Steiner diameter, maximum degree and size of a graph

TL;DR

This paper investigates the minimum size of graphs with given order, Steiner diameter, and maximum degree, focusing on cases where the Steiner parameter k is close to the order n.

Contribution

It extends previous work by analyzing the minimum size of graphs for Steiner diameter constraints when k is near n, specifically for n-3 ≤ k ≤ n.

Findings

Characterization of graphs with minimal size for given Steiner diameter and maximum degree.

Results for the cases when k equals n, n-1, n-2, and n-3.

New bounds and structural properties of such graphs.

Abstract

The Steiner diameter $sdiam_k(G)$ of a graph $G$, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical diameter. When $k=2$, $sdiam_2(G)=diam(G)$ is the classical diameter. The problem of determining the minimum size of a graph of order $n$ whose diameter is at most $d$ and whose maximum is $\ell$ was first introduced by Erd\"{o}s and R\'{e}nyi. Recently, Mao considered the problem of determining the minimum size of a graph of order $n$ whose Steiner $k$-diameter is at most $d$ and whose maximum is at most $\ell$, where $3\leq k\leq n$, and studied this new problem when $k=3$. In this paper, we investigate the problem when $n-3\leq k\leq n$.

Equations

$${\mu }_k(G)={\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^{-1}\sum _{S\subseteq V(G),|S|=k}{d}_G(S).$$

$$S{W}_k(G)=\sum _{\stackrel{S\subseteq V(G)}{|S|=k}}d(S).$$

$$e(n,\ell ,d)=\min \{e(G):G\in \mathcal{H}(n,\ell ,d)\}.$$

$$e_k(n,\ell ,d)=\min \{e(G):G\in {\mathcal{H}}_k(n,\ell ,d)\}.$$

$$e_k(n,\ell ,n-1)=n-1.$$

$$e_{n-2}(n,\ell,n-2)=\left\{\begin{array}[]{ll}n,&\mbox{{\rm if}~{}$2\leq\ell\leq n-2$;}\\ n-1,&\mbox{{\rm if}~{}$\ell=n-1$.}\end{array}\right.$$

$$\{w_1^{r+1}{w}_s^{r+2},w_1^{r+2}{w}_1^1\}\cup \{w_2^i{w}_1^{i+1}|1\le i\le r-1\}\cup \{w_2^r{w}_{\ell -1}^{r+1}\}$$

$$\cup \{v_i{v}_{i+1}|1\le i\le r+1\}\cup \{v_{r+2}{v}_1\};$$

$$\frac{1}{2}(3n+\ell -3)\le e_{n-2}(n,\ell ,n-3)\le \frac{1}{2}(3n+\ell +s-5),$$

$$e(G)$$

$$e_{n-2}(n,\ell ,n-1)=n-1.$$

$$e_{n-2}(n,\ell,n-2)=\left\{\begin{array}[]{ll}n,&\mbox{{\rm if}~{}$2\leq\ell\leq n-2$}\\ n-1,&\mbox{{\rm if}~{}$\ell=n-1$.}\end{array}\right.$$

$$\frac{1}{2}(3n+\ell -3)\le e_{n-2}(n,\ell ,n-3)\le \frac{1}{2}(3n+\ell +s-5),$$

$$e_{n-3}(n,\ell,n-2)=\left\{\begin{array}[]{ll}n,&\mbox{{\rm if}~{}$2\leq\ell\leq\lfloor\frac{n}{2}\rfloor-1$};\\ &\mbox{{\rm~{}~{}~{}or}~{}$\ell=\lfloor\frac{n}{2}\rfloor$~{}and~{}n~{}is~{}odd};\\ n-1,&\mbox{{\rm if}~{}$\lfloor\frac{n}{2}\rfloor+1\leq\ell\leq n-1$};\\ &\mbox{{\rm~{}~{}~{}or}~{}$\ell=\lfloor\frac{n}{2}\rfloor$~{}and~{}n~{}is~{}even}.\\ \end{array}\right.$$

$$2n-2-\lceil \ell /2\rceil \le e_{n-3}(n,\ell ,n-4)\le 74\lfloor \frac{n}{32}\rfloor +2i+\ell -9,$$

$$e_{n-3}(n,\ell,n-3)\leq(2\ell+3)\left\lfloor\frac{n}{\ell+1}\right\rfloor+\ell+\left\{\begin{array}[]{ll}-8,&\mbox{{\rm if}~{}$n\equiv~{}0~{}(mod~{}\ell+1)$};\\ -5,&\mbox{{\rm if}~{}$n\equiv~{}1~{}(mod~{}\ell+1)$};\\ -2,&\mbox{{\rm if}~{}$n\equiv~{}2~{}(mod~{}\ell+1)$};\\ &\mbox{{\rm~{}~{}~{}or}~{}$n\equiv~{}3~{}(mod~{}\ell+1)$};\\ 2i-7,&\mbox{{\rm if}~{}$n\equiv~{}i~{}(mod~{}\ell+1)$},\end{array}\right.$$

$$e(G)=2\ell x+3(x-1)+\ell-5+\left\{\begin{array}[]{ll}0,&\mbox{{\rm if}~{}$y=0$};\\ 3,&\mbox{{\rm if}~{}$y=1$};\\ 6,&\mbox{{\rm if}~{}$y=2$};\\ &\mbox{{\rm~{}~{}~{}or}~{}$y=3$};\\ 2(y-1)+3,&\mbox{{\rm if}~{}$4\leq y\leq\ell$},\end{array}\right.$$

$$e_{n-3}(n,\ell,n-3)\leq(2\ell+3)\left\lfloor\frac{n}{\ell+1}\right\rfloor+\ell+\left\{\begin{array}[]{ll}-8,&\mbox{{\rm if}~{}$n\equiv~{}0~{}(mod~{}\ell+1)$};\\ -5,&\mbox{{\rm if}~{}$n\equiv~{}1~{}(mod~{}\ell+1)$};\\ -2,&\mbox{{\rm if}~{}$n\equiv~{}2~{}(mod~{}\ell+1)$};\\ &\mbox{{\rm~{}~{}~{}or}~{}$n\equiv~{}3~{}(mod~{}\ell+1)$};\\ 2i-7,&\mbox{{\rm if}~{}$n\equiv~{}i~{}(mod~{}\ell+1)$},\end{array}\right.$$

$$2e(G)$$

$$e_{n-3}(n,\ell,n-2)=\left\{\begin{array}[]{ll}n,&\mbox{{\rm if}~{}$2\leq\ell\leq\lfloor\frac{n}{2}\rfloor-1$};\\ &\mbox{{\rm~{}~{}~{}or}~{}$\ell=\lfloor\frac{n}{2}\rfloor$~{}and~{}n~{}is~{}odd};\\ n-1,&\mbox{{\rm if}~{}$\lfloor\frac{n}{2}\rfloor+1\leq\ell\leq n-1$};\\ &\mbox{{\rm~{}~{}~{}or}~{}$\ell=\lfloor\frac{n}{2}\rfloor$~{}and~{}n~{}is~{}even}.\\ \end{array}\right.$$

$$2n-2-\lceil \ell /2\rceil \le e_{n-3}(n,\ell ,n-4)\le 74\lfloor \frac{n}{32}\rfloor +2i+\ell -9,$$

$$e_{n-3}(n,\ell,n-3)\leq(2\ell+3)\left\lfloor\frac{n}{\ell+1}\right\rfloor+\ell+\left\{\begin{array}[]{ll}-8,&\mbox{{\rm if}~{}$n\equiv~{}0~{}(mod~{}\ell+1)$};\\ -5,&\mbox{{\rm if}~{}$n\equiv~{}1~{}(mod~{}\ell+1)$};\\ -2,&\mbox{{\rm if}~{}$n\equiv~{}2~{}(mod~{}\ell+1)$};\\ &\mbox{{\rm~{}~{}~{}or}~{}$n\equiv~{}3~{}(mod~{}\ell+1)$};\\ 2i-7,&\mbox{{\rm if}~{}$n\equiv~{}i~{}(mod~{}\ell+1)$},\end{array}\right.$$

$$\lceil \frac{\ell +(n-1)(n-k+1)}{2}\rceil \le e_k(n,\ell ,d)\le \frac{(n-1{)}^2}{4}+\ell .$$

$$e_k(n,\ell ,d)=n-1.$$

Full text

Keywords: diameter; maximum degree; order; Steiner diameter.

