Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 424023 | 6 pages | https://doi.org/10.1155/2012/424023

On Arc Connectivity of Direct-Product Digraphs

Academic Editor: Luca Formaggia
Received20 Mar 2012
Accepted08 Jun 2012
Published30 Jul 2012

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 (π‘₯1,π‘₯2) and (𝑦1,𝑦2) are adjacent to each other in 𝐺1×𝐺2 if and only if π‘₯1𝑦1∈𝐸(𝐺1) and π‘₯2𝑦2∈𝐸(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 (π‘₯1,𝑦1)∈𝐴1 and (π‘₯2,𝑦2)∈𝐴2, there are four natural orientations of the two edges (π‘₯1,π‘₯2)(𝑦1,𝑦2) and (𝑦1,π‘₯2)(π‘₯1,𝑦2). For clarity and comparison, we depicture the Cartesian product →𝐾2░→𝐾2 and the four orientations of direct-product digraph →𝐾2×→𝐾2 in Figure 1.

A digraph is strongly connected or disconnected if any vertex is reachable from any other vertex, where a vertex 𝑒 is said to be reachable form another vertex 𝑣 if there is a directed path from 𝑣 to 𝑒. The minimum number πœ†(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 (π‘₯1,π‘₯2) to (𝑦1,𝑦2) if and only if arc (π‘₯2,𝑦2)∈𝐴(𝐷2), and (π‘₯1,𝑦1)∈𝐴(𝐷1) or (𝑦1,π‘₯1)∈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 {π‘Ž,𝑏} be the vertex set of →𝐾2. Since →𝐾2×𝐷 is strongly connected, it follows that for any two vertices 𝑒,π‘£βˆˆπ‘‰(𝐷), →𝐾2×𝐷 contains a directed walk π‘Š1 from (π‘Ž,𝑒) to (π‘Ž,𝑣). Let π‘Š1=(π‘Ž,𝑒)(𝑏,𝑣1)(π‘Ž,𝑣2)β‹―(𝑏,π‘£π‘˜)(π‘Ž,𝑣). Then it yields a directed walk 𝑒𝑣1𝑣2β‹―π‘£π‘˜π‘£ of 𝐷. The necessity follows from this observation.
Sufficiency. Since 𝐷 is strongly connected, it contains a spanning closed directed walk π‘Š=𝑒0𝑣1𝑣2β‹―π‘£π‘˜π‘’0, which corresponds two directed walks π‘Šξ…ž=(π‘Ž,𝑒0)(𝑏,𝑣1)(π‘Ž,𝑣2)β‹―(π‘₯,𝑒0) and π‘Šξ…žξ…ž=(𝑏,𝑒0)(π‘Ž,𝑣1)(𝑏,𝑣2)β‹―(𝑦,𝑒0) of →𝐾2×𝐷, where π‘₯,π‘¦βˆˆ{π‘Ž,𝑏}. It is obvious that π‘Šξ…žβˆͺπ‘Šξ…žξ…ž is a spanning subgraph of 𝑉(→𝐾2×𝐷).
Case  1. The length of π‘Š=𝑒0𝑣1𝑣2β‹―π‘£π‘˜π‘’0 is odd, say, π‘˜=2𝑛.
