Steiner Distance in Product Networks

TL;DR

This paper explores the Steiner distance and Steiner k-diameter in product graphs, providing new bounds and properties relevant to network analysis and design.

Contribution

It introduces new results on Steiner distances and k-diameters specifically for Cartesian and lexicographical product graphs, expanding understanding of these metrics in complex networks.

Findings

Derived bounds for Steiner k-diameter in product graphs

Analyzed Steiner distances in specific network classes

Extended results to practical network models

Abstract

For a connected graph $G$ of order at least $2$ and $S\subseteq V(G)$, the \emph{Steiner distance} $d_G(S)$ among the vertices of $S$ is the minimum size among all connected subgraphs whose vertex sets contain $S$. Let $n$ and $k$ be two integers with $2\leq k\leq n$. Then the \emph{Steiner $k$-eccentricity $e_k(v)$} of a vertex $v$ of $G$ is defined by $e_k(v)=\max \{d_G(S)\,|\,S\subseteq V(G), \ |S|=k, \ and \ v\in S\}$. Furthermore, the \emph{Steiner $k$-diameter} of $G$ is $sdiam_k(G)=\max \{e_k(v)\,|\, v\in V(G)\}$. In this paper, we investigate the Steiner distance and Steiner $k$-diameter of Cartesian and lexicographical product graphs. Also, we study the Steiner $k$-diameter of some networks.

Equations

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

$$d_{G\square H}((g,h),(g^{\prime },h^{\prime }))=d_G(g,g^{\prime })+d_H(h,h^{\prime }).$$

$$d_{G\square H}(S)\ge d_G(S_G)+d_H(S_H).$$

$$d_G(S_G)+d_H(S_H)$$

$$\{(g_1,h_{j_1}),(g_1,h_{j_2}),\dots ,(g_1,h_{j_s})\}\cup \{(g_1,h_{j_{s+1}}),(g_1,h_{j_{s+2}}),\dots ,(g_1,h_{j_k})\}$$

$$x=|\{H(g_i)||V(H(g_i))\cap S|=1,1\le i\le a\}|.$$

$$\left(\bigcup _{i=s+1}^{a}E(H(g_i))\right)\cup E(T(h_{j_1}))$$

$$\left(\bigcup _{i=4}^{a}E(T(g_i))\right)\cup E(P^1(g_1))\cup E(P^2(g_3))\cup E(T(h_{j_2}))$$

$$\left(\bigcup _{i=4}^{a}E(T(g_i))\right)\cup E(Q^1(g_1))\cup E(Q^2(g_2))\cup E(Q^3(g_3))\cup E(T(h^{\ast }))$$

$$d_G(S_G)+d_H(S_H)$$

$$d_{G\square H}(S)=d_G(S_G)+d_H(S_H)$$

$$sdia{m}_k(G)+sdia{m}_k(H)$$

$$n-1+sdia{m}_k(H)$$

$$n+m-2\le sdia{m}_k(G\square H)\le m-1+(k-2)(n-1).$$

$$sdia{m}_k(G\square H)=d_{G\square H}(S)\le \min \{d_G(S_G)+(k-2)d_H(S_H),(k-2)d_G(S_G)+d_H(S_H)\}.$$

$$sdia{m}_k(G\square H)=d_{G\square H}(S)$$

$$sdia{m}_k(G\square H)$$

$$sdia{m}_k(G)+sdia{m}_k(H)=d_G(S_G)+d_H(S_H)\le d_{G\square H}(S)\le sdia{m}_k(G\square H).$$

$$n-1+sdia{m}_k(H)=d_G(S_G)+d_H(S_H)\le d_{G\square H}(S)\le sdia{m}_k(G\square H).$$

$$n+m-2=(n-1)+(m-1)=d_G(S_G)+d_H(S_H)\le d_{G\square H}(S)\le sdia{m}_k(G\square H),$$

$$sdia{m}_3(G\square H)=sdia{m}_3(G)+sdia{m}_3(H).$$

$$d_{G\circ H}((g,h),(g^{\prime},h^{\prime}))=\left\{\begin{array}[]{ll}d_{G}(g,g^{\prime}),&\mbox{\rm if}\ g\neq g^{\prime};\\[5.69046pt] d_{H}(h,h^{\prime}),&\mbox{\rm if}\ g=g^{\prime}~{}and~{}deg_{G}(g)=0;\\[5.69046pt] \min\{d_{H}(h,h^{\prime}),2\},&\mbox{\rm if}\ g=g^{\prime}~{}and~{}deg_{G}(g)\neq 0.\end{array}\right.$$

$$d_{G\circ H}((g,h),(g^{\prime },h^{\prime }))\ge d_G(g,g^{\prime }).$$

$$d_{G\circ H}(S)\ge d_G(S_G).$$

$$d_{G\circ H}(S)=d_G(S_G).$$

$$f(g_{a}g_{b})=\left\{\begin{array}[]{ll}(g_{a},h_{i_{a}})(g_{b},h_{i_{b}}),&\mbox{\rm if}~{}g_{a}\in S_{G}~{}{\rm and}~{}g_{b}\in S_{G};\\[5.69046pt] (g_{a},h_{i_{a}})(g_{b},h_{1}),&\mbox{\rm if}~{}g_{a}\in S_{G}~{}{\rm and}~{}g_{b}\notin S_{G};\\[5.69046pt] (g_{a},h_{1})(g_{b},h_{1}),&\mbox{\rm if}~{}g_{a}\notin S_{G}~{}{\rm and}~{}g_{b}\notin S_{G},\end{array}\right.$$

$$e_{a}=\left\{\begin{array}[]{ll}(g_{a},h_{i_{a}})(g_{b},h_{i_{b}}),&\mbox{\rm if}~{}g_{b}\in S_{G};\\[5.69046pt] (g_{a},h_{i_{a}})(g_{b},h_{1}),&\mbox{\rm if}~{}g_{b}\notin S_{G}\end{array}\right.$$

$$d_{G\circ H}(S)=\left\{\begin{array}[]{ll}d_{H}(S_{H}),&\mbox{\rm if}\ g=g^{\prime}=g^{\prime\prime}~{}and~{}deg_{G}(g)=0;\\[5.69046pt] \min\{d_{H}(S_{H}),3\},&\mbox{\rm if}\ g=g^{\prime}=g^{\prime\prime}~{}and~{}deg_{G}(g)\neq 0;\\[5.69046pt] \infty,&\mbox{\rm if}\ g\neq g^{\prime},~{}g^{\prime}=g^{\prime\prime}~{}and~{}d_{G}(g,g^{\prime})=\infty;\\[5.69046pt] d_{G}(g,g^{\prime})+1,&\mbox{\rm if}\ g\neq g^{\prime},~{}g^{\prime}=g^{\prime\prime}~{}and~{}d_{G}(g,g^{\prime})\neq\infty;\\[5.69046pt] d_{G}(S_{G}),&\mbox{\rm if}\ g\neq g^{\prime},~{}g\neq g^{\prime\prime}~{}and~{}g^{\prime}\neq g^{\prime\prime}.\end{array}\right.$$

$$\{(g,h)(g^{\ast },h),(g,h^{\prime })(g^{\ast },h),(g,h^{\prime \prime })(g^{\ast },h)\}$$

Full text

1 Introduction

In this paper, we consider graphs that are undirected, finite and simple. We refer the readers to Bondy and Murty (2008) for graph theoretical notations and terminology that are not defined here. For a graph $G$, let $V(G)$, $E(G)$, and $\delta(G)$ denote the set of vertices, the set of edges and the minimum degree of $G$, respectively. We refer to $|V(G)|$ the order of the graph and $|E(G)|$ the size of the graph. The degree of a vertex $v$ in $G$ is denoted by $deg_{G}(v)$. In this paper, $K_{n}$, $P_{n}$, $K_{1,n-1}$ and $C_{n}$ correspond to the complete graph of order $n$, the path of order $n$, the star of order $n$, and the cycle of order $n$, respectively. If $X\subseteq V(G)$, we use $G[X]$ to denote the subgraph induced by $X$. Similarly, if $F\subseteq E(G)$, let $G[F]$ denote the subgraph induced by $F$. If $X\subseteq V(G)\cup E(G)$, we use $G-X$ to denote the subgraph of $G$ obtained from $G$ by removing all the elements of $X$ and the edges incident to vertices that are in $X$. If $X=\{x\}$, we write $G-x$ for notational simplicity. For $X,Y\subseteq V(G)$, we use $E_{G}[X,Y]$ to denote the set of edges of $G$ with one end in $X$ and the other end in $Y$. If $X=\{x\}$, we simply write $E_{G}[x,Y]$ for $E_{G}[\{x\},Y]$. 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 $G$ be a connected graph. The distance between two vertices $u$ and $v$ in $G$ is the length of a shortest path between them, and it is denoted by $d_{G}(u,v)$. The eccentricity of $v$ in $G$, denoted by $e_{G}(v)$ (or simply $e(v)$ if it is clear from the context), is $\max\{d_{G}(u,v)\,|\,u\in V(G)\}$. In addition, we define the radius $rad(G)$ and the diameter $diam(G)$ of $G$ to be $rad(G)=\min\{e(v)\,|\,v\in V(G)\}$ and $diam(G)=\max\{e(v)\,|\,v\in V(G)\}$. It is a standard exercise to check that $rad(G)\leq diam(G)\leq 2rad(G)$. The center $C(G)$ of $G$ is the subgraph induced by the vertices with eccentricity equal to the radius. For more details on distance, we refer to Buckley and Harary (1990); Goddard and Oellermann (2011).

