝐻,𝑄,𝑊,𝑍) is a partition of 𝑉(𝐷), refer to (1) of Figure 2. Since (𝐶1,𝐶2)𝑆, the arcs in this set is removed in Figure 2. If 𝐻𝑄, then the set of arcs from {(𝑎,𝑥)𝑥𝐻}{(𝑏,𝑥)𝑥𝐻} to 𝐶2 is a directed cut of 𝐾2×𝐷, but it has less arcs than 𝑆. This contradiction shows that either 𝐻= or 𝑄=. Assume without loss of generality that 𝑄=. Then 𝐾2×𝐷𝑆 can be depicted as (2) of Figure 2.
On the one hand, for every arc (𝑥,𝑦)(𝐻,𝐷𝐻), if 𝑦𝑊 then ((𝑎,𝑥),(𝑏,𝑦))𝑆; if 𝑦𝑍 then ((𝑏,𝑥),(𝑎,𝑦))𝑆. On the other hand, for every arc (𝑥,𝑦)𝐴(𝐷[𝑊]) the arc ((𝑎,𝑥),(𝑏,𝑦))𝑆 and for every arc (𝑥,𝑦)𝐴(𝐷[𝑍]) the arc ((𝑏,𝑥),(𝑎,𝑦))𝑆. It follows from these observations that ||𝑆||||[]||+||𝐴[𝑊])||+||𝐴[𝑍])||𝐻,𝐷𝐻(𝐷(𝐷min𝑗+𝛽𝑗,𝜆(𝐷)𝑗𝛿(𝐷)(4.3) where [𝐻,𝐷𝐻] represents the set of arcs with ends in 𝐻 and 𝐷𝐻, respectively. And so, the theorem follows from Claims 1 and 2.

Acknowledgments

The authors are grateful to the referees for their valuable suggestions. They are supported by National Natural Science Foundation of China (Grant no. 11126326).

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