In this case, π‘Šξ…ž=(π‘Ž,𝑒0)(𝑏,𝑣1)(π‘Ž,𝑣2)β‹―(π‘Ž,𝑣2𝑛)(𝑏,𝑒0) and π‘Šξ…žξ…ž=(𝑏,𝑒0)(π‘Ž,𝑣1)(𝑏,𝑣2)β‹―(𝑏,𝑣2𝑛)(π‘Ž,𝑒0). And so π‘Šξ…žβˆͺπ‘Šξ…žξ…ž is a spanning closed directed walk of →𝐾2×𝐷. Hence, →𝐾2×𝐷 is strongly connected.
Case  2. The length of π‘Š=𝑒0𝑣1𝑣2β‹―π‘£π‘˜π‘’0 is even, say, π‘˜=2𝑛+1.
In this case, both π‘Šβ€²=(π‘Ž,𝑒0)(𝑏,𝑣1)(π‘Ž,𝑣2)β‹―(π‘Ž,𝑣2𝑛)(π‘Ž,𝑒0) and π‘Šξ…žξ…ž=(𝑏,𝑒0)(π‘Ž,𝑣1)(𝑏,𝑣2)β‹―(𝑏,𝑣2𝑛)(𝑏,𝑒0) are closed directed walks. Since 𝐷 is nonbipartite, it contains an arc joining two vertices of π‘Š whose suffixes have same parity. Without loss of generality, let (𝑒0,𝑣2𝑙)∈𝐴(𝐷). This arc corresponds to two arcs ((π‘Ž,𝑒0),(𝑏,𝑣2𝑙)), ((𝑏,𝑒0),(π‘Ž,𝑣2𝑙)) of →𝐾2×𝐷. These two arcs make π‘Šξ…ž and π‘Šξ…žξ…ž 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 π‘Š2=𝑒0𝑒1𝑒2β‹―π‘’π‘˜π‘’0. Since 𝐷2 is bipartite, it follows that π‘˜ is odd. Let π‘Š1=𝑣0𝑣1𝑣2⋯𝑣𝑖⋯𝑣2π‘š+𝑖𝑣𝑖⋯𝑣𝑠𝑣0 be a spanning closed walk of 𝐷1, where 𝐢=𝑣𝑖⋯𝑣2π‘š+𝑖𝑣𝑖 is an odd cycle. For any vertex π‘£π‘—βˆˆπ‘‰(π‘Š1) and any one of its neighbor 𝑣𝑙 in π‘Š1, π‘Šξ…žπ‘—=(𝑣𝑗,𝑒0)(𝑣𝑙,𝑒1)(𝑣𝑗,𝑒2)β‹―(𝑣𝑙,π‘’π‘˜)(𝑣𝑗,𝑒0)β‰…π‘Š2 and 𝑉(𝐷)=βˆͺ𝑠𝑖=0𝑉(π‘Šξ…žπ‘–). Since 𝑣𝑗 and 𝑣𝑙 are adjacent in π‘Š1, it follows that 𝑙=𝑗+1 or 2π‘š+𝑗. Let π‘Šξ…žξ…ž=𝑣𝑗,𝑒0𝑣𝑗+1,𝑒1𝑣𝑗,𝑒2⋯𝑣𝑗+1,π‘’π‘˜π‘£ξ€Έξ€·π‘—,𝑒0ξ€Έ,π‘Šξ…žξ…žξ…ž=𝑣𝑗,𝑒0𝑣2π‘š+𝑗,𝑒1𝑣𝑗,𝑒2⋯𝑣2π‘š+𝑗,π‘’π‘˜π‘£ξ€Έξ€·π‘—,𝑒0ξ€Έ.(2.1)
The sum of the suffixes of each vertex in π‘Šξ…žξ…ž has the same parity as the integer 𝑗. Since 𝑣𝑗 may be any vertex of π‘Š1, 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 π‘Šξ…žξ…žξ…ž 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 (𝑒𝑖,𝑣𝑗) and (π‘’π‘˜,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 (π‘₯1,π‘₯2) to (𝑦1,𝑦2) if and only if (𝑦2,π‘₯2)∈𝐴2, and (𝑦1,π‘₯1) or (π‘₯1,𝑦1)∈𝐴1.

Orientation 3. Let 𝐷1=(𝑉1,𝐴1) and 𝐷2=(𝑉2,𝐴2) be two digraphs. 𝐷1×𝐷2 contains an arc from (π‘₯1,π‘₯2) to (𝑦1,𝑦2) if and only if (π‘₯1,𝑦1)∈𝐴1, and (π‘₯2,𝑦2) or (𝑦2,π‘₯2)∈𝐴2.

