Steiner Distance in Product Networks *footnote **footnote *Supported by the National Science Foundation of China (Nos. 11601254, 11551001, 11161037, and 11461054) and the Science Found of Qinghai Province (Nos. 2016-ZJ-948Q, and 2014-ZJ-907).

Steiner Distance in Product Networks 1

Abstract

For a connected graph of order at least and , the Steiner distance among the vertices of is the minimum size among all connected subgraphs whose vertex sets contain . Let and be two integers with . Then the Steiner -eccentricity of a vertex of is defined by . Furthermore, the Steiner -diameter of is . In this paper, we investigate the Steiner distance and Steiner -diameter of Cartesian and lexicographical product graphs. Also, we study the Steiner -diameter of some networks.
Keywords: distance; diameter; Steiner tree; Steiner distance; Steiner -diameter; Cartesian product, lexicographical product.
AMS subject classification 2010: 05C05; 05C12; 05C76.

1 Introduction

In this paper, we consider graphs that are undirected, finite and simple. We refer the readers to [6] for graph theoretical notations and terminology that are not defined here. We use standard definitions and notations. For a graph , let , , and denote the set of vertices, the set of edges and the minimum degree of , respectively. We refer to the order of the graph and the size of the graph. The degree of a vertex in is denoted by . In this paper, , , and correspond to the complete graph of order , the path of order , the star of order , and the cycle of order , respectively. If , we use to denote the subgraph induced by . Similarly, if , let denote the subgraph induced by . If , we use to denote the subgraph of obtained from by removing all the elements of and the edges incident to vertices that are in . If , we write for notational simplicity. For , we use to denote the set of edges of with one end in and the other end in . If , we simply write for . We divide our introduction into subsections to state the motivations of this paper.

1.1 Distance and its generalizations

Distance is a fundamental concept in graph theory. Let be a connected graph. The distance between two vertices and in is the length of a shortest path between them, and it is denoted by . The eccentricity of in , denoted by (or simply if it is clear from the context), is . In addition, we define the radius and the diameter of to be and . It is a standard exercise to check that . The center of is the subgraph induced by the vertices with eccentricity equal to the radius. For a recent survey paper, see Goddard and Oellermann [25].

We observe that the distance between two vertices and in is equal to the minimum size of a connected subgraph of containing both and . This suggests a generalization of the concept of distance. Indeed, such as generalization, known as the Steiner distance of a graph, was introduced by Chartrand, Oellermann, Tian and Zou in 1989. It is a natural and nice generalization. Let be a set of vertices in a graph where . We define an -Steiner tree or a Steiner tree connecting (or simply, an -tree) to be a subgraph of that is a tree with . Moreover, the Steiner distance of in (or simply the distance of ) is the minimum size among all connected subgraphs whose vertex sets contain . (Set when there is no -Steiner tree in .) It follows that if is a connected subgraph of such that and , then is a tree. We further remark that , where is subtree of . Finally, if , then is the classical distance between and . The following observation is obvious.

Observation 1.1

Let be a graph of order and be an integer with . If and , then .

Let and be two integers with . We define the Steiner -eccentricity of a vertex of to be , the Steiner -radius of to be , and the Steiner -diameter of is . We remark that for every connected graph that for all vertices of and that and . It is not difficult to see the following observation.

Observation 1.2

Let be two integers with .

If is a spanning subgraph of , then .

For a connected graph , .

Chartrand, Okamoto, Zhang [11] gave the following upper and lower bounds of .

Theorem 1.1

[11] Let be two integers with , and let be a connected graph of order . Then . Moreover, the upper and lower bounds are sharp.

Dankelmann, Swart and Oellermann [14] showed that . Ali, Dankelmann, Mukwembi [2] improved the bound and showed that where is connected. Moreover, they showed that these bounds are asymptotically best possible via a construction.