AMS subject classification 2010: 05C05; 05C12; 05C35.

1 Introduction

All graphs in this paper are undirected, finite and simple. We refer to [5] for graph theoretical notation and terminology not described here. For a graph $G$, let $V(G)$, $E(G)$, $e(G)$, $\delta(G)$, and $\overline{G}$ denote the set of vertices, the set of edges, the size, minimum degree, and the complement of $G$, respectively. The connectivity $\kappa(G)$ is defined as the order of a minimum vertex subset $S$ of $V(G)$ such that $G-S$ is disconnected or has only one vertex. In this paper, we let $K_{n}$, $P_{n}$, $K_{1,n-1}$ and $C_{n}$ be the complete graph of order $n$, the path of order $n$, the star of order $n$, and the cycle of order $n$, respectively. For any subset $X$ of $V(G)$, let $G[X]$ denote the subgraph induced by $X$; similarly, for any subset $F$ of $E(G)$, let $G[F]$ denote the subgraph induced by $F$. We use $G\setminus X$ to denote the subgraph of $G$ obtained by removing all the vertices of $X$ together with the edges incident with them from $G$; similarly, we use $G\setminus F$ to denote the subgraph of $G$ obtained by removing all the edges of $F$ from $G$. For two subsets $X$ and $Y$ of $V(G)$ we denote by $E_{G}[X,Y]$ the set of edges of $G$ with one end in $X$ and the other end in $Y$. The union $G\cup H$ of two graphs $G$ and $H$ is the graph with vertex set $V(G)\cup V(H)$ and edge set $E(G)\cup E(H)$. If $G$ is the disjoint union of $k$ copies of a graph $H$, we simply write $G=kH$. The join $G\vee H$ of two disjoint graphs $G$ and $H$ is the graph with vertex set $V(G)\cup V(H)$ and edge set $E(G)\cup E(H)\cup\{uv\,|\,u\in V(G),v\in V(H)\}$. 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 graph-theoretic subjects. If $G$ is a connected graph and $u,v\in V(G)$, then the distance $d_{G}(u,v)$ between $u$ and $v$ is the length of a shortest path connecting $u$ and $v$. If $v$ is a vertex of a connected graph $G$, then the eccentricity $e(v)$ of $v$ is defined by $e(v)=\max\{d_{G}(u,v)\,|\,u\in V(G)\}$. Furthermore, the radius $rad(G)$ and diameter $diam(G)$ of $G$ are defined by $rad(G)=\min\{e(v)\,|\,v\in V(G)\}$ and $diam(G)=\max\{e(v)\,|\,v\in V(G)\}$. These last two concepts are related by the inequalities $rad(G)\leq diam(G)\leq 2rad(G)$. The center $C(G)$ of a connected graph $G$ is the subgraph induced by the vertices $u$ of $G$ with $e(u)=rad(G)$. Goddard and Oellermann gave a survey on this subject, see [24].

The distance between two vertices $u$ and $v$ in a connected graph $G$ also equals the minimum size of a connected subgraph of $G$ containing both $u$ and $v$. 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 generalization of the concept of classical graph distance. For a graph $G(V,E)$ and a set $S\subseteq V(G)$ of at least two vertices, an $S$-Steiner tree or a Steiner tree connecting $S$ (or simply, an $S$-tree) is a subgraph $T(V^{\prime},E^{\prime})$ of $G$ that is a tree with $S\subseteq V^{\prime}$. Let $G$ be a connected graph of order at least $2$ and let $S$ be a nonempty set of vertices of $G$. Then the Steiner distance $d_{G}(S)$ among the vertices of $S$ (or simply the distance of $S$) is the minimum size among all connected subgraphs whose vertex sets contain $S$. Note that if $H$ is a connected subgraph of $G$ such that $S\subseteq V(H)$ and $|E(H)|=d_{G}(S)$, then $H$ is a tree. Observe that $d_{G}(S)=\min\{e(T)\,|\,S\subseteq V(T)\}$, where $T$ is subtree of $G$. Furthermore, if $S=\{u,v\}$, then $d_{G}(S)=d(u,v)$ is the classical distance between $u$ and $v$. Set $d_{G}(S)=\infty$ when there is no $S$-Steiner tree in $G$.

Observation 1.1

Let $G$ be a graph of order $n$ and $k$ be an integer with $2\leq k\leq n$. If $S\subseteq V(G)$ and $|S|=k$, then $d_{G}(S)\geq k-1$.

Let $n$ and $k$ be two integers with $2\leq k\leq n$. The Steiner $k$-eccentricity $e_{k}(v)$ of a vertex $v$ of $G$ is defined by $e_{k}(v)=\max\{d(S)\,|\,S\subseteq V(G),|S|=k,~{}and~{}v\in S\}$. The Steiner $k$-radius of $G$ is $srad_{k}(G)=\min\{e_{k}(v)\,|\,v\in V(G)\}$, while the Steiner $k$-diameter of $G$ is $sdiam_{k}(G)=\max\{e_{k}(v)\,|\,v\in V(G)\}$. Note for every connected graph $G$ that $e_{2}(v)=e(v)$ for all vertices $v$ of $G$ and that $srad_{2}(G)=rad(G)$ and $sdiam_{2}(G)=diam(G)$.

The following Table 1 shows how the generalization proceeds.

[TABLE]

Table 1. Classical distance parameters and Steiner distance parameters

Observation 1.2

Let $k,n$ be two integers with $2\leq k\leq n$.

$(1)$* If $H$ is a spanning subgraph of $G$, then $sdiam_{k}(G)\leq sdiam_{k}(H)$.*

$(2)$* For a connected graph $G$, $sdiam_{k}(G)\leq sdiam_{k+1}(G)$.*

In [9], Chartrand, Okamoto, Zhang obtained the following upper and lower bounds of $sdiam_{k}(G)$.

Theorem 1.1

[9]* Let $k,n$ be two integers with $2\leq k\leq n$, and let $G$ be a connected graph of order $n$. Then $k-1\leq sdiam_{k}(G)\leq n-1$. Moreover, the upper and lower bounds are sharp.*

In [13], Dankelmann, Swart and Oellermann obtained a bound on $sdiam_{k}(G)$ for a graph $G$ in terms of the order of $G$ and the minimum degree $\delta$ of $G$, that is, $sdiam_{k}(G)\leq\frac{3n}{\delta+1}+3k$. Later, Ali, Dankelmann, Mukwembi [2] improved the bound of $sdiam_{k}(G)$ and showed that $sdiam_{k}(G)\leq\frac{3n}{\delta+1}+2k-5$ for all connected graphs $G$. Moreover, they constructed graphs to show that the bounds are asymptotically best possible. In [36], Mao obtained the Nordhaus-Gaddum-type results for the parameter $sdiam_{k}(G)$.

As a generalization of the center of a graph, the Steiner $k$-center $C_{k}(G)\ (k\geq 2)$ of a connected graph $G$ is the subgraph induced by the vertices $v$ of $G$ with $e_{k}(v)=srad_{k}(G)$. Oellermann and Tian [46] showed that every graph is the $k$-center of some graph. In particular, they showed that the $k$-center of a tree is a tree and those trees that are $k$-centers of trees are characterized. The Steiner $k$-median of $G$ is the subgraph of $G$ induced by the vertices of $G$ of minimum Steiner $k$-distance. For Steiner centers and Steiner medians, we refer to [44, 45, 46].

The average Steiner distance $\mu_{k}(G)$ of a graph $G$, introduced by Dankelmann, Oellermann and Swart in [11], is defined as the average of the Steiner distances of all $k$-subsets of $V(G)$, i.e.

