The Steiner diameter of a graph ^{*}^{*}*Supported by the National Science Foundation of China (Nos. 11601254, 11551001, 11371205, 11161037, 11101232, 11461054) and the Science Found of Qinghai Province (Nos. 2016ZJ948Q, and 2014ZJ907).
Abstract
The Steiner distance of a graph, introduced by Chartrand,
Oellermann, Tian and Zou in 1989, is a natural generalization of the
concept of classical graph distance. 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 be
two integers with . Then the Steiner
eccentricity of a vertex of is defined by
. Furthermore, the Steiner diameter of is
. In 2011, Chartrand,
Okamoto and Zhang showed that . In this
paper, graphs with are characterized,
respectively.
Keywords: Diameter, Steiner tree, Steiner diameter
AMS subject classification 2010: 05C05; 05C12; 05C75.
1 Introduction
All graphs in this paper are undirected, finite and simple. We refer to [4] for graph theoretical notation and terminology not described here. For a graph , let , , , , and denote the set of vertices, the set of edges, the size, minimum degree, and the complement of , respectively. In this paper, we let , , and be the complete graph of order , the path of order , the star of order , and the cycle of order , respectively. For any subset of , let denote the subgraph induced by ; similarly, for any subset of , let denote the subgraph induced by . We use to denote the subgraph of obtained by removing all the vertices of together with the edges incident with them from ; similarly, we use to denote the subgraph of obtained by removing all the edges of from . If and , we simply write and for and , respectively. For two subsets and of we denote by the set of edges of with one end in and the other end in . If , we simply write for . We divide our introduction into the following four subsections to state the motivations and our results of this paper.
1.1 Distance and its generalizations
Distance is one of the most basic concepts of graphtheoretic subjects. For a graph , let , , and denote the set of vertices, the set of edges, and the size of , respectively. If is a connected graph and , then the distance between and is the length of a shortest path connecting and . If is a vertex of a connected graph , then the eccentricity of is defined by . Furthermore, the radius and diameter of are defined by and . These last two concepts are related by the inequalities . The center of a connected graph is the subgraph induced by the vertices of with . Recently, Goddard and Oellermann gave a survey paper on this subject, see [20].
The distance between two vertices and in a connected graph also equals the minimum size of a connected subgraph of containing both and . This observation suggests a generalization of distance. The Steiner distance of a graph, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural and nice generalization of the concept of classical graph distance. For a graph and a set of at least two vertices, an Steiner tree or a Steiner tree connecting (or simply, an tree) is a subgraph of that is a tree with . Let be a connected graph of order at least and let be a nonempty set of vertices of . Then the Steiner distance among the vertices of (or simply the distance of ) is the minimum size among all connected subgraphs whose vertex sets contain . Note that if is a connected subgraph of such that and , then is a tree. Observe that , where is subtree of . Furthermore, if , then is the classical distance between and . Set when there is no Steiner tree in .
Let and be two integers with . The Steiner eccentricity of a vertex of is defined by . The Steiner radius of is , while the Steiner diameter of is . Note for every connected graph that for all vertices of and that and . Each vertex of the graph of Figure 1 is labeled with its Steiner eccentricity, so that and .
Observation 1
Let be two integers with .
If is a spanning subgraph of , then .
For a connected graph , .
In [8], Chartrand, Okamoto, Zhang obtained the following result.
Theorem 1
[8] Let be two integers with , and let be a connected graph of order . Then . Moreover, the upper and lower bounds are sharp.
In [13], Dankelmann, Swart and Oellermann obtained a bound on for a graph in terms of the order of and the minimum degree of , that is, . Later, Ali, Dankelmann, Mukwembi [2] improved the bound of and showed that for all connected graphs . Moreover, they constructed graphs to show that the bounds are asymptotically best possible.