Orientation 4. Let 𝐷1=(𝑉1,𝐴1) and 𝐷2=(𝑉2,𝐴2) be two digraphs. 𝐷1×𝐷2 has an arc from (π‘₯1,π‘₯2) to (𝑦1,𝑦2) if and only if (𝑦1,π‘₯1)∈𝐴1, and (π‘₯2,𝑦2) or (𝑦2,π‘₯2)∈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×𝐷1≅𝐷1×𝐷2. From this observation and Corollary 3.1, Corollary 3.2 follows directly.

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 π‘‹βŠŠπ‘‰(𝐷) be a nonempty vertex set. Denote by (𝑋,𝑉(𝐷)βˆ’π‘‹) the set of arcs with tail in 𝑋 and head in 𝑉(𝐷)βˆ’π‘‹, which is called a directed cut of 𝐷. If 𝑇 is a directed cut of strongly connected digraph 𝐷, then π·βˆ’π‘‡ is connected but is not strongly connected. The size πœ†(𝐷) of minimum directed cuts of digraph 𝐷 is called its strong arc connectivity. Let 𝛽(𝐷)=min{|𝑆|βˆΆπ‘†βŠ‚π΄(𝐷)suchthatπ·βˆ’π‘†isabipartitegraph}, 𝛿(𝐷)=min{𝛿+(𝐷),π›Ώβˆ’(𝐷)} be the minimum degree of 𝐷. For every integer 𝛿(𝐷)β‰₯𝑗β‰₯πœ†(𝐷), let 𝛽𝑗=min{𝛽(𝐢)∢𝐢 be any strongly connected component of π·βˆ’π‘‡ 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 ((π‘₯1,π‘₯2),(𝑦1,𝑦2)) of 𝐷1×𝐷2, the arc (π‘₯2,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 πœ†(→𝐾2×𝐷)=min{2πœ†(𝐷),𝛽(𝐷),min{𝑗+π›½π‘—βˆΆπœ†(𝐷)≀𝑗≀𝛿}}.