[TABLE]

For more details on average Steiner distance, we refer to [11, 12].

Let $G$ be a $k$-connected graph and $u$, $v$ be any pair of vertices of $G$. Let $P_{k}(u,v)$ be a family of $k$ inner vertex-disjoint paths between $u$ and $v$, i.e., $P_{k}(u,v)=\{P_{1},P_{2},\cdots,P_{k}\}$, where $p_{1}\leq p_{2}\leq\cdots\leq p_{k}$ and $p_{i}$ denotes the number of edges of path $P_{i}$. The $k$-distance $d_{k}(u,v)$ between vertices $u$ and $v$ is the minimum $p_{k}$ among all $P_{k}(u,v)$ and the $k$-diameter $d_{k}(G)$ of $G$ is defined as the maximum $k$-distance $d_{k}(u,v)$ over all pairs $u,v$ of vertices of $G$. The concept of $k$-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 [10], Du, Lyuu and Hsu [15], Hsu [31, 32], Meyer and Pradhan [43].

1.2 Application background of Steiner distance parameters

The Steiner tree problem in networks, and particularly in graphs, was formulated in 1971-by Hakimi (see [27]) and Levi (see [33]). In the case of an unweighted, undirected graph, this problem consists of finding, for a subset of vertices $S$, a minimal-size connected subgraph that contains the vertices in $S$. The computational side of this problem has been widely studied, and it is known that it is an NP-hard problem for general graphs (see [30]). The determination of a Steiner tree in a graph is a discrete analogue of the well-known geometric Steiner problem: In a Euclidean space (usually a Euclidean plane) find the shortest possible network of line segments interconnecting a set of given points. Steiner trees have application to multiprocessor computer networks. For example, it may be desired to connect a certain set of processors with a subnetwork that uses the least number of communication links. A Steiner tree for the vertices, corresponding to the processors that need to be connected, corresponds to such a desired subnetwork.

The Wiener index $W(G)$ of the graph $G$ is defined as $W(G)=\sum_{\{u,v\}\subseteq V(G)}d_{G}(u,v)$. Details on this oldest distance–based topological index can be found in numerous surveys, e.g., in [16, 48, 49, 52]. Li et al. [34] put forward a Steiner–distance–based generalization of the Wiener index concept. According to [34], the $k$-center Steiner Wiener index $SW_{k}(G)$ of the graph $G$ is defined by

[TABLE]

For $k=2$, the above defined Steiner Wiener index coincides with the ordinary Wiener index. It is usual to consider $SW_{k}$ for $2\leq k\leq n-1$, but the above definition would be applicable also in the cases $k=1$ and $k=n$, implying $SW_{1}(G)=0$ and $SW_{n}(G)=n-1$. A chemical application of $SW_{k}$ was recently reported in [26]. Gutman [25] offered an analogous generalization of the concept of degree distance. Later, Furtula, Gutman, and Katanić [20] introduced the concept of Steiner Harary index and gave its chemical applications. Recently, Mao and Das [38] introduced the concept of Steiner Gutman index and obtained some bounds for it. For more details on Steiner distance indices, we refer to [20, 26, 25, 34, 35, 38, 40, 41, 42].

1.3 Classical extremal problem and our generalization

What is the minimal size of a graph of order $n$ and diameter $d$ ? What is the maximal size of a graph of order $n$ and diameter $d$ ? It is not surprising that these questions can be answered without the slightest effort (see [3]) just as the similar questions concerning the connectivity or the chromatic number of a graph. The class of maximal graphs of order $n$ and diameter $d$ is easy to describe and reduce every question concerning maximal graphs to a not necessarily easy question about binomial coefficient, as in [28, 29, 47, 51]. Therefore, the authors study the minimal size of a graph of order $n$ and under various additional conditions.

Erdös and Rényi [18] introduced the following problem. Let $d,\ell$ and $n$ be natural numbers, $d<n$ and $\ell<n$. Denote by $\mathscr{H}(n,\ell,d)$ the set of all graphs of order $n$ with maximum degree $\ell$ and diameter at most $d$. Put

[TABLE]

If $\mathscr{H}(n,\ell,d)$ is empty, then, following the usual convention, we shall write $e(n,\ell,d)=\infty$. For more details on this problem, we refer to [3, 4, 18, 19].

Mao [37] considered the generalization of the above problem. Let $d,\ell$ and $n$ be natural numbers, $d<n$ and $\ell<n$. Denote by $\mathscr{H}_{k}(n,\ell,d)$ the set of all graphs of order $n$ with maximum degree $\ell$ and $sdiam_{k}(G)\leq d$. Put

[TABLE]

If $\mathscr{H}_{k}(n,\ell,d)$ is empty, then, following the usual convention, we shall write $e_{k}(n,\ell,d)=\infty$. From Theorem 1.1, we have $k-1\leq d\leq n-1$.

In [37], Mao focused their attention on the case $k=3$, and studied the exact value of $e_{3}(n,\ell,d)$ for $d=n-1,n-2,n-3,2,3$. In this paper, we investigate another extreme case when $n-3\leq k\leq n-1$, and give the exact values or upper and lower bounds of $e_{k}(n,\ell,d)$ for $n-3\leq k\leq n-1$. For general $k\ (3\leq k\leq n-1)$, $d\ (k-1\leq d\leq n-1)$ and $\ell\ (2\leq\ell\leq n-1)$, we give upper and lower bounds of $e_{k}(n,\ell,d)$.

2 The case $k=n,n-1$

In the sequel, let $K_{s,t}$, $K_{n}$, $C_{n}$ and $P_{n}$ denote the complete bipartite graph of order $s+t$ with part sizes $s$ and $t$, complete graph of order $n$, cycle of order $n$ and path of order $n$, respectively.

The following observation is immediate.

Observation 2.1

[36]* $(1)$ For a cycle $C_{n}$, $sdiam_{k}(C_{n})=\left\lfloor\frac{n(k-1)}{k}\right\rfloor$;*

$(2)$* For a complete graph $K_{n}$, $sdiam_{k}(K_{n})=k-1$.*

The following result is easily proved in [37].

Lemma 2.1

[37]* For $2\leq\ell\leq n-1$ and $3\leq k\leq n$,*

[TABLE]

For $k=n$, we know that $sdiam_{n}(G)=n-1$ for a connected graph $G$, and hence $d=n-1$. From Lemma 2.1, the following result is immediate.

Proposition 2.1

For $2\leq\ell\leq n-1$, $e_{n}(n,\ell,n-1)=n-1$.

From now on, we assume that $k\leq n-1$.

Proposition 2.2

For $3\leq k\leq n$, $e_{k}(n,n-1,k)=n-1$.

Proof. Let $G=K_{1,n-1}$ be a star of order $n$. Clearly, $\Delta(G)=\ell=n-1$. Since $sdiam_{k}(G)=k$ and $e(G)=n-1$, it follows that $e_{k}(n,n-1,k)\leq n-1$. On the other hand, since we only consider connected graphs, it follows that $e(G)\geq n-1$ for a connected graph $G$ is of order $n$. So $e_{k}(n,n-1,k)=n-1$.

For $k=n-1$, we have $n-2\leq sdiam_{n-1}(G)\leq n-1$ by Theorem 1.1. So we only need to consider the case $d=n-1$ or $d=n-2$. Note that $2\leq\ell\leq n-1$.

Theorem 2.1

$(1)$* For $2\leq\ell\leq n-1$, $e_{n-1}(n,\ell,n-1)=n-1$.*

$(2)$* For $2\leq\ell\leq n-1$, $e_{n-1}(n,\ell,n-2)=n+\ell-2$.*

Proof. $(1)$ The result follows from Lemma 2.1.

$(2)$ For $\ell=2$, we let $C_{n}$ be the cycle of order $n$. From Observation 2.1, we have $sdiam_{n-1}(G)=n-2$. Since $e(G)=n$, it follows that $e_{n-1}(n,2,n-2)\leq n$. Let $G$ be graph of order $n$ with $\Delta(G)=2$ and $sdiam_{n-1}(G)=n-2$. Since $\Delta(G)=2$, it follows that $G=P_{n}$ or $G=C_{n}$. If $G=P_{n}$, then $sdiam_{n-1}(G)=n-1$, a contradiction. Therefore, $G=C_{n}$ and hence $e(G)\geq n$. So $e_{n-1}(n,2,n-2)=n$.