As a generalization of the center of a graph, the Steiner center of a connected graph is the subgraph induced by the vertices of with . Oellermann and Tian [41] showed that every graph is the center of some graph. In particular, they showed that the center of a tree is a tree and those trees that are centers of trees are characterized. The Steiner median of is the subgraph of induced by the vertices of of minimum Steiner distance. For Steiner centers and Steiner medians, we refer to [39, 40, 41].
The average Steiner distance of a graph , introduced by Dankelmann, Oellermann and Swart in [11], is defined as the average of the Steiner distances of all subsets of , i.e.
For more details on average Steiner distance, we refer to [11, 12].
Let be a connected graph and , be any pair of vertices of . Let be a family of inner vertexdisjoint paths between and , i.e., , where and denotes the number of edges of path . The distance between vertices and is the minimum among all and the diameter of is defined as the maximum distance over all pairs of vertices of . The concept of diameter emerges rather naturally when one looks at the performance of routing algorithms. Its applications to network routing in distributed and parallel processing are studied and discussed by various authors including Chung [9], Du, Lyuu and Hsu [16], Hsu [24, 25], Meyer and Pradhan [34].
1.2 Application background of Steiner distance
Let be a connected graph and , be any pair of vertices of . Let be a family of internally vertexdisjoint paths between and , i.e. , where and denotes the number of edges of path . The distance between vertices and is the minimum among all and the diameter of is defined as the maximum distance over all pairs of vertices of . The concept of diameter emerges rather naturally when one looks at the performance of routing algorithms. Its applications to network routing in distributed and parallel processing are studied and discussed by various authors including Chung [9], Du, Lyuu and Hsu [16], Hsu [24, 25], Meyer and Pradhan [34].
The Wiener index of the graph is defined as . Details on this oldest distance–based topological index can be found in numerous surveys, e.g., in [37, 38, 15, 42]. Li et al. [28] put forward a Steiner–distance–based generalization of the Wiener index concept. According to [28], the center Steiner Wiener index of the graph is defined by
(1.1) 
For , the above defined Steiner Wiener index coincides with the ordinary Wiener index. It is usual to consider for , but the above definition would be applicable also in the cases and , implying and . A chemical application of was recently reported in [22]. Gutman [21] offered an analogous generalization of the concept of degree distance. Later, Furtula, Gutman, and Katanić [17] introduced the concept of Steiner Harary index and gave its chemical applications. For more details on Steiner distance indices, we refer to [17, 22, 21, 28, 29, 31, 32, 33].
1.3 Our results
From Theorem 1, we have . In [30], Mao characterized the graphs with , respectively, and studied the NordhausGaddumtype problem of the parameter .
In this paper, graphs with are characterized, respectively.
Theorem 2
Let be a connected graph of order .
If , then ;
If , then if and only if and is not a subgraph of .
A graph is defined as a connected graph of order obtained from a with vertex set and four stars by identifying the center of one star and one vertex in , where , , and ; see Figure 1.3.
A graph is defined as a connected graph of order obtained from with vertex set , and two stars , by identifying the center of a star and one vertex in , and then adding the paths , where , , and ; see Figure 1.1.
A graph is defined as a connected graph of order obtained from a cycle by adding the paths and the paths , where , and ; see Figure 1.1.
A graph is defined as a connected graph of order obtained from a star with vertex set and a star by identifying and the center of , where is the center of , and then adding the vertices and the edges , where , and ; see Figure 1.1.
Theorem 3
Let be a connected graph of order . Then if and only if satisfies one of the following conditions.
and is a subgraph of ;
and each is not a spanning subgraph of (see Figure 1.3).
We now define some graph classes.