We observe that the distance between two vertices $u$ and $v$ in $G$ is equal to the minimum size of a connected subgraph of $G$ containing both $u$ and $v$. This suggests a generalization of the concept of distance. The Steiner distance of a graph, introduced by Chartrand et al. (1989) in 1989, is such a natural and nice generalization. Let $S$ be a set of vertices in a graph $G(V,E)$ where $|S|\geq 2$. We define an $S$-Steiner tree or a Steiner tree connecting $S$ (or simply, an $S$-tree) to be a subgraph $T(V^{\prime},E^{\prime})$ of $G$ that is a tree with $S\subseteq V^{\prime}$. Moreover, the Steiner distance $d_{G}(S)$ of $S$ in $G$ (or simply the distance of $S$) is the minimum size among all connected subgraphs whose vertex sets contain $S$. (Set $d_{G}(S)=\infty$ when there is no $S$-Steiner tree in $G$.) We remark 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. We further remark that $d_{G}(S)=\min\{e(T)\,|\,S\subseteq V(T)\}$, where $T$ is subtree of $G$. Finally, if $S=\{u,v\}$, then $d_{G}(S)=d(u,v)$ is the classical distance between $u$ and $v$. The following observation is obvious.

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 $2\leq k\leq n$. We define the Steiner $k$-eccentricity $e_{k}(v)$ of a vertex $v$ of $G$ to be $e_{k}(v)=\max\{d(S)\,|\,S\subseteq V(G),|S|=k,~{}and~{}v\in S\}$, the Steiner $k$-radius of $G$ to be $srad_{k}(G)=\min\{e_{k}(v)\,|\,v\in V(G)\}$, and the Steiner $k$-diameter of $G$ is $sdiam_{k}(G)=\max\{e_{k}(v)\,|\,v\in V(G)\}$. We remark that 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 $sdiam_{2}(G)=diam(G)$. It is not difficult to see the following observation.

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 $G$, $sdiam_{k}(G)\leq sdiam_{k+1}(G)$.*

Chartrand et al. (2010) obtained the following upper and lower bounds of $sdiam_{k}(G)$.

Theorem 1.3

Chartrand et al. (2010)* 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.*

Dankelmann et al. (1999) showed that $sdiam_{k}(G)\leq\frac{3|V(G)|}{\delta(G)+1}+3k$. Ali et al. (2012) improved the bound and showed that $sdiam_{k}(G)\leq\frac{3|V(G)|}{\delta(G)+1}+2k-5$ where $G$ 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. As a generalization of the center of a graph, one defines the Steiner $k$-center $C_{k}(G)\ (k\geq 2)$ of a connected graph $G$ to be the subgraph induced by the vertices $v$ of $G$ where $e_{k}(v)=srad_{k}(G)$. Oellermann and Tian (1990) showed that every graph is the $k$-center of some graph. Moreover, they showed that the $k$-center of a tree is a tree and they characterized those trees that are $k$-centers of trees. The Steiner $k$-median of $G$ is the subgraph of $G$ induced by the vertices of $G$ of minimum Steiner $k$-distance. The papers Oellermann (1995, 1999); Oellermann and Tian (1990) contain important results for Steiner centers and Steiner medians. For more details on the Steiner distance parameters, we refer to the survey paper Mao and papers Ali (2013); Cáceresa et al. (2008); D’Atri and Moscarini (1988); Dankelmann and Entringer (2000); Dankelmann et al. (1999); Day et al. (1994); Goddard and Oellrmann (1994); Mao et al. (2018).

Let $G$ be a $k$-connected graph and $u$, $v$ be a pair of vertices of $G$. Let $P_{k}(u,v)=\{P_{1},P_{2},\cdots,P_{k}\}$ be a family of $k$ internally vertex-disjoint paths between $u$ and $v$ and $l(P_{k}(u,v))$ be the length of the longest path in $P_{k}(u,v)$. Then the $k$-distance $d_{k}(u,v)$ between vertices $u$ and $v$ is the smallest $l(P_{k}(u,v))$ among all $P_{k}(u,v)$’s and the $k$-diameter $d_{k}(G)$ of $G$ is the maximum $k$-distance $d_{k}(u,v)$ over all pairs $u,v$ of vertices of $G$. The concept of $k$-diameter has its origin in the analysis of routings in networks as described by Chung (1987); Du et al. (1993); Hsu (1994); Hsu and Łuczak (1994); Meyer and Pradhan (1987).

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 Hakimi (1971); Levi (1971) is as follows: Given a graph and a set of vertices $S$, find a connected subgraph with minimum number of edges that contains $S$. This is, in general, an NP-hard problem Hwang et al. (1992). 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. 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. (2016) gave such a concept. They defined the $k$-center Steiner Wiener index $SW_{k}(G)$ of the graph $G$ to be

[TABLE]

For $k=2$, it coincides with the ordinary Wiener index. One usually considers $SW_{k}$ for $2\leq k\leq n-1$. However, the above definition can be extended to $k=1$ and $k=n$ as well where $SW_{1}(G)=0$ and $SW_{n}(G)=n-1$. There are other related concepts such as the Steiner Harary index. Both indices have chemical applications Furtula et al. (2016); Gutman et al. (2015). In addition, Gutman (2016) gave a generalization of the concept of degree distance, and then Mao and Das (2018) gave a generalization of the concept of Gutman index. We refer the readers to Furtula et al. (2016); Gutman et al. (2015); Gutman (2016); Li et al. (2016, 2017); Mao and Das (2018); Mao et al. (2016, 2017a, 2017b) for details.

1.3 Products of graphs

The main focus of this paper is Steiner $k$-diameter of two products of graphs, namely, the Cartesian product and the lexicographic product. These are well-known products. See Hammack et al. (2011).

$\bullet$ The Cartesian product of two graphs $G$ and $H$, written as $G\Box H$, is the graph with vertex set $V(G)\times V(H)$, in which two vertices $(g,h)$ and $(g^{\prime},h^{\prime})$ are adjacent if and only if $g=g^{\prime}$ and $(h,h^{\prime})\in E(H)$, or $h=h^{\prime}$ and $(g,g^{\prime})\in E(G)$.

$\bullet$ The lexicographic product of two graphs $G$ and $H$, written as $G\circ H$, is defined as follows: $V(G\circ H)=V(G)\times V(H)$, and two distinct vertices $(g,h)$ and $(g^{\prime},h^{\prime})$ of $G\circ H$ are adjacent if and only if either $(g,g^{\prime})\in E(G)$ or $g=g^{\prime}$ and $(h,h^{\prime})\in E(H)$.

It is easy to see that the Cartesian product is commutative, that is, $G\Box H$ is isomorphic to $H\Box G$. 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 Bao et al. (1998); Day and Al-Ayyoub (1997); Hammack et al. (2011); Ku et al. (2003).

Gologranc (2018) obtained a sharp lower bound for Steiner distance of Cartesian product graphs. We continue this study in Section $2$ by obtaining a sharp upper bound for Steiner distance. In addition, we will also present sharp upper and lower bounds for Steiner $k$-diameter of Cartesian product graphs. In Section $3$, we derive the results for Steiner distance and Steiner $k$-diameter of lexicographic product graphs, which strengthen a result given by Anand et al. (2012). In Section $4$, 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 $G$ and $H$ be two graphs with $V(G)=\{g_{1},g_{2},\ldots,g_{n}\}$ and $V(H)=\{h_{1},h_{2},\ldots,h_{m}\}$, respectively. Then $V(G\ast H)=\{(g_{i},h_{j})\,|\,1\leq i\leq n,\ 1\leq j\leq m\}$, where $\ast$ denotes the Cartesian product operation or lexicographical product operation. For $h\in V(H)$, we use $G(h)$ to denote the subgraph of $G\ast H$ induced by the vertex set $\{(g_{i},h)\,|\,1\leq i\leq n\}$. Similarly, for $g\in V(G)$, we use $H(g)$ to denote the subgraph of $G\circ H$ induced by the vertex set $\{(g,h_{j})\,|\,1\leq j\leq m\}$.

The following observation can be easily seen.

Observation 2.1

Let $G$ be a connected graph, and let $S\subseteq V(G)$ and $|S|=3$. Let $T$ be a minimal $S$-Steiner tree in $G$. Then the tree $T$ satisfies one of the following conditions.

$\bullet$* $T$ is a path;*

$\bullet$* $T$ is a subdivision of $K_{1,3}$.*

We start with the following basic result.

Lemma 2.2

Hammack et al. (2011)* Let $G$ and $H$ be two graphs, and let $(g,h)$ and $(g^{\prime},h^{\prime})$ be two vertices of $G\Box H$. Then*

[TABLE]

2.1 Steiner distance of Cartesian product graphs

Gologranc (2018) obtained the following lower bound for Steiner distance.

Lemma 2.3

Gologranc (2018)* Let $k\geq 2$ be an integer, and let $G,H$ be two connected graphs. Let $S=\{(g_{i_{1}},h_{j_{1}}),$ $(g_{i_{2}},h_{j_{2}}),\ldots,(g_{i_{k}},h_{j_{k}})\}$ be a set of distinct vertices of $G\Box H$. Let $S_{G}=\{g_{i_{1}},g_{i_{2}},\ldots,g_{i_{k}}\}$ and $S_{H}=\{h_{j_{1}},h_{j_{2}},\ldots,h_{j_{k}}\}$. Then*

[TABLE]

We will show that the inequality in Lemma 2.3 can be equality if $k=3$; shown in following Corollary 2.6. But, for general $k\ (k\geq 4)$, from Lemma 2.3 and Corollary 2.6, one may conjecture that for two connected graphs $G,H$, $d_{G\Box H}(S)=d_{G}(S_{G})+d_{H}(S_{H})$, where $S=\{(g_{i_{1}},h_{j_{1}}),(g_{i_{2}},h_{j_{2}}),\ldots,(g_{i_{k}},h_{j_{k}})\}\subseteq V(G\Box H)$, $S_{G}=\{g_{i_{1}},g_{i_{2}},\ldots,g_{i_{k}}\}\subseteq V(G)$ and $S_{H}=\{h_{j_{1}},h_{j_{2}},\ldots,h_{j_{k}}\}\subseteq V(H)$.