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)β‰₯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 π·βˆ’πΉ=(𝑋,π‘Œ;𝐸) is a bipartite graph with bipartition 𝑋 and π‘Œ. Then 𝛽(𝐷)=|𝐹| and, πœ†(𝐷)=|𝑇|β‰₯1 by Lemma 2.1.
Claim  1. πœ†(→𝐾2×𝐷)≀min{2πœ†(𝐷),𝛽(𝐷),min{𝑗+π›½π‘—βˆΆπœ†(𝐷)≀𝑗≀𝛿}.
Let 𝐴(→𝐾2)={(π‘Ž,𝑏)}. By Theorem 1.1, →𝐾2Γ—(π·βˆ’πΉ) consists of two components. The vertex sets of these two components are {(π‘Ž,π‘₯)∢π‘₯βˆˆπ‘‹}βˆͺ{(𝑏,𝑦)βˆΆπ‘¦βˆˆπ‘Œ} and {(𝑏,π‘₯)∢π‘₯βˆˆπ‘‹}βˆͺ{(π‘Ž,𝑦)βˆΆπ‘¦βˆˆπ‘Œ}. It is not difficult to see that {((π‘Ž,π‘₯),(𝑏,𝑦))∢(π‘₯,𝑦)∈𝐹∩𝐴(𝐷[𝑋])}βˆͺ{((𝑏,π‘₯),(π‘Ž,𝑦))∢(π‘₯,𝑦)∈𝐹∩𝐴(𝐷[π‘Œ])} is a directed cut of →𝐾2×𝐷, since its removal makes {(𝑏,π‘₯)∢π‘₯βˆˆπ‘‹}βˆͺ{(π‘Ž,𝑦)βˆΆπ‘¦βˆˆπ‘Œ} not reachable from {(π‘Ž,π‘₯)∢π‘₯βˆˆπ‘‹}βˆͺ{(𝑏,𝑦)βˆΆπ‘¦βˆˆπ‘Œ}. And so, πœ†(←𝐾2×𝐷)≀|𝐹|=𝛽(𝐷).
By Lemma 4.1, π·βˆ’π‘‡ contains a strongly connected component 𝐷1 that has no outer neighbors. By the minimality of 𝑇, we have (𝐷1,𝐷1)=𝑇. Obviously, {((π‘Ž,π‘₯),(𝑏,𝑦))∢(π‘₯,𝑦)βˆˆπ‘‡}βˆͺ{((π‘Ž,𝑦),(𝑏,π‘₯))∢(π‘₯,𝑦)βˆˆπ‘‡} is a directed cut of →𝐾2×𝐷. Hence πœ†(→𝐾2×𝐷)≀2|𝑇|=2πœ†(𝐷).
Let 𝑇𝑗 be a directed cut of 𝐷 that has size 𝑗, 𝐢𝑗 be a strongly connected component of π·βˆ’π‘‡π‘— with (𝐢𝑗,π·βˆ’π‘‡π‘—βˆ’πΆπ‘—)=βˆ… (by Lemma 4.1 such components exist), 𝐹𝑗 be an arc set of 𝐢𝑗 such that πΆπ‘—βˆ’πΉπ‘—=(𝑋𝑗,π‘Œπ‘—;𝐸𝑗) is a bipartite subgraph with bipartition (𝑋𝑗,π‘Œπ‘—). Then →𝐾2Γ—(πΆπ‘—βˆ’πΉπ‘—) consists of two components, whose vertex sets are {(π‘Ž,π‘₯)∢π‘₯βˆˆπ‘‹π‘—}βˆͺ{(𝑏,𝑦)βˆΆπ‘¦βˆˆπ‘Œπ‘—} and {(𝑏,π‘₯)∢π‘₯βˆˆπ‘‹π‘—}βˆͺ{(π‘Ž,𝑦)βˆΆπ‘¦βˆˆπ‘Œπ‘—}. It’s not difficult to see that the union of {((π‘Ž,π‘₯),(𝑏,𝑦))∢(π‘₯,𝑦)βˆˆπΉπ‘—βˆ©π΄(𝐷[𝑋𝑗])}, {((𝑏,π‘₯),(π‘Ž,𝑦))∢(π‘₯,𝑦)βˆˆπΉπ‘—βˆ©π΄(𝐷[π‘Œπ‘—])}, {((π‘Ž,π‘₯),(𝑏,𝑦))∢(π‘₯,𝑦)∈(𝑋𝑗,π·βˆ’πΆπ‘—)} and {((𝑏,π‘₯),(π‘Ž,𝑦))∢(π‘₯,𝑦)∈(π‘Œπ‘—,π·βˆ’πΆπ‘—)} is