1.2 Related concepts

Although we will not consider these related concepts in this paper, they provide a context of problems related to Steiner distance. The first such concept is a generalization of the center of a graph. One defines the Steiner -center of a connected graph to be the subgraph induced by the vertices of where . Oellermann and Tian [48] showed that every graph is the -center of some graph. More importantly, they showed that the -center of a tree is a tree and they characterized those trees that are -centers of trees. The Steiner -median of is the subgraph of induced by the vertices of of minimum Steiner -distance. The papers [46, 47, 48] contain important results for Steiner centers and Steiner medians.

Let be a -connected graph and , be a pair of vertices of . Let be a family of internally vertex-disjoint paths between and and be the length of the longest path in . Then the -distance between vertices and is the smallest among all ’s and the -diameter of is the maximum -distance over all pairs of vertices of . The concept of -diameter has its origin in the analysis of routings in networks. We refer the readers to Chung [12], Du, Lyuu and Hsu [20], Hsu [31, 32], Meyer and Pradhan [45] for additional information.

Perhaps the most famous Steiner type problem is the Steiner tree problem. The original Steiner tree problem was stated for the Euclidean plane: Given a set of points on the plane, the goal is to connect these points, and possibly additional points, by line segments between some pairs of these points such that the total length of these line segments is minimized. The graph theoretical version [29, 37] is as follows: Given a graph and a set of vertices , find a connected subgraph with minimum number of edges that contains . This is, in general, an NP-hard problem [33]. There is also a corresponding weighted version. Obviously, this has applications in computer science and electrical engineering. For example, a graph can be a computer network with vertices being computers and edges being links between them, or it can be the underlying graph topology of a supercomputer such as the IBM Blue Gene. Here the Steiner tree problem is to find a subnetwork containing these computers with the least number of links. We can replace processors by electrical stations for applications in electrical networks.

Li et al. [38] gave a related concept. They defined the -center Steiner Wiener index of the graph to be

(1.1)

For , it coincides with the ordinary Wiener index. One usually considers for . However, the above definition can be extended to and as well where and . There are other related concepts such as the Steiner Harary index. Both indices have chemical applications [28, 21]. In addition, Gutman [27] gave a generalization of the concept of degree distance. We refer the readers to [21, 28, 27, 38, 39, 42, 43, 44] for details.

1.3 Products of graphs

The main focus of this paper is Steiner -diameter of two products of graphs, namely, the Cartesian product and the lexicographic product. These are well-known products. See [30].

The Cartesian product of two graphs and , written as , is the graph with vertex set , in which two vertices and are adjacent if and only if and , or and .

The lexicographic product of two graphs and , written as , is defined as follows: , and two distinct vertices and of are adjacent if and only if either or and .

It is easy to see that the Cartesian product is commutative, that is, is isomorphic to . However, the lexicographic product is non-commutative.

Product networks are important as often the resulting graph inherits properties from its factors. Both the lexicographical product and the Cartesian product are important concepts. See [4, 17, 30, 36].

Gologranc [26] obtained a sharp lower bound for Steiner distance of Cartesian product graphs. We continue this study in Section by obtaining a sharp upper bound for Steiner distance. In addition, we will also present sharp upper and lower bounds for Steiner -diameter of Cartesian product graphs. In Section , we derive the results for Steiner distance and Steiner -diameter of lexicographic product graphs, which strengthen a result given by Anand et al. [3]. In Section , we give some applications of our main results, and study the Steiner diameter of some important networks.

2 Results for Cartesian product

In this paper, let and be two graphs with and , respectively. Then , where denotes the Cartesian product operation or lexicographical product operation. For , we use to denote the subgraph of induced by the vertex set . Similarly, for , we use to denote the subgraph of induced by the vertex set .

The following observation can be easily seen.

Observation 2.1

Let be a connected graph, and let . Let be a minimal -Steiner tree. Then the tree satisfies one of the following conditions.

is a path;

is a subdivision of .

We start with the following basic result.

Lemma 2.1

[30] Let and be two vertices of . Then

2.1 Steiner distance of Cartesian product graphs

In [26], Gologranc obtained the following lower bound for Steiner distance.