Remark 1: Actually, the equality $d_{G\Box H}(S)=d_{G}(S_{G})+d_{H}(S_{H})$ is not true for $|S|\geq 4$. For example, let $G$ be a tree with degree sequence $(3,2,1,1,1)$ and $H$ be a path of order $5$. Let $S=\{(g_{1},h_{1}),(g_{2},h_{2}),(g_{3},h_{3}),(g_{4},h_{4})\}$ be a vertex set of $G\Box H$ shown in Fig.1. Then $d_{G}(S_{G})=4$ for $S_{G}=\{g_{1},g_{2},g_{3},g_{4}\}$, and $d_{H}(S_{H})=4$ for $S_{H}=\{h_{1},h_{2},h_{3},h_{4}\}$. One can check that there is no $S$-Steiner tree of size $8$ in $G\Box H$, which implies $d_{G\Box H}(S)\geq 9$.

Although the conjecture of such an ideal formula is not correct, it is possible to give a strong upper bound for general $k\ (k\geq 3)$. Remark 1 also indicates that obtaining a nice formula for the general case may be difficult. We now give such an upper bound of $d_{G\Box H}(S)$ for $S\subseteq V(G\Box H)$ and $|S|=k$.

Theorem 2.4

Let $k,m,n$ be three integers with $3\leq k\leq mn$, and let $G,H$ be two connected graphs with $V(G)=\{g_{1},g_{2},\ldots,g_{n}\}$ and $V(H)=\{h_{1},h_{2},\ldots,h_{m}\}$. Let $S=\{(g_{i_{1}},h_{j_{1}}),(g_{i_{2}},h_{j_{2}}),\ldots,(g_{i_{k}},h_{j_{k}})\}$ be a set of distinct vertices of $G\Box H$, $S_{G}=\{g_{i_{1}},g_{i_{2}},$ $\ldots,g_{i_{k}}\}$, and $S_{H}=\{h_{j_{1}},h_{j_{2}},\ldots,h_{j_{k}}\}$, where $S_{G}\subseteq V(G)$, $S_{H}\subseteq V(H)$ ($S_{G},S_{H}$ are both multi-sets). Then

[TABLE]

where $r,t\ (0\leq r,t\leq k-3)$ are defined as follows.

$\bullet$* Let $X_{G}^{i}\ (1\leq i\leq{k\choose 3})$ be all the $(k-3)$-multi-subsets of $\{g_{i_{1}},g_{i_{2}},,\ldots,g_{i_{k}}\}$ in $G$, and let $r_{i}$ be the numbers of distinct vertices in $X_{G}^{i}\ (1\leq i\leq{k\choose 3})$, and let $r=\min\{r_{i}\,|\,1\leq i\leq{k\choose 3}\}$.*

$\bullet$* Let $Y_{H}^{j}\ (1\leq j\leq{k\choose 3})$ be all the $(k-3)$-multi-subsets of $\{h_{j_{1}},h_{j_{2}},\ldots,h_{j_{k}}\}$ in $H$, and let $t_{j}$ be the numbers of distinct vertices in $Y_{H}^{j}\ (1\leq j\leq{k\choose 3})$, and let $t=\min\{t_{j}\,|\,1\leq j\leq{k\choose 3}\}$.*

Proof. From Lemma 2.3, we have $d_{G\Box H}(S)\geq d_{G}(S_{G})+d_{H}(S_{H})$. By symmetry, we only need to show $d_{G\Box H}(S)\leq d_{G}(S_{G})+(r+1)d_{H}(S_{H})$. Recall that $V(G)=\{g_{1},g_{2},\ldots,g_{n}\}$ and $V(H)=\{h_{1},h_{2},\ldots,h_{m}\}$. Without loss of generality, we assume that $H(g_{1}),H(g_{2}),\ldots,$ $H(g_{a})$ be the $H$ copies such that $|V(H(g_{i}))\cap S|\neq 0$, $1\leq i\leq a$. Then $(g_{i_{1}},h_{j_{1}}),(g_{i_{2}},h_{j_{2}}),$ $\ldots,(g_{i_{k}},h_{j_{k}})\in\bigcup_{i=1}^{a}V(H(g_{i}))$, and hence we have the following cases to consider.

Case 1. For each $H(g_{i})\ (1\leq i\leq a)$, $|V(H(g_{i}))\cap S|\geq 2$.

Without loss of generality, let $V(H(g_{1}))\cap S=\{(g_{i_{1}},h_{j_{1}}),(g_{i_{2}},h_{j_{2}}),\ldots,(g_{i_{s}},h_{j_{s}})\}$, where $s\geq 2$. Thus, we have $(g_{i_{p}},h_{j_{p}})=(g_{1},h_{j_{p}})$ for each $p\ (1\leq p\leq s)$, and $(g_{i_{s+1}},h_{j_{s+1}}),(g_{i_{s+2}},h_{j_{s+2}}),\ldots,$ $(g_{i_{k}},h_{j_{k}})\in\bigcup_{i=2}^{a}V(H(g_{i}))$. Note that $(g_{1},h_{j_{1}}),(g_{1},h_{j_{2}}),\ldots,(g_{1},h_{j_{s}})\in V(H(g_{1}))$. On one hand, since there is an $S_{H}$-Steiner tree of size $d_{H}(S_{H})$ in $H$, it follows that there exists an Steiner tree of size $d_{H}(S_{H})$ connecting

[TABLE]

in $H(g_{1})$, say $T(g_{1})$. For each $i\ (2\leq i\leq k)$, let $T(g_{i})$ be the Steiner tree in $H(g_{i})$ corresponding to $T(g_{1})$ in $H(g_{1})$. Note that $T(g_{i})\ (1\leq i\leq k)$ is the Steiner tree of size $d_{H}(S_{H})$ connecting $\{(g_{i},h_{j_{1}}),(g_{i},h_{j_{2}}),\ldots,(g_{i},h_{j_{s}}),(g_{i},h_{j_{s+1}}),(g_{i},h_{j_{s+2}}),\ldots,(g_{i},h_{j_{k}})\}$ in $H(g_{i})$. One can see that $(g_{i_{s+1}},h_{j_{s+1}}),\ldots,(g_{i_{k}},h_{j_{k}})\in\bigcup_{i=2}^{a}V(T(g_{i}))$. On the other hand, since there is an $S_{G}$-Steiner tree of size $d_{G}(S_{G})$ in $G$, it follows that there exists an Steiner tree of size $d_{G}(S_{G})$ connecting $\{(g_{1},h_{j_{1}}),$ $(g_{2},h_{j_{1}}),\ldots,(g_{a},h_{j_{1}})\}$ in $G(h_{j_{1}})$, say $T(h_{j_{1}})$. Furthermore, the subgraph induced by the edges in $\left(\bigcup_{i=1}^{a}E(T(g_{i}))\right)\cup E(T(h_{j_{1}}))$ is an $S$-Steiner tree in $G\Box H$ (see Fig.$2$ $(a)$), and hence $d_{G\Box H}(S)\leq d_{G}(S_{G})+ad_{H}(S_{H})$.

From the definition of $r$, if $|V(H(g_{i}))\cap S|\geq 4$ for each $H(g_{i})\ (1\leq i\leq a)$, then $r=a$ and $d_{G\Box H}(S)\leq d_{G}(S_{G})+rd_{H}(S_{H})$. If there exists some $H(g_{i})\ (1\leq i\leq a)$ such that $2\leq|V(H(g_{i}))\cap S|\leq 3$ for $H(g_{i})\ (1\leq i\leq a)$, then $r=a-1$ and $d_{G\Box H}(S)\leq d_{G}(S_{G})+(r+1)d_{H}(S_{H})$.

Case 2. There exists some $H(g_{i})$ such that $|V(H(g_{i}))\cap S|=1$, where $1\leq i\leq a$.

Without loss of generality, we assume that $|V(H(g_{i}))\cap S|=1$ for each $i\ (1\leq i\leq x)$, where $1\leq x\leq a$. For $x\neq a$, we have $|V(H(g_{i}))\cap S|\geq 2$ for each $i\ (x+1\leq i\leq a)$. One can see that

[TABLE]

Subcase 2.1. $x\geq 3$.

If $|\{h_{j_{1}},h_{j_{2}},\cdots,h_{j_{x}}\}|=1$, then $h_{j_{1}}=h_{j_{2}}=\cdots=h_{j_{x}}$. Since there is an $S_{G}$-Steiner tree of size $d_{G}(S_{G})$ in $G$, it follows that there exists an Steiner tree of size $d_{G}(S_{G})$ connecting $\{(g_{1},h_{j_{1}}),(g_{2},h_{j_{1}}),$ $\ldots,(g_{a},h_{j_{1}})\}$ in $G(h_{j_{1}})$, say $T(h_{j_{1}})$. Since there is an $S_{H}$-Steiner tree of size $d_{H}(S_{H})$ in $H$, it follows that there exists an Steiner tree of size $d_{H}(S_{H})$ connecting $\{(g_{x+1},h_{j_{1}})\}\cup\{(g_{x+1},h_{j_{x+1}}),(g_{x+1},h_{j_{x+2}}),$ $\ldots,(g_{x+1},h_{j_{k}})\}$ in $H(g_{x+1})$, say $T(g_{x+1})$. For each $i\ (x+2\leq i\leq a)$, let $T(g_{i})$ be the Steiner tree in $H(g_{i})$ corresponding to $T(g_{x+1})$ in $H(g_{x+1})$. Note that $T(g_{i})\ (x+1\leq i\leq a)$ is the Steiner tree of size $d_{H}(S_{H})$ connecting $\{(g_{i},h_{j_{x+1}}),(g_{i},h_{j_{x+2}}),\ldots,(g_{i},h_{j_{k}})\}$ in $H(g_{i})$. Furthermore, the subgraph induced by the edges in $\left(\bigcup_{i=x+1}^{a}E(T(g_{i}))\right)\cup E(T(h_{j_{1}}))$ is an $S$-Steiner tree (see Fig.2 $(b)$), and hence $d_{G\Box H}(S)\leq d_{G}(S_{G})+(a-x)d_{H}(S_{H})\leq d_{G}(S_{G})+(a-3)d_{H}(S_{H})$. From the definition of $r$, we have $r=a-3$, and hence $d_{G\Box H}(S)\leq d_{G}(S_{G})+rd_{H}(S_{H})\leq d_{G}(S_{G})+(r+1)d_{H}(S_{H})$, as desired.