a directed cut of →𝐾2×𝐷, since its removal makes {(π‘Ž,𝑦)βˆΆπ‘¦βˆˆπ‘Œπ‘—}βˆͺ{(𝑏,π‘₯)∢π‘₯βˆˆπ‘‹π‘—} not reachable from {(π‘Ž,π‘₯)∢π‘₯βˆˆπ‘‹π‘—}βˆͺ{(𝑏,𝑦)βˆΆπ‘¦βˆˆπ‘Œπ‘—}. Noticing that |{((π‘Ž,π‘₯),(𝑏,𝑦))∢(π‘₯,𝑦)βˆˆπΉπ‘—βˆ©π΄(𝐷[𝑋𝑗])}βˆͺ{((𝑏,π‘₯),(π‘Ž,𝑦))∢(π‘₯,𝑦)βˆˆπΉπ‘—βˆ©π΄(𝐷[π‘Œπ‘—])}|=|𝐹𝑗|, |{((π‘Ž,π‘₯),(𝑏,𝑦))∢(π‘₯,𝑦)∈(𝑋𝑗,π·βˆ’πΆπ‘—)}βˆͺ{((𝑏,π‘₯),(π‘Ž,𝑦))∢(π‘₯,𝑦)∈(π‘Œπ‘—,π·βˆ’πΆπ‘—)}|=|𝑇𝑗| and |𝑇𝑗βˆͺ𝐹𝑗|=|𝑇𝑗|+|𝐹𝑗|=𝑗+𝛽𝑗, we deduce that πœ†(→𝐾2×𝐷)≀min{𝑗+π›½π‘—βˆΆπœ†(𝐷)≀𝑗≀𝛿}. And so, Claim  1 follows.
Claim  2. πœ†(→𝐾2×𝐷)β‰₯min{2πœ†(𝐷),𝛽(𝐷),min{𝑗+π›½π‘—βˆΆπœ†(𝐷)≀𝑗≀𝛿}.
Let 𝑆 be a minimum direct cut of →𝐾2×𝐷, 𝑉1={π‘₯βˆˆπ‘‰(𝐷): vertices (π‘Ž,π‘₯) and (𝑏,π‘₯) lie in common strongly connected component of →𝐾2Γ—π·βˆ’π‘†}, 𝑉2={π‘₯βˆˆπ‘‰(𝐷): vertices (π‘Ž,π‘₯) and (𝑏,π‘₯) lie in different components of →𝐾2Γ—(π·βˆ’π‘†)}. Then (𝑉1,𝑉2) is a partition of 𝑉(𝐷). From the minimality of 𝑆 it follows that →𝐾2Γ—π·βˆ’π‘† consists of two strongly connected components, say 𝐢1 and 𝐢2. Noticing that (𝐢1,𝐢2)βŠ†π‘† or (𝐢2,𝐢1)βŠ†π‘†, we assume without loss of generality that (𝐢1,𝐢2)βŠ†π‘†. Now three different cases occur: 𝑉1β‰ βˆ…=𝑉2; 𝑉1=βˆ…β‰ π‘‰2; 𝑉1β‰ βˆ…β‰ π‘‰2.
Consider at first the case when 𝑉1β‰ βˆ…=𝑉2. By Lemma 2.1, we deduce that {π‘₯βˆˆπ‘‰(𝐷)∢(π‘Ž,π‘₯)βˆˆπ‘‰(𝐢1)} induces a strongly connected component of 𝐷 as well as {π‘¦βˆˆπ‘‰(𝐷)∢(π‘Ž,𝑦)βˆˆπ‘‰(𝐢2)} in this case. Furthermore, ({π‘₯βˆˆπ‘‰(𝐷)∢(π‘Ž,π‘₯)βˆˆπ‘‰(𝐢1)},{π‘¦βˆˆπ‘‰(𝐷)∢(π‘Ž,𝑦)βˆˆπ‘‰(𝐢2)}) is a directed cut of 𝐷 and ({(π‘Ž,π‘₯)∢(π‘Ž,π‘₯)βˆˆπ‘‰(𝐢1)},{(𝑏,𝑦)∢(𝑏,𝑦)βˆˆπ‘‰(𝐢2)})βˆͺ({(𝑏,π‘₯)∢(𝑏,π‘₯)βˆˆπ‘‰(𝐢1)},{(π‘Ž,𝑦)∢(π‘Ž,𝑦)βˆˆπ‘‰(𝐢2)})βŠ†π‘†. It follows from these observations that ||𝑆||β‰₯||𝐢(π‘Ž,π‘₯)βˆˆπ‘‰1,𝐢(𝑏,𝑦)βˆˆπ‘‰2||+||ξ€½(𝐢𝑏,π‘₯)βˆˆπ‘‰1,ξ€½(ξ€·πΆξ€Έξ€Ύπ‘Ž,𝑦)βˆˆπ‘‰2||ξ€Έξ€Ύβ‰₯2πœ†(𝐷).(4.1)