Let be a tree of order obtained from three paths of length respectively by identifying the th vertex of and one endvertex of , and then identifying the th vertex of and one endvertex of (Note that and can be the same vertex);

Let be an unicyclic graph of order obtained from three paths of length respectively by identifying the th vertex of and one endvertex of , and then identifying the th vertex of and one endvertex of , and then adding an edge (Note that and can be the same vertex).

Let be an bicyclic graph of order obtained from three paths of length respectively by identifying the th vertex of and one endvertex of , and then identifying the th vertex of and one endvertex of , and then adding two edges and (Note that and can be the same vertex).

Let be a graph of order obtained from a cycle of order and four paths of length respectively by identifying each vertex of this cycle with an endvertex of one of the four paths.

Let be a graph of order obtained from and four paths of length respectively by identifying each vertex of with an endvertex of one of the four paths, where denotes the graph obtained from a clique of order by deleting one edge.
Theorem 4
Let be a connected graph of order . Then if and only if or or or or or .
2 Proofs of Theorem and
In this section, we characterize graphs with and give the proofs of Theorems and .
Lemma 1
Let be a connected graph of order , and let be an integer with . Then if and only if the number of noncut vertices in is at most .
Proof. Let be the number of noncut vertices in . Suppose . We claim that . Assume, to the contrary, that . For any with , there exists a noncut vertex in , say , such that . Then is connected, and hence contains a spanning tree of size . From the arbitrariness of , we have , a contradiction. So , as desired.
Conversely, we suppose . Let be all the noncut vertices in . Then the remaining vertices are all cut vertices of . Choose . Set . Note that each vertex in is a cut vertex of . Therefore, any Steiner tree occupies all the vertices of , and hence . From Theorem 1, we have , as desired.
The following corollary is immediate from the above lemma.
Corollary 1
Let be a connected graph of order , and let be an integer with . Then if and only if the number of noncut vertices in is at least .
Mao [30] obtained the following result, which will be used later.
Lemma 2
[30] Let be two integers with , and let be a connected graph of order . If , then , namely, .
Proof of Theorem 2: If , then . So we assume that . Suppose . For Lemma 2, if , then . We claim that is not a subgraph of . Assume, to the contrary, that is a subgraph of . Choose . Since is not connected, it follows that any Steiner tree must contain one vertex in , and hence , a contradiction. So is not a subgraph of .
Conversely, we suppose that and
is not a subgraph of . Since
, it follows that is a graph obtained
from the complete graph of order by deleting some independent
paths and cycles. For any , since
is not a subgraph of , it follows
that or or
or or
or . Then
or or or or
or , where is the
graph obtained from a star by adding an edge. Since is a connected graph,
it follows that . From the arbitrariness of , we
have and hence by Theorem
1. The proof is complete.
Proof of Theorem 3: Suppose that is a graph with . From Theorem 2, we have and is a subgraph of , or . For the former, we have and is a subgraph of , as desired. Suppose . It suffices to prove that each is not a spanning subgraph of , and we have the following claims.
Claim 1. is not a spanning subgraph of .
Proof of Claim 1. Assume, to the contrary, that is a spanning subgraph of . Choose . Then the subgraph in induced by the vertices in is a complete graph of order , and hence is not connected. Therefore, any Steiner tree must occupy a vertex in , say . Because is a spanning subgraph of , we have or or or . Thus, the Steiner tree must occupy another vertex in , and hence the tree must occupy at least two vertices in . Then , and hence , a contradiction. So is not a spanning subgraph of , as desired.
Claim 2. is not a spanning subgraph of .
Proof of Claim 2. Assume, to the contrary, that is a spanning subgraph of . Choose . Since is not connected, it follows that any Steiner tree must occupy a vertex in , say . From the structure of , since is a spanning subgraph of , we have or or . If , then there are at most three edges in belonging to . In order to connect to or , the Steiner tree uses at least two vertex of . If , then there are at most three edges in belonging to . In order to connect to , the Steiner tree must use at least two vertex of . The same is true for . Therefore, and , which results in , a contradiction. So is not a spanning subgraph of .
Claim 3. is not a spanning subgraph of .
Proof of Claim 3. Assume, to the contrary, that is a spanning subgraph of . Choose . Since is not connected, it follows that any Steiner tree must occupy a vertex in , say . From the structure of , since is a spanning subgraph of , we have or . If , then there are at most four edges in belonging to . In order to connect to or , the Steiner tree uses at least two vertex of . If , then there are at most four edges in belonging to . In order to connect to or , the Steiner tree must use at least two vertex of . Therefore, and , which results in , a contradiction. So is not a spanning subgraph of .
Claim 4. is not a spanning subgraph of .
Proof of Claim 4. Assume, to the contrary, that is a spanning subgraph of . Choose . Since is not connected, it follows that any Steiner tree must occupy a vertex in , say . From the structure of , since is a spanning subgraph of , we have or . If , then there are at most six edges in belonging to . In order to connect to , the Steiner tree uses at least two vertex of . If , then there are at most four edges in belonging to . In order to connect to or or , the Steiner tree uses at least two vertex of . Therefore, and , which results in , a contradiction. So is not a spanning subgraph of .
From the above argument, we know that the result holds.
Conversely, suppose that is a connected graph satisfying one of the following conditions.
and is a subgraph of ;
and is not a spanning subgraph of .
Suppose that and is a subgraph of . Since , it follows that is a graph obtained from the complete graph of order by deleting some pairwise independent paths and cycles. Then is a union of pairwise independent paths, cycles, and isolated vertices. For any , since contains as its subgraph, it follows that or or or or or or . Then or or or or or or , where is the graph obtained from by adding an edge. If , then for any in , since . Thus, we have . For the other cases, is connected, and so . From the arbitrariness of , we have , as desired.
Suppose and each is not a spanning subgraph of . For any and , if there exists a vertex such that , then , and hence the tree induced by the four edges in is an Steiner tree in , and hence , as desired. From now on, we assume for any and , and any , .
From the definition of and Theorem 2, it suffices to show that for any set and . It is clear that . If , then is connected, and hence contains a spanning tree, which is an Steiner tree in . So , as desired. From now on, we assume .
If , then . Since for any , it follows that is a spanning subgraph of , a contradiction.
Suppose . Set . Without loss of generality, let and . Then , and hence is a graph obtained from by deleting one edge. Since is not a spanning subgraph of , it follows that there exists a vertex such that but or but . By symmetry, we only to consider the former case. Clearly, . Combining this with , the tree induced by the edges in is an Steiner tree in and hence , as desired.
Suppose . Without loss of generality, we can assume that or . First, we consider the case . Clearly, , and hence . Note that for any and , and any