If $|\{h_{j_{1}},h_{j_{2}},\cdots,h_{j_{x}}\}|=2$, then we can assume that $h_{j_{1}}=h_{j_{2}}=\ldots=h_{j_{s}}$, $h_{j_{s+1}}=h_{j_{s+2}}=\ldots=h_{j_{x}}$, and $h_{j_{1}}\neq h_{j_{x}}$. Furthermore, we can assume that $s\geq 2$. Since there is an $S_{H}$-Steiner tree of size $d_{H}(S_{H})$ in $H$, it follows that there is a Steiner tree of size $d_{H}(S_{H})$ connecting $\{(g_{s+1},h_{j_{s+1}}),$ $(g_{s+1},h_{j_{x+2}}),\ldots,(g_{s+1},h_{j_{k}})\}$ in $H(g_{s+1})$, say $T(g_{s+1})$. For each $i\ (s+2\leq i\leq a)$, let $T(g_{i})$ be the Steiner tree in $H(g_{i})$ corresponding to $T(g_{s+1})$ in $H(g_{s+1})$. Since there is an $S_{G}$-Steiner tree of size $d_{G}(S_{G})$ in $G$, it follows that there exists an Steiner tree of size $d_{G}(S_{G})$ connecting $\{(g_{1},h_{j_{1}}),(g_{2},h_{j_{1}}),$ $\ldots,(g_{a},h_{j_{1}})\}$ in $G(h_{j_{1}})$, say $T(h_{j_{1}})$. Then the subgraph induced by the edges in

[TABLE]

is an $S$-Steiner tree in $G\Box H$, and hence $d_{G\Box H}(S)\leq d_{G}(S_{G})+(a-s)d_{H}(S_{H})\leq d_{G}(S_{G})+(a-2)d_{H}(S_{H})$. Since $r=a-3$, it follows that $d_{G\Box H}(S)\leq d_{G}(S_{G})+(a-2)d_{H}(S_{H})=d_{G}(S_{G})+(r+1)d_{H}(S_{H})$.

From now on, we assume $|\{h_{j_{1}},h_{j_{2}},\cdots,h_{j_{x}}\}|\geq 3$. Note that there is an $S_{H}$-Steiner tree of size $d_{H}(S_{H})$ in $H$, say $T$. Without loss of generality, let $h_{j_{1}}\neq h_{j_{2}}\neq h_{j_{3}}$. Since $h_{j_{1}},h_{j_{2}},h_{j_{3}}\in V(T)$, it follows that there is a minimal subtree $T^{\prime}$ connecting $\{h_{j_{1}},h_{j_{2}},h_{j_{3}}\}$ in $T$. From Observation 2.1, $T^{\prime}$ is a path or $T^{\prime}$ is a subdivision of $K_{1,3}$. If $T^{\prime}$ is a path, then without loss of generality, we can assume $h_{j_{2}}$ is the interval vertex of $T^{\prime}$. Therefore, there are a unique $(h_{j_{1}},h_{j_{2}})$-path, say $P^{1}$, and a unique $(h_{j_{2}},h_{j_{3}})$-path, say $P^{2}$, in $T^{\prime}$. If $T^{\prime}$ is a subdivision of $K_{1,3}$, then there exists a vertex in $T^{\prime}$, say $h^{*}\in V(H)\setminus\{h_{j_{1}},h_{j_{2}},h_{j_{3}}\}$, such that there are three paths $Q^{1},Q^{2},Q^{3}$ connecting $h^{*}$ and $h_{j_{1}},h_{j_{2}},h_{j_{3}}$, respectively, in $T^{\prime}$.

We first consider the case that $T^{\prime}$ is a path. On one hand, for each $i\ (1\leq i\leq k)$, let $T(g_{i})$ be the Steiner tree in $H(g_{i})$ corresponding to $T$ in $H$. Note that $T(g_{i})$ is the Steiner tree of size $d_{H}(S_{H})$ connecting $\{(g_{i},h_{j_{1}}),(g_{i},h_{j_{2}}),\ldots,(g_{i},h_{j_{k}})\}$ in $H(g_{i})$. For each $i\ (1\leq i\leq 3)$, let $P^{1}(g_{i})$ be the path in $H(g_{i})$ corresponding to $P^{1}$ in $H$, and let $P^{2}(g_{i})$ be the path in $H(g_{i})$ corresponding to $P^{2}$ in $H$. On the other hand, since there is an $S_{G}$-Steiner tree of size $d_{G}(S_{G})$ in $G$, it follows that there exists an Steiner tree of size $d_{G}(S_{G})$ connecting $\{(g_{1},h_{j_{2}}),(g_{2},h_{j_{2}}),$ $\ldots,(g_{k},h_{j_{2}})\}$ in $G(h_{j_{2}})$, say $T(h_{j_{2}})$. Furthermore, the subgraph induced by the edges in

[TABLE]

is an $S$-Steiner tree in $G\Box H$ (see Fig.3 $(a)$), and hence $d_{G\Box H}(S)\leq d_{G}(S_{G})+(a-2)d_{H}(S_{H})$. Since $r=a-3$, it follows that $d_{G\Box H}(S)\leq d_{G}(S_{G})+(r+1)d_{H}(S_{H})$.

Next, we consider the case that $T^{\prime}$ is a subdivision of $K_{1,3}$. On one hand, for each $i\ (1\leq i\leq k)$, let $T(g_{i})$ be the tree in $H(g_{i})$ corresponding to $T$ in $H$. Note that $T(g_{i})$ is the Steiner tree of size $d_{H}(S_{H})$ connecting $\{(g_{i},h_{j_{1}}),(g_{i},h_{j_{2}}),\ldots,(g_{i},h_{j_{k}})\}$ in $H(g_{i})$. For each $i\ (1\leq i\leq 3)$, let $Q^{1}(g_{i})$ be the path in $H(g_{i})$ corresponding to $Q^{1}$ in $H$, and let $Q^{2}(g_{i})$ be the path in $H(g_{i})$ corresponding to $Q^{2}$ in $H$, and let $Q^{3}(g_{i})$ be the path in $H(g_{i})$ corresponding to $Q^{3}$ in $H$. For each $i\ (1\leq i\leq k)$, let $(g_{i},h^{*})$ be the path in $H(g_{i})$ corresponding to $h^{*}$ in $H$.

On the other hand, since there is an $S_{G}$-Steiner tree of size $d_{G}(S_{G})$ in $G$, it follows that there exists an Steiner tree of size $d_{G}(S_{G})$ connecting $\{(g_{1},h^{*}),(g_{2},h^{*}),$ $\ldots,(g_{k},h^{*})\}$ in $G(h^{*})$, say $T(h^{*})$. Furthermore, the subgraph induced by the edges in

[TABLE]

is an $S$-Steiner tree in $G\Box H$ (see Fig.3 $(b)$), and hence $d_{G\Box H}(S)\leq d_{G}(S_{G})+(a-2)d_{H}(S_{H})$. Since $r=a-3$, it follows that $d_{G\Box H}(S)\leq d_{G}(S_{G})+(r+1)d_{H}(S_{H})$.

Subcase 2.2. $x=1$ or $x=2$.

Without loss of generality, let $|V(H(g_{1}))\cap S|=1$ and $(g_{i_{1}},h_{j_{1}})=(g_{1},h_{j_{1}})$. Since there is an $S_{G}$-Steiner tree of size $d_{G}(S_{G})$ in $G$, it follows that there exists an Steiner tree of size $d_{G}(S_{G})$ connecting $\{(g_{1},h_{j_{1}}),(g_{2},h_{j_{1}}),\ldots,(g_{a},h_{j_{1}})\}$ in $G(h_{j_{1}})$, say $T(h_{j_{1}})$. Since there is an $S_{H}$-Steiner tree of size $d_{H}(S_{H})$ in $H$, it follows that there exists an Steiner tree of size $d_{H}(S_{H})$ connecting $\{(g_{2},h_{j_{1}}),(g_{2},h_{j_{2}}),$ $\ldots,(g_{2},h_{j_{k}})\}$ in $H(g_{2})$, say $T(g_{2})$. For each $i\ (3\leq i\leq a)$, let $T(g_{i})$ be the Steiner tree in $H(g_{i})$ corresponding to $T(g_{2})$ in $H(g_{2})$. Note that $T(g_{i})\ (2\leq i\leq a)$ is the Steiner tree of size $d_{H}(S_{H})$ connecting $\{(g_{i},h_{j_{1}}),(g_{i},h_{j_{2}}),\ldots,(g_{i},h_{j_{k}})\}$ in $H(g_{i})$. Furthermore, the subgraph induced by the edges in $\left(\bigcup_{i=2}^{a}E(T(g_{i}))\right)\cup E(T(h_{j_{1}}))$ is an $S$-Steiner tree, and hence $d_{G\Box H}(S)\leq d_{G}(S_{G})+(a-1)d_{H}(S_{H})$. From the definition of $r$, we have $r=a-2$ or $r=a-1$, and hence $d_{G\Box H}(S)\leq d_{G}(S_{G})+(r+1)d_{H}(S_{H})$, as desired.

From the above argument, we conclude that $d_{G\Box H}(S)\leq\min\{d_{G}(S_{G})+(r+1)d_{H}(S_{H}),$ $d_{H}(S_{H})+(t+1)d_{G}(S_{G})\}$, as desired. \qed

The following corollaries are immediate from Theorem 2.4.

Corollary 2.5

Let $G,H$ be two connected graphs of order $n,m$, respectively. Let $k$ be an integer with $3\leq k\leq mn$. Let $S=\{(g_{i_{1}},h_{j_{1}}),(g_{i_{2}},h_{j_{2}}),\ldots,(g_{i_{k}},h_{j_{k}})\}$ be a set of distinct vertices of $G\Box H$. Let $S_{G}=\{g_{i_{1}},g_{i_{2}},\ldots,g_{i_{k}}\}$ and $S_{H}=\{h_{j_{1}},h_{j_{2}},\ldots,h_{j_{k}}\}$. Then

[TABLE]

Corollary 2.6

Let $G,H$ be two connected graphs, and let $(g,h)$, $(g^{\prime},h^{\prime})$ and $(g^{\prime\prime},h^{\prime\prime})$ be three vertices of $G\Box H$. Let $S_{G}=\{g,g^{\prime},g^{\prime\prime}\}$, $S_{H}=\{h,h^{\prime},h^{\prime\prime}\}$, and $S=\{(g,h),(g^{\prime},h^{\prime}),$ $(g^{\prime\prime},h^{\prime\prime})\}$. Then

[TABLE]

To show the sharpness of the above upper and lower bound, we consider the following example.