Suppose $3\leq\ell\leq n-1$. Let $G$ be a graph obtained from a cycle $C_{n}=v_{1}v_{2}\cdots v_{n}$ by adding the edges $v_{1}v_{j}\ (3\leq j\leq\ell)$. From Observation 1.1, we have $sdiam_{n-1}(G)\leq sdiam_{n-1}(C_{n})\leq n-2$. Since $\Delta(G)=\ell$ and $e(G)=n+\ell-2$, it follows that $e_{n-1}(n,\ell,n-2)\leq n+\ell-2$. Conversely, let $G$ be a graph such that $sdiam_{n-1}(G)=n-2$ and $\Delta(G)=\ell\ (3\leq\ell\leq n-1)$. Then there exists a vertex $u$ in $G$ such that $d_{G}(u)=\ell$. Choose $S=V(G)-u$. Since $sdiam_{n-1}(G)=n-2$, it follows that there exists an $S$-Steiner tree in $G-u$, say $T$. Then $e(G-u)\geq e(T)=n-2$ and hence $e(G)=e(G-u)+\ell\geq n-2+\ell$, which implies that $e_{n-1}(n,\ell,n-2)\geq n+\ell-2$.

From the above arguments, we conclude that $e_{n-1}(n,\ell,n-2)=n+\ell-2$ for $2\leq\ell\leq n-1$.

3 The case $k=n-2$

From Theorem 1.1, we have $n-3\leq sdiam_{n-2}(G)\leq n-1$. Mao et al. [39] characterized the graphs with $sdiam_{n-2}(G)=d\ (n-3\leq d\leq n-1)$.

Lemma 3.1

[39]* Let $G$ be a connected graph of order $n\ (n\geq 5)$. Then*

$(1)$* $sdiam_{n-2}(G)=n-3$ if and only if $\kappa(G)\geq 3$.*

$(2)$* $sdiam_{n-2}(G)=n-2$ if and only if $\kappa(G)=2$ or $G$ contains only one cut vertex.*

$(3)$* $sdiam_{n-2}(G)=n-1$ if and only if there are at least two cut vertices in $G$.*

For $d=n-2$, we have the following.

Proposition 3.1

For $2\leq\ell\leq n-1$ and $n\geq 5$,

[TABLE]

Proof. For $\ell=n-1$, from Proposition 2.2, we have $e_{n-2}(n,\ell,n-2)=n-1$. From now on, we suppose $2\leq\ell\leq n-2$. Let $G$ be a graph obtained by a cycle $C_{n-\ell+2}$ and a star $K_{1,\ell-2}$ by identifying a vertex of $C_{n-\ell+2}$ and the center of $K_{1,\ell-2}$. Clearly, $\Delta(G)=\ell$ and there is exactly one cut vertex in $G$. From $(2)$ of Lemma 3.1, we have $sdiam_{n-2}(G)=n-2$, and hence $e_{n-2}(n,\ell,n-2)\leq n$. It suffices to show that $e_{n-2}(n,\ell,n-2)\geq n$. Let $G$ be a graph of order $n$ with $sdiam_{n-2}(G)\leq n-2$ and $\Delta(G)=\ell$. If $G$ is a tree, then $G$ contains at least two cut vertices, since $\Delta(G)=\ell\leq n-2$. From $(3)$ of Lemma 3.1, $sdiam_{n-2}(G)=n-1$, a contradiction. So $G$ contains at least one cycle, and hence $e(G)\geq n$. Therefore, we have $e_{n-2}(n,\ell,n-2)=n$, as desired.

Let $P_{j}^{i}$ be a path of order $j$, where $1\leq i\leq r+2$. We call the graph $K_{1}\vee(K_{1}\cup P_{j}^{i})$ as a $(u_{i},v_{i},P_{j}^{i})$-Fan; see Figure 1 $(a)$. For $1\leq i\leq r$, we choose $(u_{i},v_{i},P_{2}^{i})$-Fan, and choose $(u_{r+1},v_{r+1},P_{\ell-1}^{r+1})$-Fan and $(u_{r+2},v_{r+2},P_{s}^{r+2})$-Fan. Let $H_{n}$ be a graph obtained from the above $(r+2)$ Fans by adding the edges in

[TABLE]

see Figure 1 $(b)$, where $4r+\ell+s+3=n$, $2\leq s\leq 5$ and $1\leq r\leq\frac{n-\ell-4}{4}$.

Proposition 3.2

For $6\leq\ell\leq n-9$,

[TABLE]

where $2\leq s\leq 5$. Furthermore, if $s=2$, then $e_{n-2}(n,\ell,n-3)=\frac{1}{2}(3n+\ell-3)$.

Proof. Let $H_{n}$ be the graph constructed in Figure 1 $(b)$. Clearly, $H_{n}$ is $3$-connected. From Lemma 3.1, $sdiam_{n-2}(H_{n})=n-3$. Since $6\leq\ell\leq n-9$ and $2\leq s\leq 5$, it follows that $\Delta(H_{n})=\ell$. Then

[TABLE]

and hence $e_{n-2}(n,\ell,n-3)\leq\frac{1}{2}(3n+\ell+s-5)$.

Conversely, we suppose that $G$ is a graph with $|V(G)|=n$, $\Delta(G)=\ell$, and $sdiam_{n-2}(G)=n-3$. Then there is a vertex in $G$, say $u$, such that $d_{G}(u)=\ell$. Since $sdiam_{n-2}(G)=n-3$, it follows from Lemma 3.1 that $\kappa(G)\geq 3$, and hence $d_{G}(v)\geq 3$ for any $v\in V(G)\setminus u$. Therefore, $e(G)\geq\frac{1}{2}(3n+\ell-3)$, and hence $e_{n-2}(n,\ell,n-3)\geq\frac{1}{2}(3n+\ell-3)$.

For $n-8\leq\ell\leq n-1$, we have the following result.

Lemma 3.2

Let $k,\ell$ be two integers with $n-8\leq\ell\leq n-1$. Then

$(i)$* $e_{n-2}(n,n-1-i,n-3)=2n-2$ for $n\geq 5+i$ and $i=0,1$;*

$(ii)$* $e_{n-2}(n,n-3-i,n-3)=2n-3$ for $n\geq 7+2i$ and $i=0,1$;*

$(iii)$* $e_{n-2}(n,n-5-i,n-3)=2n-4$ for $n\geq 11+2i$ and $i=0,1$;*

$(iv)$* $e_{n-2}(n,n-7-i,n-3)=2n-5$ for $n\geq 15+2i$ and $i=0,1$.*

Proof. For $(i)$, we first consider the case $i=0$. For $\ell=n-1$, let $G_{n}^{1}$ be a wheel of order $n$. From Lemma 3.1, $sdiam_{n-2}(G_{n}^{1})=n-3$ and $\Delta(G_{n}^{1})=n-1$, and hence $e_{n-2}(n,n-1,n-3)\leq 2n-2$. Conversely, we suppose that $G$ is a graph of order $n$ such that $sdiam_{n-2}(G)=n-3$ and $\Delta(G)=n-1$. Then there exists a vertex $u$ such that $d_{G}(u)=n-1$. Since $\kappa(G)\geq 3$, it follows that $\kappa(G-u)\geq 2$, and hence $e(G-u)\geq n-1$. Then $e(G)\geq 2n-2$, and hence $e_{n-2}(n,n-1,n-3)\geq 2n-2$. So $e_{n-2}(n,n-1,n-3)=2n-2$.