Lemma 2.2

[26] Let be an integer, and let be two connected graphs. Let be a set of distinct vertices of . Let and . Then

We will show that the inequality in Lemma 2.2 can be equality if ; shown in following Corollary 2.2. But, for general , from Lemma 2.2 and Corollary 2.2, one may conjecture that for two connected graphs , , where , and .

Figure 1: Graphs for Remark 1.

Remark 1: Actually, the equality is not true for . For example, let be a tree with degree sequence and be a path of order . Let be a vertex set of shown in Figure 1. Then for , and for . One can check that there is no -Steiner tree of size in , which implies .


Although the conjecture of such an ideal formula is not correct, it is possible to give a strong upper bound for general . Remark 1 also indicates that obtaining a nice formula for the general case may be difficult. We now give such an upper bound of for and .

Theorem 2.1

Let be three integers with , and let be two connected graphs with and . Let be a set of distinct vertices of , , and , where , ( are both multi-sets). Then

where are defined as follows.

Let be all the -multi-subsets of in , and let be the numbers of distinct vertices in , and let .

Let be all the -multi-subsets of in , and let be the numbers of distinct vertices in , and let .

Proof. From Lemma 2.2, we have . By symmetry, we only need to show . Recall that and . Without loss of generality, we assume that be the copies such that , . Then , and hence we have the following cases to consider.

Case 1. For each , .

Without loss of generality, let , where . Thus, we have for each , and . Note that . On one hand, since there is an -Steiner tree of size in , it follows that there exists an Steiner tree of size connecting

in , say . For each , let be the Steiner tree in corresponding to in . Note that is the Steiner tree of size connecting in . One can see that . On the other hand, since there is an -Steiner tree of size in , it follows that there exists an Steiner tree of size connecting in , say . Furthermore, the subgraph induced by the edges in is an -Steiner tree in (see Figure 2 ), and hence .

From the definition of , if for each , then and . If there exists some such that for each , then and .

Figure 2: Graphs for Cases 1 and 2 in the proof of Theorem 2.1.

Case 2. There exists some such that , where .

Without loss of generality, we assume that for each , where . For , we have for each . One can see that

Subcase 2.1. .

If , then . Since there is an -Steiner tree of size in , it follows that there exists an Steiner tree of size connecting in , say . Since there is an -Steiner tree of size in , it follows that for each , there exists an Steiner tree of size connecting

in , say . For each , let be the Steiner tree in corresponding to in . Note that is the Steiner tree of size connecting in . Furthermore, the subgraph induced by the edges in is an -Steiner tree (see Figure 2 ), and hence . From the definition of , we have , and hence , as desired.

If , then we can assume that , , and . Furthermore, we can assume that . Since there is an -Steiner tree of size in , it follows that for each , there is the Steiner tree of size connecting in . Since there is an -Steiner tree of size in , it follows that there exists an Steiner tree of size connecting in , say . Then the subgraph induced by the edges in

is an -Steiner tree in , and hence . Since , it follows that .

From now on, we assume . Note that there is an -Steiner tree of size in , say . Without loss of generality, let . Since , it follows that there is a minimal subtree connecting in . From Observation 2.1, is a path or is a subdivision of . If is a path, then without loss of generality, we can assume is the interval vertex of . Therefore, there are a unique -path, say , and a unique -path, say , in . If is a subdivision of , then there exists a vertex in , say , such that there are three paths connecting and , respectively, in .

We first consider the case that is a path. On one hand, for each , let be the Steiner tree in corresponding to in . Note that is the Steiner tree of size connecting in . For each , let be the path in corresponding to in , and let be the path in corresponding to in . On the other hand, since there is an -Steiner tree of size in , it follows that there exists an Steiner tree of size connecting in , say . Furthermore, the subgraph induced by the edges in

is an -Steiner tree in (see Figure 3 ), and hence . Since , it follows that .

Figure 3: Graphs for Case 3 in the proof of Theorem 2.1.

Next, we consider the case that is a subdivision of . On one hand, for each , let be the tree in corresponding to in . Note that is the Steiner tree of size connecting