Example 1: $(1)$ For $k=3$, from Corollary 2.6, we have $d_{G\Box H}(S)=d_{G}(S_{G})+d_{H}(S_{H})$, which implies that the upper and lower bounds in Corollary 2.5 and Theorem 2.4 are sharp.

$(2)$ Let $G=P_{n}$ and $H=K_{1,m-1}$, where $P_{n}=g_{1}g_{2}\cdots g_{n}$, $h_{1},h_{2},\cdots,h_{m-1}$ are the leaves of $H$, and $h_{m}$ is the center of $H$. Choose $S=\{(g_{1},h_{1}),(g_{1},h_{2}),(g_{1},h_{m})\}\cup\{(g_{n},h_{1}),(g_{n},h_{2}),(g_{n},h_{m})\}\cup\{(g_{i},h_{1}),(g_{i},h_{2}),(g_{i},h_{m})\,|\,2\leq i\leq x-2\}$, where $4\leq x\leq n$. Then $d_{G}(S_{G})=n-1$, $d_{H}(S_{H})=2$, $r=x-1$, $t=3$ and $d_{G\Box H}(S)=n-1+2x=n-1+2+\min\{2(x-1),3(n-1)\}=d_{G}(S_{G})+d_{H}(S_{H})+\min\{rd_{H}(S_{H}),td_{G}(S_{G})\}$, which implies that the upper bound in Corollary 2.5 are sharp.

2.2 Steiner diameter of Cartesian product graphs

For Steiner $k$-diameter, we have the following.

Theorem 2.7

Let $k,m,n$ be an integer with $3\leq k\leq mn$ and $n\leq m$. Let $G,H$ be two connected graphs of order $n,m$, respectively.

$(1)$* If $k\leq n$, then*

[TABLE]

$(2)$* If $n<k\leq m$, then*

[TABLE]

$(3)$* If $m<k\leq mn$, then*

[TABLE]

$(4)$* If $mn-\kappa(G\Box H)+1\leq k\leq mn$, then $sdiam_{k}(G\Box H)=k-1$.*

Proof. We first consider all the upper bounds in this theorem. From the definition of $sdiam_{k}(G\Box H)$, there exists a vertex subset $S\subseteq V(G\Box H)$ with $|S|=k$ such that $d_{G\Box H}(S)=sdiam_{k}(G\Box H)$. Let $S=\{(g_{i_{1}},h_{j_{1}}),(g_{i_{2}},h_{j_{2}}),\ldots,(g_{i_{k}},h_{j_{k}})\}$, and let $S_{G}=\{g_{i_{1}},g_{i_{2}},\ldots,g_{i_{k}}\}$ and $S_{H}=\{h_{j_{1}},h_{j_{2}},\ldots,h_{j_{k}}\}$. From Corollary 2.5, we have

[TABLE]

For $(1)$, since $k\leq n$, it follows that $d_{G}(S_{G})\leq sdiam_{k}(G)$ and $d_{H}(S_{H})\leq sdiam_{k}(H)$, and hence

[TABLE]

For $(2)$, since $n<k\leq m$, it follows that $d_{G}(S_{G})\leq n-1$ and $d_{H}(S_{H})\leq sdiam_{k}(H)$, and hence

[TABLE]

For $(3)$, since $m<k\leq mn$, it follows that $d_{G}(S_{G})\leq n-1$ and $d_{H}(S_{H})\leq m-1$, and hence

[TABLE]

Next, we consider the lower bounds in this theorem. For $(1)$, we suppose $k\leq n\leq m$. From the definition of $sdiam_{k}(G)$, it follows that there exists a vertex subset $S_{G}\subseteq V(G)$ with $|S_{G}|=k$ such that $d_{G}(S_{G})=sdiam_{k}(G)$. Similarly, there exists a vertex subset $S_{H}\subseteq V(H)$ with $|S_{H}|=k$ such that $d_{H}(S_{H})=sdiam_{k}(H)$. Without loss of generality, let $S_{G}=\{g_{1},g_{2},\ldots,g_{k}\}$ and $S_{H}=\{h_{1},h_{2},\ldots,h_{k}\}$. Then $S=\{(g_{1},h_{1}),(g_{2},h_{2}),\ldots,(g_{k},h_{k})\}\subseteq V(G\Box H)$ and $|S|=k$. From Lemma 2.3 and the definition of Steiner $k$-diameter, we have

[TABLE]

For $(2)$, we suppose $n<k\leq m$. Let $S=\{(g_{i_{1}},h_{j_{1}}),(g_{i_{2}},h_{j_{2}}),\ldots,(g_{i_{k}},h_{j_{k}})\}$ be a set of distinct vertices of $G\Box H$ such that $V(G)\subseteq\{g_{i_{1}},g_{i_{2}},\ldots,g_{i_{k}}\}=S_{G}$ and $d_{H}(S_{H})=sdiam_{k}(H)$, where $S_{H}=\{h_{j_{1}},h_{j_{2}},\ldots,h_{j_{k}}\}$. From Lemma 2.3, we have

[TABLE]

For $(3)$, we suppose $m<k\leq mn$. Let $S=\{(g_{i_{1}},h_{j_{1}}),(g_{i_{2}},h_{j_{2}}),\ldots,(g_{i_{k}},h_{j_{k}})\}$ be a set of distinct vertices of $G\Box H$ such that $V(G)\subseteq S_{G}$ and $V(H)\subseteq S_{H}$, where $S_{G}=\{g_{i_{1}},g_{i_{2}},\ldots,g_{i_{k}}\}$ and $S_{H}=\{h_{j_{1}},h_{j_{2}},\ldots,h_{j_{k}}\}$. From Lemma 2.3, we have

[TABLE]

as desired.

For $(4)$, we suppose $mn-\kappa(G\Box H)+1\leq k\leq mn$. For any $S\subseteq V(G\Box H)$ with $|S|=k$, we have $|V(G)|-|S|\leq\kappa(G\Box H)-1$, and hence $G[S]$ is connected. Therefore, we have $d_{G\Box H}(S)\leq k-1$, and hence $sdiam_{k}(G\Box H)\leq k-1$ by the arbitrariness of $S$. So, we have $sdiam_{k}(G\Box H)=k-1$. \qed

The following corollary is immediate from Theorem 2.7.

Corollary 2.8

Let $G,H$ be two connected graphs of order at least $3$. Then

[TABLE]

To show the sharpness of the above upper and lower bound, we consider the following example.

Example 2: $(1)$ For $k=3$, from Corollary 2.8, we have $sdiam_{k}(G\Box H)=sdiam_{k}(G)+sdiam_{k}(H)$, which implies that the upper and lower bounds in Theorem 2.7 are sharp.

$(2)$ Let $G=P_{n}$ and $H=P_{m}$ with $5\leq n\leq m$. Then $sdiam_{4}(G)=n-1$, $sdiam_{4}(H)=m-1$ and $sdiam_{4}(G\Box H)=2(n-1)+(m-1)$, which implies that all the upper bounds in Theorem 2.7 are sharp.

3 Results for lexicographic product

From the definition, the lexicographic product graph $G\circ H$ is the graph obtained by replacing each vertex of $G$ by a copy of $H$ and replacing each edge of $G$ by a complete bipartite graph $K_{m,m}$, where $m=|V(H)|$.

Lemma 3.1

Hammack et al. (2011)* Let $G$ and $H$ be two graphs, and let $(g,h)$ and $(g^{\prime},h^{\prime})$ be two vertices of $G\circ H$. Then*

[TABLE]

A weak homomorphism $\varphi:G\rightarrow H$ is a map $\varphi:V(G)\rightarrow V(H)$ for which $uv\in E(G)$ implies $\varphi(u)\varphi(v)\in E(H)$ or $\varphi(u)=\varphi(v)$. Observe that the projection $p:G\circ H\rightarrow G$ is a weak homomorphism. For more details, we refer to Hammack et al. (2011) (p.32,p.57).

Lemma 3.2

Hammack et al. (2011)* Let $G$ and $H$ be two graphs, and let $(g,h)$ and $(g^{\prime},h^{\prime})$ be two vertices of $G\circ H$. Then*

[TABLE]

3.1 Steiner distance of lexicographic product graphs

The following lemma is a generalization of Lemma 3.2, which is a natural lower bound of $d_{G\circ H}(S)$ for $S\subseteq V(G\circ H)$ and $|S|=k$.

Lemma 3.3

Let $k\geq 2$ be an integer, $G$ be a connected graph, and $H$ be a graph. Let $S=\{(g_{i_{1}},h_{j_{1}}),$ $(g_{i_{2}},h_{j_{2}}),\ldots,(g_{i_{k}},h_{j_{k}})\}$ be a set of distinct vertices of $G\circ H$. Let $S_{G}=\{g_{i_{1}},g_{i_{2}},\ldots,g_{i_{k}}\}$. Then

[TABLE]

Proof. We note that $g_{i_{1}},g_{i_{2}},\ldots,g_{i_{k}}$ are not necessarily distinct. Let $T$ be a minimum $S$-Steiner tree in $G\circ H$. So $T$ has $d_{G\circ H}(S)$ edges. Let $Z$ be the minor obtained from $G\circ H$ by contracting edge in $H(g)$ for every $g$ of $G$. (Equivalently, identifying all the vertices in $H(g)$ into a single vertex $g$ and delete multiple edges in the resulting graph.) Then $Z$ is isomorphic to $G$. Now $T$ becomes $Y$, a connected subgraph of $Z$ containing the vertices corresponding to $g_{i_{1}},g_{i_{2}},\ldots,g_{i_{k}}$ in $G$. Thus $E(Y)\geq d_{G}(S_{G})$. Since $E(T)\geq E(Y)$, the result follows.\qed

Anand et al. (2012) obtained the following formula.

