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 and , denoted by , is defined on vertex set , where two vertices and are adjacent to each other in if and only if and . 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 and be connected nonempty graphs. Then is connected if and only if at least one of them is nonbipartite. Furthermore, if both and are bipartite, then has exactly two components.

Let and 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 and , there are four natural orientations of the two edges and . For clarity and comparison, we depicture the Cartesian product and the four orientations of direct-product digraph in Figure 1.

Figure 1 

Orientations of direct-product digraph .

A digraph is strongly connected or disconnected if any vertex is reachable from any other vertex, where a vertex is said to be reachable form another vertex if there is a directed path from to . The minimum number 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 contains an arc from vertex to if and only if arc , and or .

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

Proof. Necessity. If is bipartite, then by Theorem 1.1 the underlying graph of is disconnected. This contradiction shows that is nonbipartite. Let be the vertex set of . Since is strongly connected, it follows that for any two vertices , contains a directed walk from to . Let . Then it yields a directed walk of . The necessity follows from this observation.
Sufficiency. Since is strongly connected, it contains a spanning closed directed walk , which corresponds two directed walks and of , where . It is obvious that is a spanning subgraph of .
Case  1. The length of is odd, say, .
In this case, and . And so is a spanning closed directed walk of . Hence, is strongly connected.
Case  2. The length of is even, say, .
In this case, both and are closed directed walks. Since is nonbipartite, it contains an arc joining two vertices of whose suffixes have same parity. Without loss of generality, let . This arc corresponds to two arcs , of . These two arcs make and reachable from each other. Therefore, is strongly connected.

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

Proof. Sufficiency. If is nonbipartite and strongly connected, by Lemma 2.1, is strongly connected.
If is strongly connected and bipartite, but is nonbipartite, then contains a spanning closed directed walk . Since is bipartite, it follows that is odd. Let be a spanning closed walk of , where is an odd cycle. For any vertex and any one of its neighbor in , and . Since and are adjacent in , it follows that or . Let
The sum of the suffixes of each vertex in has the same parity as the integer . Since may be any vertex of , it follows that the vertices of whose suffix sum has same parity induce a strong component (a strongly connected vertex-induced subgraph with as many as possible vertices). Since contains vertices with odd suffix sum as well vertices with even suffix sum, it follows that is strongly connected.
Necessity. Since is strongly connected, every two vertices and are reachable from each other in . It follows that and are reachable from each other in . Hence, is strongly connected. If is strongly connected, but both and are bipartite, then 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 and be two digraphs. contains an arc from to if and only if , and or .

Orientation 3. Let and be two digraphs. contains an arc from to if and only if , and or .

Orientation 4. Let and be two digraphs. has an arc from to if and only if , and or .

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 has Orientation  1 (refer to (2) of Figure 1) and has Orientation 3 then . From this observation and Corollary 3.1, Corollary 3.2 follows directly.

Figure 2 

Partition of  .

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

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

4. Strong Arc Connectivity

Let be a strongly arc connected digraph and be a nonempty vertex set. Denote by the set of arcs with tail in and head in , which is called a directed cut of . If is a directed cut of strongly connected digraph , then is connected but is not strongly connected. The size of minimum directed cuts of digraph is called its strong arc connectivity. Let , be the minimum degree of . For every integer , let be any strongly connected component of and is an arbitrary directed cut of of size .

Let be oriented according to the first orientation of Figure 1. For every arc of , the arc is called its projection on . The following Lemma 4.1 is immediate, so we omit its proof herein.

Lemma 4.1. If directed graph 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 is strongly connected, then ,.

Proof . By Lemma 2.1, Theorem 4.2 is clearly true in the case when is bipartite, and so we assume in what follows that and is nonbipartite. Let be a minimum direct cut of and let be a minimum arc-set of such that is a bipartite graph with bipartition and . Then and, by Lemma 2.1.
Claim  1. .
Let . By Theorem 1.1, consists of two components. The vertex sets of these two components are and . It is not difficult to see that is a directed cut of , since its removal makes not reachable from . And so, .
By Lemma 4.1, contains a strongly connected component that has no outer neighbors. By the minimality of , we have . Obviously, is a directed cut of . Hence .
Let be a directed cut of that has size , be a strongly connected component of with (by Lemma 4.1 such components exist), be an arc set of such that is a bipartite subgraph with bipartition . Then consists of two components, whose vertex sets are and . It’s not difficult to see that the union of , , and is a directed cut of , since its removal makes not reachable from . Noticing that , and , we deduce that . And so, Claim  1 follows.
Claim  2. .
Let be a minimum direct cut of , : vertices and lie in common strongly connected component of , : vertices and lie in different components of . Then is a partition of . From the minimality of it follows that consists of two strongly connected components, say and . Noticing that or , we assume without loss of generality that . Now three different cases occur: ; ; .
Consider at first the case when . By Lemma 2.1, we deduce that induces a strongly connected component of as well as in this case. Furthermore, is a directed cut of and . It follows from these observations that
Consider secondly the case when . Let and . Then is a partition of and . Since is a bipartite subgraph of , it follows that . Recalling that , we have .
Consider finally the case when . Let
Then is a partition of , refer to (1) of Figure 2. Since , the arcs in this set is removed in Figure 2. If , then the set of arcs from to is a directed cut of , but it has less arcs than . This contradiction shows that either or . Assume without loss of generality that . Then can be depicted as (2) of Figure 2.
On the one hand, for every arc , if then ; if then . On the other hand, for every arc the arc and for every arc the arc . It follows from these observations that where represents the set of arcs with ends in and , 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).