Next, we consider the case $i=1$. For $\ell=n-2$, let $G_{n}^{2}$ be a graph of order $n$ obtained by $G_{n-1}^{1}$ by deleting the edge $w_{2}w_{3}$, and then adding a new vertex $w$ and three edges $ww_{1},ww_{2},ww_{3}$; see Figure 2 $(b)$. Since $\kappa(G_{n}^{2})=3$, it follows from Lemma 3.1 that $sdiam_{n-2}(G_{n}^{2})=n-3$. From this together with $\Delta(G_{n}^{2})=n-2$, we have $e_{n-2}(n,n-2,n-3)\leq 2n-2$. Conversely, we suppose that $G$ is a graph of order $n$ such that $sdiam_{n-2}(G)=n-3$ and $\Delta(G)=n-2$. Then there exists a vertex $u$ such that $d_{G}(u)=n-2$. Since $sdiam_{n-2}(G)=n-3$, it follows that $\kappa(G)\geq 3$, and hence $\kappa(G-u)\geq 2$. Clearly, $e(G-u)\geq n-1$. If $e(G-u)=n-1$, then $G-u$ is a cycle of order $n-1$, say $G-u=v_{1}v_{2}\ldots v_{n-1}v_{1}$. Since $d_{G}(u)=n-2$, it follows that there exists some vertex $v_{i}$ in $G-u$ such that $uv_{i}\notin E(G)$, and hence $d_{G}(v_{i})=2$, which contradicts to the fact $\kappa(G)\geq 3$. Then $e(G-u)\geq n$, and hence $e(G)\geq 2n-2$, and hence $e_{n-2}(n,n-2,n-3)\geq 2n-2$. So, we have $e_{n-2}(n,n-2,n-3)=2n-2$.

For $(ii)$, we first consider the case $i=0$. For $\ell=n-3$, let $G_{n}^{3}$ be a graph of order $n$ obtained by $G_{n-1}^{2}$ by deleting the edge $v_{1}w_{1},w_{1}w_{n-3}$, and then adding a new vertex $w$ and three edges $wv_{1},ww_{1},ww_{n-3}$; see Figure 3 $(b)$. Since $\kappa(G_{n}^{3})=3$, it follows from Lemma 3.1 that $sdiam_{n-2}(G_{n}^{3})=n-3$. Note that $\Delta(G_{n}^{3})=n-3$. Therefore, we have $e_{n-2}(n,n-3,n-3)\leq 2n-3$. Conversely, we suppose that $G$ is a graph of order $n$ such that $sdiam_{n-2}(G)=n-3$ and $\Delta(G)=n-3$. Then there exists a vertex $u$ such that $d_{G}(u)=n-3$. Since $sdiam_{n-2}(G)=n-3$, it follows that $\kappa(G)\geq 3$, and hence $\kappa(G-u)\geq 2$. Clearly, $e(G-u)\geq n-1$. If $e(G-u)=n-1$, then $G-u$ is a cycle of order $n-1$, say $G-u=v_{1}v_{2}\ldots v_{n-1}v_{1}$. Since $d_{G}(u)=n-3$, it follows that there exists some vertex $v_{i}$ in $G-u$ such that $uv_{i}\notin E(G)$, and hence $d_{G}(v_{i})=2$, which contradicts to the fact $\kappa(G)\geq 3$. Then $e(G-u)\geq n$, and hence $e(G)\geq 2n-3$, and hence $e_{n-2}(n,n-3,n-3)\geq 2n-3$. So, we have $e_{n-2}(n,n-3,n-3)=2n-3$.

Next, we consider the case $i=1$. For $\ell=n-4$, let $G_{n}^{4}$ be a graph of order $n$ obtained by $G_{n-1}^{3}$ by deleting the edge $v_{2}w_{n-4}$, and then adding a new vertex $w$ and three edges $wv_{2},ww_{n-4},ww_{n-5}$; see Figure 3 $(d)$. Since $\kappa(G_{n}^{4})=3$, it follows from Lemma 3.1 that $sdiam_{n-2}(G_{n}^{4})=n-3$. Note that $\Delta(G_{n}^{3})=n-4$. Therefore, we have $e_{n-2}(n,n-2,n-3)\leq 2n-3$. Conversely, we suppose that $G$ is a graph of order $n$ such that $sdiam_{n-2}(G)=n-3$ and $\Delta(G)=n-4$. Then there exists a vertex $u$ such that $d_{G}(u)=n-4$. Since $sdiam_{n-2}(G)=n-3$, it follows that $\kappa(G)\geq 3$, and hence $\kappa(G-u)\geq 2$. Clearly, $e(G-u)\geq n-1$. If $e(G-u)=n-1$, then $G-u$ is a cycle of order $n-1$, say $G-u=v_{1}v_{2}\ldots v_{n-1}v_{1}$. Since $d_{G}(u)=n-3$, it follows that there exists some vertex $v_{i}$ in $G-u$ such that $uv_{i}\notin E(G)$, and hence $d_{G}(v_{i})=2$, which contradicts to the fact $\kappa(G)\geq 3$. If $e(G-u)=n$, then $G-u$ is a graph obtained from a cycle $C=v_{1}v_{2}\ldots v_{n-1}v_{1}$ by adding some edge $v_{p}v_{q}\ (1\leq p\neq q\leq n-1)$. Since $d_{G}(u)=n-4$, it follows that there exists some vertex $v_{i}\ (1\leq i\leq n-1,\ i\neq p,\ i\neq q)$ in $G-u$ such that $uv_{i}\notin E(G)$, and hence $d_{G}(v_{i})=2$, which contradicts to the fact $\kappa(G)\geq 3$. Then $e(G-u)\geq n+1$, and hence $e(G)\geq 2n-3$, and hence $e_{n-2}(n,n-4,n-3)\geq 2n-3$. So, we have $e_{n-2}(n,n-4,n-3)=2n-3$.

For $(iii)$ and $(iv)$, we only give the graph construction operation (see Figure 3), and omit the proof of them.

From Propositions 3.1 and 3.2, and Lemmas 3.2 and 2.1, we have the following theorem.

Theorem 3.1

$(1)$* For $2\leq\ell\leq n-1$,*

[TABLE]

$(2)$* For $2\leq\ell\leq n-1$ and $n\geq 5$,*

[TABLE]

$(3)$* For $n-8\leq\ell\leq n-1$, $e_{n-2}(n,n-1-i,n-3)=2n-2$ for $n\geq 5+i$ and $i=0,1$; $e_{n-2}(n,n-3-i,n-3)=2n-3$ for $n\geq 7+2i$ and $i=0,1$; $e_{n-2}(n,n-5-i,n-3)=2n-4$ for $n\geq 11+2i$ and $i=0,1$; $e_{n-2}(n,n-7-i,n-3)=2n-5$ for $n\geq 15+2i$ and $i=0,1$. For $6\leq\ell\leq n-9$,*

[TABLE]

where $2\leq s\leq 5$. Furthermore, if $s=2$, then $e_{n-2}(n,\ell,n-3)=\frac{1}{2}(3n+\ell-3)$.

4 The case $k=n-3$

Mao et al. [36] derived the following results for Steiner $(n-3)$-diameter.

Lemma 4.1

[36]* Let $G$ be a connected graph of order $n$. Then $sdiam_{n-3}(G)=n-4$ if and only if $\kappa(G)\geq 4$, and $sdiam_{n-3}(G)=n-1$ if and only if $G$ contains at least $3$ cut vertices.*

Wang et al. [50] obtained the structural properties of graphs with $sdiam_{k}(G)=n-1$.

Lemma 4.2

[50]* Let $k,n$ be two integers with $3\leq k\leq n-1$. Let $G$ be a connected graph of order $n$. Then $sdiam_{k}(G)=n-1$ if and only if the number of non-cut vertices in $G$ is at most $k$.*

The following corollary is immediate from Lemma 2.1.

Corollary 4.1

For $2\leq\ell\leq n-1$, $e_{n-3}(n,\ell,n-1)=n-1$.

Let $uv$ be an edge in $G$. A double-star on $uv$ is a maximal tree in $G$ which is the union of stars centered at $u$ or $v$ such that each star contains the edge $uv$.