Lemma 3.4

Anand et al. (2012)* Let $k\geq 2$. Let $G,H$ be two graphs such that $G$ is connected. Let $S=\{(g_{i_{1}},h_{j_{1}}),$ $(g_{i_{2}},h_{j_{2}}),\ldots,(g_{i_{k}},h_{j_{k}})\}$ be a set of distinct vertices of $G\circ H$ such that $g_{i_{p}}\neq g_{i_{q}}\ (1\leq p,q\leq k)$. Let $S_{G}=\{g_{i_{1}},g_{i_{2}},\ldots,g_{i_{k}}\}$. Then*

[TABLE]

For general case, we have the following formula for Steiner distance of lexicographic product graphs.

Theorem 3.5

Let $k,n,m$ be three integers with $2\leq k\leq mn$. Let $G$ be a connected graph of order $n$, and $H$ be a graph of order $m$. Let $S=\{(g_{i_{1}},h_{i_{1}}),(g_{i_{2}},h_{i_{2}}),\ldots,(g_{i_{k}},h_{i_{k}})\}$ be a set of distinct vertices of $G\circ H$. Let $S_{G}=\{g_{i_{1}},g_{i_{2}},\ldots,g_{i_{k}}\}$ and $S_{H}=\{h_{j_{1}},h_{j_{2}},\ldots,h_{j_{k}}\}$ (note that $S_{G},S_{H}$ are both multi-sets). Let $r$ be the number of distinct vertices in $S_{G}$, where $1\leq r\leq k$.

$(1)$* If $r=1$ and $H[S_{H}]$ is connected in $H$, then $d_{G\circ H}(S)=k-1$.*

$(2)$* If $r=1$ and $H[S_{H}]$ is not connected in $H$, then $d_{G\circ H}(S)=k$.*

$(3)$* If $r\geq 2$, then $d_{G\circ H}(S)=d_{G}(S_{G})+k-r$.*

Proof. $(1)$ Since $r=1$, it follows that $g_{i_{1}}=g_{i_{2}}=\ldots=g_{i_{k}}$, and hence $S=\{(g_{i_{1}},h_{i_{1}}),(g_{i_{2}},h_{i_{2}}),\ldots,$ $(g_{i_{k}},h_{i_{k}})\}=\{(g_{i_{1}},h_{i_{1}}),(g_{i_{1}},h_{i_{2}}),\ldots,(g_{i_{1}},h_{i_{k}})\}\subseteq V(H(g_{i_{1}}))$. Since $H[S_{H}]$ is connected in $H$, it follows that the subgraph induced by the vertices in $\{(g_{i_{1}},h_{i_{1}}),$ $(g_{i_{2}},h_{i_{2}}),\ldots,(g_{i_{k}},h_{i_{k}})\}$ is connected in $H(g_{i_{1}})$, and hence $d_{G\circ H}(S)=k-1$.

$(2)$ Since $H[S_{H}]$ is not connected in $H$, it follows that the subgraph induced by the vertices in $\{(g_{i_{1}},h_{i_{1}}),$ $(g_{i_{2}},h_{i_{2}}),\ldots,(g_{i_{k}},h_{i_{k}})\}$ is not connected in $H(g_{i_{1}})$, and hence $d_{G\circ H}(S)\geq k$. Since $G$ is a connected graph of order at least $2$, it follows that there exists a vertex $g^{*}\in V(G)$ such that $g_{i_{1}}g^{*}\in E(G)$. From the structure of $G\circ H$, the tree induced by the edges in $\{(g_{i_{p}},h_{i_{p}})(g^{*},h_{1})\,|\,1\leq p\leq k\}=\{(g_{i_{1}},h_{i_{p}})(g^{*},h_{1})\,|\,1\leq p\leq k\}$ is an $S$-Steiner tree in $G\circ H$, and hence $d_{G\circ H}(S)\leq k$. So, we have $d_{G\circ H}(S)=k$.

$(3)$ Since $r\geq 2$, it follows that the vertices in $S$ belong to at least two copies of $H$ in $G\circ H$. From the definition of $r$, we can assume that $H(g_{1}),H(g_{2}),\ldots,H(g_{r})$ satisfy $S\cap V(H(g_{i}))\neq\emptyset$ for each $g_{i}\ (1\leq i\leq r)$, and $S\cap V(H(g_{i}))=\emptyset$ for each $g_{i}\ (r+1\leq i\leq n)$. Let $S_{G}^{\prime}=\{g_{1},g_{2},\ldots,g_{r}\}$. Then $S_{G}^{\prime}=S_{G}$ when we regard $S_{G}$ as a normal set, not a multi-set. Clearly, $d_{G}(S_{G})=d_{G}(S_{G}^{\prime})$, and $S=\{(g_{i_{1}},h_{i_{1}}),(g_{i_{2}},h_{i_{2}}),\ldots,(g_{i_{k}},h_{i_{k}})\}\subseteq\bigcup_{i=1}^{r}V(H(g_{i}))$. Without loss of generality, we can assume $(g_{i_{a}},h_{i_{a}})\in V(H(g_{a}))$ for each $a\ (1\leq a\leq r)$. Then $(g_{i_{a}},h_{i_{a}})=(g_{a},h_{i_{a}})$ for each $a\ (1\leq a\leq r)$. Let $S^{\prime}=\{(g_{a},h_{i_{a}})\,|\,1\leq a\leq r\}$. Then $(g_{i_{r+1}},h_{i_{r+1}}),(g_{i_{r+2}},h_{i_{r+2}}),\ldots,(g_{i_{k}},h_{i_{k}})\in(\bigcup_{i=1}^{r}V(H(g_{i}))\setminus S^{\prime}$. Note that there exists an $S_{G}^{\prime}$-Steiner tree $T_{G}$ of size $d_{G}(S_{G}^{\prime})=d_{G}(S_{G})$ in $G$. Without loss of generality, let $V(T_{G})=\{g_{1},g_{2},\ldots,g_{t}\}$, where $r\leq t\leq n$. In order to select $d_{G}(S_{G}^{\prime})$ edges in $G\circ H$ to form an $S^{\prime}$-Steiner tree $T^{\prime}$ in $G\circ H$ isomorphic to $T_{G}$ in $G$ such that $V(T^{\prime})\subseteq\bigcup_{i=1}^{t}V(H(g_{i}))$, we define a function $f:E(T_{G})\longrightarrow E(T^{\prime})$:

[TABLE]

for each $g_{a}g_{b}\in E(T_{G})\ (1\leq a\neq b\leq t)$. Note that $T^{\prime}$ is an $S^{\prime}$-Steiner tree in $G\circ H$.

We now extend the tree $T^{\prime}$ to an $S$-Steiner tree $T$ by adding $|S|-|S^{\prime}|=k-r$ edges in $G\circ H$. For each vertex $(g_{i_{a}},h_{j_{a}})\in S\setminus S^{\prime}\ (r+1\leq a\leq k)$, since there exists a vertex $g_{i_{b}}\in V(T_{G})\ (1\leq b\neq a\leq t)$ in $G$ such that $g_{i_{a}}g_{i_{b}}\in E(T_{G})$, we select an edge

[TABLE]

in $G\circ H$, and then add it into $T^{\prime}$. Observe that the tree induced by the edges in $E(T^{\prime})\cup\{e_{a}\,|\,r+1\leq i\leq k\}$ is an $S$-Steiner tree $T$ in $G\circ H$. Since $|E(T)|=d_{G}(S_{G})+k-r$, it follows that $d_{G\circ H}(S)\leq d_{G}(S_{G})+k-r$.

It remains us to show that $d_{G\circ H}(S)\geq d_{G}(S_{G})+k-r$. Recall that $V(G)=\{g_{1},g_{2},\ldots,g_{n}\}$. Without loss of generality, we assume that $H(g_{1}),H(g_{2}),\ldots,H(g_{r})$ be the $H$ copies such that $V(H(g_{i}))\cap S\neq\emptyset$, $1\leq i\leq r$. Clearly, $S=\{(g_{i_{1}},h_{i_{1}}),(g_{i_{2}},h_{i_{2}}),\ldots,(g_{i_{k}},h_{i_{k}})\}\subseteq\bigcup_{i=1}^{r}V(H(g_{i}))$. Set $|S\cap V(H(g_{i}))|=x_{i}$. Then $\sum_{i=1}^{r}x_{i}=k$. Without loss of generality, let $S_{i}=S\cap V(H(g_{i}))=\{(g_{i},h_{j})\,|\,1\leq j\leq x_{i}\}$ for each $g_{i}\ (1\leq i\leq r)$. In order to find an $S$-Steiner tree $T$ in $G\circ H$, we need the edges between some $H(g_{i})$ and $H(g_{j})$, $1\leq i\neq j\leq r$. Note that $S_{i}\subseteq V(H(g_{i}))$ for each $i\ (1\leq i\leq r)$. Clearly, there exists a subtree $T^{\prime}$ connecting $S^{\prime}$ in $T$ such that $E(T^{\prime})\cap(\bigcup_{i=1}^{r}E(H(g_{i})))=\emptyset$, where $|S^{\prime}\cap S_{i}|=1\ (1\leq i\leq r)$. Since $|E(T^{\prime})|\geq d_{G}(S_{G})$ and $|S|-|S^{\prime}|=k-r$, it follows that $T$ is an $S$-Steiner tree of size $d_{G}(S_{G})+k-r$ in $G\circ H$, and hence $d_{G\circ H}(S)\geq d_{G}(S_{G})+k-r$.

From the above argument, we conclude that $d_{G\circ H}(S)=d_{G}(S_{G})+k-r$.\qed

In Theorem 3.5, we assume that $G$ is a connected graph. For $k=3$, we have the following by assuming that $G$ is not connected.