Consider secondly the case when 𝑉1=βˆ…β‰ π‘‰2. Let 𝑀={π‘₯βˆˆπ‘‰(𝐷)∢(π‘Ž,π‘₯)βˆˆπ‘‰(𝐢1)} and 𝑁={π‘¦βˆˆπ‘‰(𝐷)∢(𝑏,𝑦)βˆˆπ‘‰(𝐢1)}. Then (𝑀,𝑁) is a partition of 𝑉(𝐷) and (𝐢1,𝐢2)={((π‘Ž,π‘₯),(𝑏,𝑦))∢(π‘₯,𝑦)∈𝐴(𝐷[𝑀])}βˆͺ{((𝑏,π‘₯),(π‘Ž,𝑦))∢(π‘₯,𝑦)∈𝐴(𝐷[𝑁])}. Since π·βˆ’π΄(𝐷[𝑀])βˆ’π΄(𝐷[𝑁]) is a bipartite subgraph of 𝐷, it follows that |(𝐢1,𝐢2)|=|𝐴(𝐷[𝑀])|+|𝐴(𝐷[𝑁])|β‰₯𝛽(𝐷). Recalling that (𝐢1,𝐢2)βŠ†π‘†, we have |𝑆|β‰₯|(𝐢1,𝐢2)|β‰₯𝛽(𝐷).
Consider finally the case when 𝑉1β‰ βˆ…β‰ π‘‰2. Let 𝐢𝐻=π‘₯βˆˆπ‘‰(𝐷)∢(π‘Ž,π‘₯)βˆˆπ‘‰1ξ€Έ,π‘₯βˆˆπ‘‰1ξ€Ύ,𝐢𝑄=π‘₯βˆˆπ‘‰(𝐷)∢(π‘Ž,π‘₯)βˆˆπ‘‰2ξ€Έ,π‘₯βˆˆπ‘‰1ξ€Ύ,ξ€½ξ€·πΆπ‘Š=π‘₯βˆˆπ‘‰(𝐷)∢(π‘Ž,π‘₯)βˆˆπ‘‰1ξ€Έ,π‘₯βˆˆπ‘‰2ξ€Ύ,𝐢𝑍=π‘₯βˆˆπ‘‰(𝐷)∢(π‘Ž,π‘₯)βˆˆπ‘‰2ξ€Έ,π‘₯βˆˆπ‘‰2ξ€Ύ.(4.2)
Then (𝐻,𝑄,π‘Š,𝑍) is a partition of 𝑉(𝐷), refer to (1) of Figure 2. Since (𝐢1,𝐢2)βŠ†π‘†, the arcs in this set is removed in Figure 2. If π»β‰ βˆ…β‰ π‘„, then the set of arcs from {(π‘Ž,π‘₯)∢π‘₯∈𝐻}βˆͺ{(𝑏,π‘₯)∢π‘₯∈𝐻} to 𝐢2 is a directed cut of →𝐾2×𝐷, but it has less arcs than 𝑆. This contradiction shows that either 𝐻=βˆ… or 𝑄=βˆ…. Assume without loss of generality that 𝑄=βˆ…. Then →𝐾2Γ—π·βˆ’π‘† can be depicted as (2) of Figure 2.
On the one hand, for every arc (π‘₯,𝑦)∈(𝐻,π·βˆ’π»), if π‘¦βˆˆπ‘Š then ((π‘Ž,π‘₯),(𝑏,𝑦))βˆˆπ‘†; if π‘¦βˆˆπ‘ then ((𝑏,π‘₯),(π‘Ž,𝑦))βˆˆπ‘†. On the other hand, for every arc (π‘₯,𝑦)∈𝐴(𝐷[π‘Š]) the arc ((π‘Ž,π‘₯),(𝑏,𝑦))βˆˆπ‘† and for every arc (π‘₯,𝑦)∈𝐴(𝐷[𝑍]) the arc ((𝑏,π‘₯),(π‘Ž,𝑦))βˆˆπ‘†. It follows from these observations that ||𝑆||β‰₯||[]||+ξ€·||𝐴[π‘Š])||+||𝐴[𝑍])||𝐻,π·βˆ’π»(𝐷(𝐷β‰₯min𝑗+𝛽𝑗,βˆΆπœ†(𝐷)≀𝑗≀𝛿(𝐷)(4.3) where [𝐻,π·βˆ’π»] represents the set of arcs with ends in 𝐻 and π·βˆ’π», 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).

References

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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.

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