Proposition 4.1

For $2\leq\ell\leq n-1$ and $n\geq 4$,

[TABLE]

Proof. For $2\leq\ell\leq\lfloor\frac{n}{2}\rfloor$, we let $G$ be a graph of order $n$ obtained from a cycle $C_{n-\ell+2}$ and a star $K_{1,\ell-2}$ by identifying the center and one vertex of $C_{n-\ell+2}$. Clearly, $\Delta(G)=\ell$, and $G$ contains only one cut vertex. From Lemma 4.2, we have $sdiam_{n-3}(G)\leq n-2$, and hence $e_{n-3}(n,\ell,n-2)\leq n$. It suffices to show $e_{n-3}(n,\ell,n-2)=n$ if $2\leq\ell\leq\lfloor\frac{n}{2}\rfloor-1$, or $\ell=\lfloor\frac{n}{2}\rfloor$ and $n$ is odd. Let $G$ be a graph of order $n$ such that $sdiam_{n-3}(G)\leq n-2$ and $\Delta(G)=\ell$, where $2\leq\ell\leq\lfloor\frac{n}{2}\rfloor-1$, or $\ell=\lfloor\frac{n}{2}\rfloor$ and $n$ is odd. Then we have the following claim.

Claim 1. $G$ is not a tree.

Proof of Claim 1. Assume, to the contrary, that $G$ is a tree. Since $sdiam_{n-3}(G)\leq n-2$, it follows from Lemma 4.2 that $G$ contains at most two cut vertices. Then $G=K_{1,n-1}$ or $G$ is a double star of order $n$. If $G=K_{1,n-1}$, then $\Delta(G)=n-1>\lfloor\frac{n}{2}\rfloor$, a contradiction. Suppose that $G$ is a double star of order $n$. Let $u,v$ be the two centers of $G$. Then $d_{G}(u)\geq\lceil\frac{n}{2}\rceil$ or $d_{G}(v)\geq\lceil\frac{n}{2}\rceil$, and hence $\ell\geq\lceil\frac{n}{2}\rceil$, a contradiction.

From Claim 1, $G$ is not a tree. Then $e(G)\geq n$, and hence $e_{n-3}(n,\ell,n-2)=n$ if $2\leq\ell\leq\lfloor\frac{n}{2}\rfloor-1$, or $\ell=\lfloor\frac{n}{2}\rfloor$ and $n$ is odd.

We now show that if $\lfloor\frac{n}{2}\rfloor+1\leq\ell\leq n-1$, or $\ell=\lfloor\frac{n}{2}\rfloor$ and $n$ is even, then $e_{n-3}(n,\ell,n-2)=n-1$. Let $G$ be a double star of order $n$ such that $d_{G}(u)=\ell$ and $d_{G}(v)=n-\ell$, where $u,v$ are the two centers of $G$. From Lemma 4.2, we have $sdiam_{n-3}(G)\leq n-2$. Since $\lfloor\frac{n}{2}\rfloor+1\leq\ell\leq n-1$, or $\ell=\lfloor\frac{n}{2}\rfloor$ and $n$ is even, it follows that $\Delta(G)=\ell$, and hence $e_{n-3}(n,\ell,n-2)\leq n-1$. So, we have $e_{n-3}(n,\ell,n-2)=n-1$.

A graph is said to be minimally $k$-connected if it is $k$-connected but omitting any of the edges the resulting graph is no longer $k$-connected.

Let $A_{32}$ be a minimally $4$-connected graph shown in Figure 4 $(a)$ (see [3], Page 18). We now give a graph $H_{n}$ of order $n\ (n\geq 96)$ such that $\Delta(H_{n})=\ell$ and $sdiam_{n-3}(H_{n})=n-4$ constructed by the following steps.

• Step 1: For each $i\ (1\leq i\leq x)$, we let $A_{32}^{i}$ be the copy of $A_{32}$, where $n=32x+y$, $x=\lfloor n/32\rfloor$, and $0\leq y\leq 31$. Let $V(A_{32}^{i})=\{u^{i}_{j}\,|\,1\leq j\leq 12\}\cup\{v^{i}_{j}\,|\,1\leq j\leq 20\}$ such that $d_{G}(u^{i}_{j})=5$ for $1\leq j\leq 12$, and $d_{G}(v^{i}_{j})=4$ for $1\leq j\leq 20$; see Figure 4 $(a)$. Let $B_{32x}$ be a graph obtained from $A_{32}^{i}\ (1\leq i\leq x)$ by adding the edges in $\{v_{5}^{i}v_{1}^{i+1}\,|\,1\leq i\leq x-1\}\cup\{v_{6}^{i}v_{2}^{i+1}\,|\,1\leq i\leq x-1\}\cup\{v_{7}^{i}v_{3}^{i+1}\,|\,1\leq i\leq x-1\}\cup\{v_{8}^{i}v_{4}^{i+1}\,|\,1\leq i\leq x-1\}$; see Figure 4 $(b)$.
• Step 2: Let $y=4z+a$, where $z=\lfloor y/4\rfloor$, $0\leq a\leq 3$. For each $j\ (1\leq j\leq z)$, we let $K_{4}^{j}$ be the complete graph of order $4$. Furthermore, let $K_{4}^{j,*}$ be the graph obtained from $K_{4}^{j}$ by adding four pendant vertices $w_{1}^{j},w_{2}^{j},w_{3}^{j},w_{4}^{j}$ with four pendant edges such that another end vertex of each pendant edge is attached on only one vertex in $K_{4}^{j}$; see Figure 4 $(c)$. For each $j\ (1\leq j\leq a)$, we let $S_{5}^{j}$ be the star of order $5$ with its leaves $p_{1}^{j},p_{2}^{j},p_{3}^{j},p_{4}^{j}$. Since $n\geq 96$, it follows that $A_{32}^{1},A_{32}^{2},A_{32}^{3}$ all exist. Set $S_{1}=\{v_{j}^{1}\,|\,1\leq j\leq 4\}\cup\{v_{j}^{1}\,|\,9\leq j\leq 20\}\subseteq V(A_{32}^{1})$, and $S_{2}=\{v_{j}^{2}\,|\,9\leq j\leq 20\}\subseteq V(A_{32}^{2})$, and $S_{3}=\{v_{j}^{3}\,|\,9\leq j\leq 20\}\subseteq V(A_{32}^{3})$. Then $|S_{1}\cup S_{2}\cup S_{3}|=40$. If $n\equiv 0~{}(mod~{}32)$, then $D_{n}=B_{32x}$. If $n\neq 0~{}(mod~{}32)$ and $n-32x\equiv 0~{}(mod~{}4)$, then $D_{n}$ is a graph obtained from $B_{32x}$ and $K_{4}^{1,*},K_{4}^{2,*},\ldots,K_{4}^{z,*}$ by identifying each vertex in $S^{\prime}=\{w_{j}^{i}\,|\,1\leq i\leq 4,\ 1\leq j\leq z\}$ and only one vertex in $S_{1}\cup S_{2}\cup S_{3}$. Since $|S^{\prime}|=4z<40=|S_{1}\cup S_{2}\cup S_{3}|$, for any vertex in $S^{\prime}$, we can find a vertex in $S_{1}\cup S_{2}\cup S_{3}$ and then identify the two vertices. If $n\neq 0~{}(mod~{}32)$ and $n-32x\neq 0~{}(mod~{}4)$, then $D_{n}$ is a graph obtained from $B_{32x}$, $K_{4}^{1,*},K_{4}^{2,*},\ldots,K_{4}^{z,*}$ and $S_{5}^{1},S_{5}^{2},\ldots,S_{5}^{a}$ by identifying each vertex in $S^{\prime}=\{w_{j}^{i}\,|\,1\leq i\leq 4,\ 1\leq j\leq z\}\cup\{p_{j}^{i}\,|\,1\leq i\leq 4,\ 1\leq j\leq a\}$ and only one vertex in $S_{1}\cup S_{2}\cup S_{3}$. Since $|S^{\prime}|=4z+4a\leq 28+12=40=|S_{1}\cup S_{2}\cup S_{3}|$, for any vertex in $S^{\prime}$, we can find a vertex in $S_{1}\cup S_{2}\cup S_{3}$ and then identify the two vertices.
• Step 3: Let $H_{n}$ be the graph $D_{n}$ by adding $\ell-5$ edges between $u_{12}^{1}$ and $V(G)-u_{12}^{1}$.