Proposition 3.6

Let $G$ and $H$ be two graphs such that $G$ is connected, and let $(g,h)$, $(g^{\prime},h^{\prime})$ and $(g^{\prime\prime},h^{\prime\prime})$ be three vertices of $G\circ H$. Let $S=\{(g,h),(g^{\prime},h^{\prime}),(g^{\prime\prime},h^{\prime\prime})\}$, $S_{G}=\{g,g^{\prime},g^{\prime\prime}\}$ and $S_{H}=\{h,h^{\prime},h^{\prime\prime}\}$. Then

[TABLE]

Proof. Suppose that $g=g^{\prime}=g^{\prime\prime}$ and $deg_{G}(g)=0$. Since $g$ is isolated, it follows that $H(g)$ is a component of $G\circ H$, and hence $d_{G\circ H}(S)=d_{H}(S_{H})$.

Suppose that $g=g^{\prime}=g^{\prime\prime}$ and $deg_{G}(g)\geq 1$. Since $deg_{G}(g)\geq 1$, there exists a vertex $g^{*}$ in $G$ such that $gg^{*}\in E(G)$, and hence the tree induced by the edges in

[TABLE]

is an $S$-Steiner tree. Therefore, $d_{G\circ H}(S)\leq 3$. On the other hand, from Observation 1.1, $d_{G\circ H}(S)\geq 2$. So $d_{G\circ H}(S)=2$ or $d_{G\circ H}(S)=3$. Since $d_{H}(S_{H})\geq 2$ by Observation 1.1, it follows that $d_{G\circ H}(S)=\min\{d_{H}(S_{H}),3\}$.

Suppose that $g\neq g^{\prime}$, $g^{\prime}=g^{\prime\prime}$ and $d_{G}(g,g^{\prime})=\infty$. Then there is no path connecting $g$ and $g^{\prime}$ in $G$. Note that $(g,h)\in V(H(g))$ and $(g^{\prime},h^{\prime}),(g^{\prime\prime},h^{\prime\prime})\in V(H(g^{\prime}))$. Clearly, there is no $S$-Steiner tree in $G\circ H$. Therefore, $d_{G\circ H}(S)=\infty$.

Suppose that $g\neq g^{\prime}$, $g^{\prime}=g^{\prime\prime}$ and $d_{G}(g,g^{\prime})\neq\infty$. Set $d_{G}(g,g^{\prime})=\ell$. Let $P=gg_{1}g_{2}\cdots g_{\ell-1}g^{\prime}$ be a path connecting $g$ and $g^{\prime}$ in $G$. Then the tree induced by the edges in

[TABLE]

is an $S$-Steiner tree. Therefore, $d_{G\circ H}(S)\leq\ell+1$. It suffices to show $d_{G\circ H}(S)\geq\ell+1$. From Observation 2.1, any minimal $S$-Steiner tree $T$ is a path or there exists a vertex $(g^{*},h^{*})\in V(G\circ H)\setminus S$ such that the tree $T$ consists of three paths connecting $(g^{*},h^{*})$ and $(g,h),(g^{\prime},h^{\prime}),(g^{\prime\prime},h^{\prime\prime})$, respectively. If $T$ is a path, then we can assume that $(g^{\prime},h^{\prime})$ be the internal vertex of the path $T$. Since $g^{\prime}=g^{\prime\prime}$, it follows that $(g^{\prime},h^{\prime}),(g^{\prime\prime},h^{\prime\prime})\in V(H(g^{\prime}))$. One can see that the length of the path from $(g^{\prime},h^{\prime})$ to $(g^{\prime\prime},h^{\prime\prime})$ is at least $1$. By Lemma 3.2, $d_{G\circ H}(S)=d_{G\circ H}((g,h)(g^{\prime},h^{\prime}))+1\geq d_{G}(g,g^{\prime})+1=\ell+1$, as desired. Suppose that $T$ is a tree and there exists a vertex $(g^{*},h^{*})\in V(G\circ H)\setminus S$ such that $T$ consists of three paths connecting $(g^{*},h^{*})$ and $(g,h),(g^{\prime},h^{\prime}),(g^{\prime\prime},h^{\prime\prime})$, respectively. Then

[TABLE]

Suppose that $g\neq g^{\prime}$, $g\neq g^{\prime\prime}$ and $g^{\prime}\neq g^{\prime\prime}$. From Lemma 3.4, we have $d_{G\circ H}(S)=d_{G}(S_{G})$, as desired. The proof is now complete. \qed

3.2 Steiner diameter of lexicographic product graphs

By Theorem 3.5, we can derive the following results for Steiner diameter of lexicographic product graphs.

Theorem 3.7

Let $k,n,m$ be three integers with $2\leq k\leq mn$. Let $G$ be a connected graph of order $n$, and $H$ be a graph of order $m$. Then

$(1)$**

[TABLE]

Furthermore, if $n\geq 3$, then $sdiam_{k}(G\circ H)\leq n+k-3$.

$(2)$**

[TABLE]

Moreover, if

[TABLE]

then

[TABLE]

Proof. $(1)$ From the definition of $sdiam_{k}(G\circ H)$, there exists a vertex subset $S\subseteq V(G\circ H)$ with $|S|=k$ such that $d_{G\circ H}(S)=sdiam_{k}(G\circ H)$. Let $S=\{(g_{i_{1}},h_{i_{1}}),(g_{i_{2}},h_{i_{2}}),\ldots,(g_{i_{k}},h_{i_{k}})\}$, and $S_{G}=\{g_{i_{1}},g_{i_{2}},\ldots,g_{i_{k}}\}$. Let $s$ be the number of distinct vertices in $S_{G}$. We apply Theorem 3.5. (Here $s$ plays the role of $r$ in Theorem 3.5.) If $s\geq 2$, then $sdiam_{k}(G\circ H)=d_{G\circ H}(S)=d_{G}(S_{G})+k-s\leq d_{G}(S_{G})+k-2$. Furthermore, if $k\leq n$, then $d_{G}(S_{G})\leq sdiam_{k}(G)$, and hence $sdiam_{k}(G\circ H)\leq d_{G}(S_{G})+k-2\leq sdiam_{k}(G)+k-2$. If $n<k\leq mn$, then $d_{G}(S_{G})\leq n-1$, and hence $sdiam_{k}(G\circ H)\leq d_{G}(S_{G})+k-2\leq(n-1)+k-2=n+k-3$. Note that if $s=1$, then $k\leq m$, and hence $sdiam_{k}(G\circ H)=d_{G\circ H}(S)\leq k$. From the above argument, we conclude that $sdiam_{k}(G\circ H)\leq sdiam_{k}(G)+k-2$ if $k\leq n$, and $sdiam_{k}(G\circ H)\leq\max\{n+k-3,k\}$ if $n<k\leq mn$, as desired.

$(2)$ If $2\leq k\leq n$, then we let $S=\{(g_{i_{1}},h_{i_{1}}),$ $(g_{i_{2}},h_{i_{2}}),\ldots,(g_{i_{k}},h_{i_{k}})\}$ be a set of distinct vertices of $G\circ H$ such that $d_{G}(S_{G})=sdiam_{k}(G)$, where $S_{G}=\{g_{i_{1}},g_{i_{2}},\ldots,g_{i_{k}}\}$. From Lemma 3.3, we have $sdiam_{k}(G)=d_{G}(S_{G})\leq d_{G\circ H}(S)\leq sdiam_{k}(G\circ H)$. If $n\leq k\leq nm$, then it follows from Observation 1.1 that $k-1\leq d_{G\circ H}(S)\leq sdiam_{k}(G\circ H)$ for any $S\subseteq V(G\circ H)$ and $|S|=k$.

Now for the “moreover” part of the result. Let $r=\min_{2\leq x\leq n}\{x\,|\,sdiam_{x}(G)=n-1\}$. Suppose $sdiam_{r}(G)=n-1\ (2\leq r\leq n)$. If $2\leq k\leq r$, then $2\leq k\leq r\leq n$, and hence $sdiam_{k}(G\circ H)\geq sdiam_{k}(G)$. Suppose $r<k\leq rm$. Since $sdiam_{r}(G)=n-1$, it follows that there exists a vertex set $S^{\prime}=\{g_{1},g_{2},\ldots,g_{r}\}\subseteq V(G)$ such that $d_{G}(S^{\prime})=n-1=sdiam_{k}(G)$. Let $S=S_{1}\cup S_{2}\subseteq\bigcup_{i=1}^{r}V(H(g_{i}))$ such that $S_{1}=\{(g_{i},h_{1})\,|\,1\leq i\leq r\}$ and $S_{2}\subseteq\bigcup_{i=1}^{r}V(H(g_{i}))-S_{1}$ and $|S_{2}|=k-r$. Since $r\geq 2$ and $sdiam_{r}(G)=n-1$, it follows that $sdiam_{k}(G\circ H)\geq d_{G\circ H}(S)=d_{G}(S_{G})+k-r=d_{G}(S^{\prime})+k-r=n-1+k-r$, as desired. Suppose $rm<k\leq nm$. Since $sdiam_{r}(G)=n-1$, it follows that there exists a vertex set $S^{\prime}=\{g_{1},g_{2},\ldots,g_{r}\}\subseteq V(G)$ such that $d_{G}(S^{\prime})=n-1=sdiam_{k}(G)$. Let $S=S_{1}\cup S_{2}\subseteq\bigcup_{i=1}^{r}V(H(g_{i}))$ such that $S_{1}=\bigcup_{i=1}^{r}V(H(g_{i}))$ and $S_{2}\subseteq\bigcup_{i=r+1}^{x}V(H(g_{i}))$ and $|S_{2}|=k-rm$, where $x=\lceil\frac{k-rm}{m}\rceil$. Then $sdiam_{k}(G\circ H)\geq d_{G\circ H}(S)=d_{G}(S^{\prime})+r(m-1)+\lfloor\frac{k-rm}{m}\rfloor(m-1)+\max\{k-(r+\lfloor\frac{k-rm}{m}\rfloor)m-1,0\}=n-1+r(m-1)+\lfloor\frac{k-rm}{m}\rfloor(m-1)+\max\{k-(r+\lfloor\frac{k-rm}{m}\rfloor)m-1,0\}$. \qed

To show the sharpness of the upper and lower bounds in Theorem 3.7, we consider the following example.