We now in a position to give the upper and lower bounds of $e_{n-3}(n,\ell,n-4)$.

Proposition 4.2

Let $\ell,n$ be two integers with $2\leq\ell\leq n-1$ and $n\geq 96$. Then

[TABLE]

where $n\equiv~{}i~{}(mod~{}32)$, $1\leq i\leq 31$.

Proof. Let $G$ be a graph of order $n$ such that $sdiam_{n-3}(G)=n-4$ and $\Delta(G)=\ell$, where $2\leq\ell\leq n-1$. Since $\Delta(G)=\ell$, it follows that there exists a vertex $v$ in $G$ such that $d_{G}(v)=\ell$. Since $sdiam_{n-3}(G)=n-4$, it follows from Lemma 4.2 that $\delta(G)\geq\kappa(G)\geq 4$. For any vertices in $V(G)-v$, its degree is at least $4$. Then $2e(G)\geq\ell+4(n-1)$, and hence $e_{n-3}(n,\ell,n-3)\geq 2n-2-\lceil\ell/2\rceil$.

It suffices to show $e_{n-3}(n,\ell,n-3)\leq 74\left\lfloor\frac{n}{32}\right\rfloor+2i+\ell-9$, where $n\equiv~{}i~{}(mod~{}32)$, and $1\leq i\leq 31$. Let $G=H_{n}$. Clearly, $\Delta(G)=\ell$. Since $\kappa(G)\geq 4$, it follows from Lemma 4.2 that $sdiam_{n-3}(G)=n-4$, and hence $e_{n-3}(n,\ell,n-3)\leq e(G)=74\left\lfloor\frac{n}{32}\right\rfloor+2i+\ell-9$, where $n\equiv~{}i~{}(mod~{}32)$, $1\leq i\leq 31$.

For the remain case $d=n-3$, we have the following.

Proposition 4.3

Let $\ell,n$ be two integers with $2\leq\ell\leq n-1$ and $n\geq 5$.

$(i)$* If $\lceil n/2\rceil+1\leq\ell\leq n-1$, then $e_{n-3}(n,\ell,n-3)\leq 2n-\ell+1$.*

$(ii)$* If $5\leq\ell\leq\lceil n/2\rceil$, then*

[TABLE]

where $n=(\ell+1)x+i$ and $0\leq i\leq\ell$.

$(iii)$* If $2\leq\ell\leq n-1$, then $e_{n-3}(n,\ell,n-3)\geq\max\{n-1+\lceil\ell/2\rceil,\lceil\frac{3n-\ell-3}{2}\rceil\}$.*

Proof. $(i)$ Let $G$ be a graph obtained from a wheel $W_{n-\ell+3}$ with center $w$ and a star $K_{1,\ell-3}$ by identifying the center of the star and one vertex of $W_{n-\ell+3}-w$. Note that $W_{n-\ell+3}-w$ is a cycle of order $n-\ell+2$, say $C=u_{1}u_{2}\ldots u_{n-\ell+2}u_{1}$. Let $u_{1}$ be the identifying vertex. Clearly, $d_{G}(w)=n-\ell+2$, $d_{G}(u_{1})=\ell$ and $d_{G}(u_{i})=3$ for each $i\ (2\leq i\leq n-\ell+2)$. Since $\lceil n/2\rceil+1\leq\ell\leq n-1$, it follows that $\ell\geq n-\ell+2$, and hence $\Delta(G)=\ell$. Since $G$ contains only one cut vertex, one can easily check that $sdiam_{n-3}(G)\leq n-3$. So, we have $e_{n-3}(n,\ell,n-3)\leq e(G)=2n-\ell+1$.

$(ii)$ Let $n=(\ell+1)x+y$, where $x=\lfloor n/(\ell+1)\rfloor$, $0\leq y\leq\ell$. For each $i\ (1\leq i\leq x)$, we let $W_{\ell+1}^{i}$ be the wheel of order $\ell+1$, with its center $w_{i}$. Note that $W_{\ell+1}^{i}-w_{i}$ is a cycle of order $\ell$, say $v^{i}_{1}v^{i}_{2}\ldots v^{i}_{\ell}v^{i}_{1}$. Let $F_{(\ell+1)x}$ be a graph obtained from $W_{\ell+1}^{i}\ (1\leq i\leq x)$ by adding the edges in $\{v_{j}^{i}v_{j}^{i+1}\,|\,1\leq i\leq x-1,\ 1\leq j\leq 3\}$. We now give a graph $D_{n}$ of order $n\ (n\geq\ell+1)$ constructed in the following way.

• •

If $n\equiv~{}0~{}(mod~{}\ell+1)$, then $D_{n}=F_{(\ell+1)x}$.

• •

If $n\equiv~{}1~{}(mod~{}\ell+1)$, then $D_{n}$ is the graph obtained from $F_{(\ell+1)x}$ by adding a new vertex $u$ and three edges $uv^{1}_{1},uv^{1}_{2},uv^{1}_{3}$.

• •

If $n\equiv~{}2~{}(mod~{}\ell+1)$, then $D_{n}$ is the graph obtained from $F_{(\ell+1)x}$ by adding two new vertices $u_{1},u_{2}$ and six edges $\{u_{1}u_{2},u_{1}v^{1}_{1},u_{1}v^{1}_{2},u_{2}v^{1}_{3},u_{2}v^{1}_{4},u_{2}v^{1}_{5}\}$.

• •

If $n\equiv~{}3~{}(mod~{}\ell+1)$, then $D_{n}$ is the graph obtained from $F_{(\ell+1)x}$ by adding two new vertices $u_{1},u_{2},u_{3}$ and six edges $\{u_{1}u_{2},u_{1}u_{3},u_{2}u_{3},u_{1}v^{1}_{1},u_{2}v^{1}_{2},u_{3}v^{1}_{3}\}$.

• •

If $n\equiv~{}y~{}(mod~{}\ell+1)\ (4\leq y\leq\ell)$, then $D_{n}$ is the graph obtained from $F_{(\ell+1)x}$ and a new wheel $W^{*}_{y}$ with center $w^{*}$ and $y$ vertices by adding three edges $\{v^{*}_{2}v^{1}_{1},v^{*}_{2}v^{1}_{2},v^{*}_{3}v^{1}_{3}\}$, where $V(W^{*}_{y})-w^{*}=\{v^{*}_{2},v^{*}_{2},\ldots,v^{*}_{y-1}\}$.

Let $H_{n}$ be the graph $D_{n}$ by adding $\ell-5$ edges between $v_{1}^{1}$ and $V(G)-v_{1}^{1}$. Clearly, $\Delta(H_{n})=\ell$. Since $H_{n}$ is $3$-connected, one can easily check that $sdiam_{n-3}(G)\leq n-3$. Since

[TABLE]

it follows that

[TABLE]

where $n=(\ell+1)x+i$ and $0\leq i\leq\ell$.

$(iii)$ Let $G$ be a graph of order $n$ such that $sdiam_{n-3}(G)\leq n-3$ and $\Delta(G)=\ell$, where $2\leq\ell\leq n-1$. Then $G$ is $2$-connected, or $G$ contains only one cut vertex. If $G$ is $2$-connected, then there exists a vertex $v$ in $G$ such that $d_{G}(v)=\ell$, since $\Delta(G)=\ell$. For any vertices in $V(G)-v$, its degree is at least $2$. Then $2e(G)\geq\ell+2(n-1)$, and hence $e_{n-3}(n,\ell,n-3)\geq n-1+\lceil\ell/2\rceil$. Suppose that $G$ contains only one cut vertex, say $v$. Then each connected component of $G\setminus v$ is a connected subgraph of order at least $3$, or an edge of $G$, or an isolated vertex. Let $w_{1},w_{2},\ldots,w_{r}$ be the isolated vertices, $e_{1},e_{2},\ldots,e_{s}$ be the edges, and $C_{1},C_{2},\ldots,C_{t}$ be the connected components of order at least $3$ in $G\setminus v$. Then we have the following claim.

Claim 1. For any $w\in\bigcup_{i=1}^{t}V(C_{i})$, if $d_{G}(w)=2$, then $wv\in E(G)$.

Proof of Claim 1. Assume, to the contrary, that $wv\notin E(G)$ for any $w\in\bigcup_{i=1}^{t}V(C_{i})$ with $d_{G}(w)=2$. Without loss of generality, let $w\in V(C_{1})$. Then there exist two vertices $u_{1},u_{2}$ in $C_{1}$ such that $u_{1}w\in E(C_{1})$ and $u_{2}w\in E(C_{1})$. Choose $S\subseteq V(G)$ with $|S|=n-3$ such that $w\in S$ but $u_{1},u_{2},v\notin S$. Then any $S$-Steiner tree must occupy $v$ and one of $u_{1},u_{2}$, and hence $d_{G}(S)\geq n-2$, a contradiction.

From Claim 1, we suppose that there are $x$ vertices $u_{1},u_{2},\ldots,u_{x}$ in $\bigcup_{i=1}^{t}V(C_{i})$ such that its degree is $2$. Then for any vertex in $(\bigcup_{i=1}^{t}V(C_{i}))\setminus\{u_{1},u_{2},\ldots,u_{x}\}$, its degree is at least $3$.

If $d_{G}(v)=\ell$, then $x\leq\ell-r-2s$ and $r\leq\ell$. Furthermore, we have

[TABLE]

and hence $e(G)\geq\lceil\frac{3n-\ell-3}{2}\rceil$.

If $d_{G}(v)\neq\ell$, then there exists a vertex $u\in V(G)-v$ such that $d_{G}(u)=\ell$. Clearly, $r+2s+t\leq d_{G}(v)\leq\ell$. Since $x+2s+r\leq\ell-1$, it follows that

[TABLE]

and hence $e(G)\geq\lceil\frac{3n-5}{2}\rceil$.

From the above argument, we conclude that $e_{n-3}(n,\ell,n-3)\geq\max\{n-1+\lceil\ell/2\rceil,\lceil\frac{3n-\ell-3}{2}\rceil\}$.

From Propositions 4.1, 4.2 and 4.3, we have the following theorem.

Theorem 4.1

$(i)$* For $2\leq\ell\leq n-1$, $e_{n-1}(n,\ell,n-1)=n-1$.*

$(ii)$* For $2\leq\ell\leq n-1$ and $n\geq 4$,*

[TABLE]

$(iii)$* For $2\leq\ell\leq n-1$ and $n\geq 96$,*

[TABLE]

where $n\equiv~{}i~{}(mod~{}32)$, $1\leq i\leq 31$.

$(iv)$* If $2\leq\ell\leq n-1$, then $e_{n-3}(n,\ell,n-3)\geq\max\{n-1+\lceil\ell/2\rceil,\frac{3n-\ell-5}{2}\}$. If $\lceil n/2\rceil+1\leq\ell\leq n-1$, then $e_{n-3}(n,\ell,n-3)\leq 2n-\ell+1$. If $5\leq\ell\leq\lceil n/2\rceil$, then*

[TABLE]

where $n=(\ell+1)x+i$ and $0\leq i\leq\ell$.

5 For general $k$

In [39], Mao et al. obtained the following result.

Lemma 5.1

Let $\ell,n$ be two integers with $1\leq\ell\leq n-2$, and let $G$ be a graph of order $n$. Then $\kappa(G)\geq\ell$ if and only if $sdiam_{n-\ell+1}(G)=n-\ell$.

In this section, we construct a graph and give an upper bound of $e_{k}(n,\ell,d)$ for general $k$, $\ell$, and $d$.

Theorem 5.1

Let $k,\ell,d$ be three integers with $2\leq k\leq n$, $2\leq\ell\leq n-1$, and $k-1\leq d\leq n-1$.

$(i)$* If $d=k-1$, $\lceil\frac{n+1}{2}\rceil\leq k\leq n$, and $\max\{n-k+1,\lceil\frac{n}{2}\rceil\}<\ell\leq n-1$, then*

[TABLE]

$(ii)$* If $2\leq k\leq d$, $k\leq d\leq n-1$, and $2+\lceil\frac{n-d+k-3}{d-k+1}\rceil\leq\ell\leq n-1$, then*

[TABLE]

Proof. $(i)$ We first consider the lower bound. From Lemma 5.1, for a connected graph $G$ of order $n$, $sdiam_{k}(G)=k-1$ if and only if $\kappa(G)\geq n-k+1$. Let $G$ be a graph of order $n$ such that $sdiam_{k}(G)=k-1$ and $\Delta(G)=\ell$, where $2\leq\ell\leq n-1$. Since $sdiam_{k}(G)=k-1$, it follows that $\delta(G)\geq\kappa(G)\geq n-k+1$. Since $\Delta(G)=\ell$, it follows that there exists a vertex $v$ in $G$ such that $d_{G}(v)=\ell$. For any vertex in $V(G)-v$, its degree is at least $n-k+1$. Then $2e(G)\geq\ell+(n-1)(n-k+1)$, and hence $e_{k}(n,\ell,k-1)\geq\lceil\frac{\ell+(n-1)(n-k+1)}{2}\rceil$.

Next, we consider the upper bound. For $\max\{n-k+1,\lceil\frac{n}{2}\rceil\}\leq\ell\leq n-1$, we let $K_{a,b}$ be a complete bipartite graph of order $n=a+b$ with $a\geq b$. Let $U,V=\{v_{1},v_{2},\ldots,v_{b}\}$ be the parts of order $a,b$ in $K_{a,b}$. Let $G$ be a graph obtained from $K_{a,b}$ by adding edges $v_{1}v_{i}\ (2\leq i\leq\ell-a)$, where $\ell\leq a+b-1=n-1$. Then $\Delta(G)=a+(\ell-a)=\ell$. Since $\max\{n-k+1,\lceil\frac{n}{2}\rceil\}<\ell\leq n-1$ and $a\geq b$, it follows that for any $S\subseteq V(G)$ and $|S|=k$, we have $S\cap U\neq\emptyset$ and $S\cap U\neq\emptyset$, and hence $sdiam_{k}(G)=k-1$. So $e_{k}(n,\ell,d)\leq ab+\ell-a=a(n-a)-a+\ell\leq\frac{(n-1)^{2}}{4}+\ell$.

$(ii)$ Let $P_{d-k+3}=v_{1}v_{2}\ldots v_{d-k+3}$ be a path of order $d-k+3$. Set $x=\lceil\frac{n-d+k-3}{d-k+1}\rceil$, $U_{i}=\{u_{i,1},u_{i,2},\ldots,u_{i,x}\,|\,2\leq i\leq d-k+1\}$, and $U_{d-k+2}=\{u_{d-k+2,1},u_{d-k+2,2},\ldots,u_{d-k+2,p}\}$, where $p=n-(d-k)(x+1)+3$. Let $T^{\prime}$ be a tree of order $n$ obtained from $P_{d-k+3}$ by adding all the vertices in $\bigcup_{i=2}^{d-k+2}U_{i}$, and then adding the edges in $\bigcup_{i=2}^{d-k+2}E_{T^{\prime}}[v_{i},U_{i}]$. Let $T$ be a tree obtained from $T^{\prime}$ by gradually deleting $\ell-x-2$ vertices in $V(T^{\prime})\setminus(\{v_{1},v_{2}\}\bigcup U_{2})$ and then adding $\ell-x-2$ pendant edges at $v_{2}$. Clearly, $d_{T}(v_{2})=\ell=\Delta(T)$. Since $sdiam_{k}(T)\leq d$, it follows that $e_{k}(n,\ell,d)=n-1$.