Example 3: Let $G=P_{n}$, and $H$ be a graph of order $m$. If $k\leq\min\{2m,n\}$, then $sdiam_{k}(G\circ H)=n+k-3=sdiam_{k}(G)+k-2$. If $\max\{n,m+1\}\leq k\leq 2m$, then $sdiam_{k}(G\circ H)=n+k-3=\max\{n+k-3,k\}$. These implies that the upper bounds in Theorem 3.7 are sharp.

Example 4: Let $G=K_{n}$ and $H=K_{m}$. Then $G\circ H$ is a complete graph of order $mn$. If $2\leq k\leq n$, then $sdiam_{k}(G)=k-1=sdiam_{k}(G\circ H)$. If $n\leq k\leq nm$, then $sdiam_{k}(G\circ H)=k-1$. These implies that the lower bounds in Theorem 3.7 are sharp.

Example 5: Let $G=P_{n}\ (n\geq 3)$, and $H$ be a graph of order $m$. From the definition of $r$, we have $r=2$. For $2\leq k\leq r$, we have $k=r=2$, and hence $sdiam_{2}(G)=n-1=sdiam_{2}(G\circ H)$. For $r<k\leq rm$, we have $n-1+k-2\leq sdiam_{k}(G\circ H)\leq n+k-3$, and hence $sdiam_{k}(G\circ H)=n+k-3$. Let $G^{\prime}=P_{n}\ (n\geq 3)$, and $H^{\prime}=P_{2}$. For $rm<k\leq nm$, we let $k=2t$. From Theorem 3.7, we have $sdiam_{k}(G\circ H)\geq n-1+t$. One can easily check that $sdiam_{k}(G\circ H)=n-1+t$. These implies that the lower bounds for parameter $r$ in Theorem 3.7 are sharp.

The following result is immediate from Proposition 3.6.

Proposition 3.8

Let $G,H$ be two connected graphs. Then

[TABLE]

4 Applications

In this section, we demonstrate the usefulness of the proposed constructions by applying them to some instances of Cartesian and lexicographical product networks.

The following results are immediate.

Proposition 4.1

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

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

$(2)$* For a path $P_{n}$, $sdiam_{k}(P_{n})=n-1$;*

$(3)$* For a cycle $C_{n}$, sdiam_{k}(C_{n})=\big{\lfloor}\frac{n(k-1)}{k}\big{\rfloor}.*

4.1 Two-dimensional grid graph

A two-dimensional grid graph $G_{n,m}$ is the Cartesian product graph $P_{n}\Box P_{m}$ of path graphs on $m$ and $n$ vertices. For more details on grid graph, we refer to Calkin and Wilf (1998); Itai and Rodeh (1988). The network $P_{n}\circ P_{m}$ is the lexicographical product of $P_{n}$ and $P_{m}$; see Mao (2016).

Proposition 4.2

Let $k,n,m$ be three integers with $3\leq k\leq mn$, $n\geq 3$, and $m\geq 3$.

$(1)$* For network $P_{n}\Box P_{m}$,*

[TABLE]

$(2)$* For network $P_{n}\circ P_{m}$,*

[TABLE]

Proof. $(1)$ From $(2)$ of Proposition 4.1, we have $sdiam_{k}(P_{n})=n-1$ and $sdiam_{k}(P_{m})=m-1$. By Theorem 2.7, $sdiam_{k}(P_{n}\Box P_{m})\geq sdiam_{k}(P_{n})+sdiam_{k}(P_{m})=m+n-2$ and $sdiam_{k}(P_{n}\Box P_{m})\leq m+n-2+(k-3)\min\{m-1,n-1\}$.

$(2)$ Set $G=P_{n}$ and $H=P_{m}$. From Theorem 3.7, the result holds.\qed

4.2 $r$-dimensional mesh

An $r$-dimensional mesh is the Cartesian product of $r$ paths. By this definition, two-dimensional grid graph is a $2$-dimensional mesh. An $r$-dimensional hypercube is a special case of an $r$-dimensional mesh, in which the $r$ linear arrays are all of size $2$; see Johnsson and Ho (1989).

Proposition 4.3

Let $k,m_{1},m_{2},\cdots,m_{r}$ be the integers with $m_{1}\geq m_{2}\geq\cdots\geq m_{r}$ and $3\leq k\leq\prod_{i=1}^{r}m_{i}$.

$(1)$* For an $r$-dimensional mesh $P_{m_{1}}\Box P_{m_{2}}\Box\cdots\Box P_{m_{r}}$,*

[TABLE]

$(2)$* For an $r$-dimensional network $P_{m_{1}}\circ P_{m_{2}}\circ\cdots\circ P_{m_{r}}$,*

[TABLE]

Proof. $(1)$ From $(2)$ of Proposition 4.1, $sdiam_{k}(P_{m_{i}})=m_{i}-1$ for each $i\ (1\leq i\leq r)$. From Theorem 2.7, we have $sdiam_{k}(P_{m_{1}}\Box P_{m_{2}}\Box\cdots\Box P_{m_{r}})\geq\sum_{i=1}^{r}sdiam_{k}(P_{m_{i}})=\sum_{i=1}^{r}m_{i}-r$, and $sdiam_{k}(G\Box H)\leq sdiam_{k}(G)+sdiam_{k}(H)+(k-3)\min\{sdiam_{k}(G),sdiam_{k}(H)\}$ for two connected graphs $G$ and $H$, and hence

[TABLE]

[TABLE]

$(2)$ From Theorem 2.7, the result holds.\qed

4.3 $r$-dimensional torus

An $r$-dimensional torus is the Cartesian product of $r$ cycles $C_{m_{1}},C_{m_{2}},\cdots,C_{m_{r}}$ of size at least three. The cycles $C_{m_{i}}$ are not necessary to have the same size. Ku et al. (2003) showed that there are $r$ edge-disjoint spanning trees in an $r$-dimensional torus. The network $C_{m_{1}}\circ C_{m_{2}}\circ\cdots\circ C_{m_{r}}$ is investigated in Mao (2016). Here, we consider the networks constructed by $C_{m_{1}}\Box C_{m_{2}}\Box\cdots\Box C_{m_{r}}$ and $C_{m_{1}}\circ C_{m_{2}}\circ\cdots\circ C_{m_{r}}$.

Proposition 4.4

Let $k,m_{1},m_{2},\cdots,m_{r}$ be the integers with $m_{1}\geq m_{2}\geq\cdots\geq m_{r}\geq 3$ and $3\leq k\leq\prod_{i=1}^{r}m_{i}$.

$(1)$* For network $C_{m_{1}}\Box C_{m_{2}}\Box\cdots\Box C_{m_{r}}$,*

[TABLE]

where $m_{i}$ is the order of $C_{m_{i}}$ and $1\leq i\leq n$.

$(2)$* For network $C_{m_{1}}\circ C_{m_{2}}\circ\cdots\circ C_{m_{r}}$,*

[TABLE]

and

[TABLE]

Proof. $(1)$ From $(3)$ of Proposition 4.1, $sdiam_{k}(C_{m_{i}})=\left\lfloor\frac{(k-1)m_{i}}{k}\right\rfloor$ for each $i\ (1\leq i\leq r)$. By Theorem 2.7, we have

[TABLE]

and

[TABLE]

$(2)$ The result follows from Theorem 3.7.\qed

4.4 $r$-dimensional generalized hypercube

Let $K_{m}$ be a clique of $m$ vertices, $m\geq 2$. An $r$-dimensional generalized hypercube or Hamming graph Day and Al-Ayyoub (1997); Fragopoulou et al. (1996) is the product of $r$ cliques. We have the following:

Proposition 4.5

Let $k,m_{1},m_{2},\cdots,m_{r}$ be the integers with $m_{1}\geq m_{2}\geq\cdots\geq m_{r}\geq k\geq 2$.

$(1)$* For network $K_{m_{1}}\Box K_{m_{2}}\Box\cdots\Box K_{m_{r}}$,*

[TABLE]

$(2)$* For network $K_{m_{1}}\circ K_{m_{2}}\circ\cdots\circ K_{m_{r}}$,*

[TABLE]

Proof. $(1)$ From $(1)$ of Proposition 4.1, $sdiam_{k}(K_{m_{i}})=k-1$ for each $i\ (1\leq i\leq r)$. From Theorem 2.7, we have

[TABLE]

and

[TABLE]

$(2)$ From the definition of lexicographical product, $K_{m_{1}}\circ K_{m_{2}}\circ\cdots\circ K_{m_{n}}$ is a complete graph, and hence $sdiam_{k}(K_{m_{1}}\circ K_{m_{2}}\circ\cdots\circ K_{m_{n}})=k-1$.\qed

4.5 $n$-dimensional hyper Petersen network

An $n$-dimensional hyper Petersen network $HP_{n}\ (n\geq 3)$ is defined as follows (see Das et al. (1995)).

• •

$HP_{3}$ is the Petersen graph (see Fig.4 $(a)$);

• •

$HP_{n}$ is the Cartesian product of the Petersen graph $PG$ and an $(n-3)$-dimensional hypercube $Q_{n-3}$, that is, $HP_{n}=PG\Box Q_{n-3}$, where $n\geq 4$.

The hyper Petersen network $HP_{4}$ are depicted in Fig.4 $(b)$.

The network $HL_{n}\ (n\geq 3)$ is defined as follows (see Mao (2016)).

• •

$HL_{3}$ is the Petersen graph;

• •

$HL_{n}$ is the lexicographic product of the Petersen graph $PG$ and an $(n-3)$-dimensional hypercube $Q_{n-3}$, that is, $HP_{n}=PG\circ Q_{n-3}$, where $n\geq 4$.

Note that $HL_{4}$ is a graph obtained from two copies of the Petersen graph by add one edge between one vertex in a copy of the Petersen graph and one vertex in another copy. See Figure 4 $(c)$ for an example (We only show the edges $v_{1}u_{i}\ (1\leq i\leq 10)$).

Similarly to the proof of $(4)$ of Theorem 2.7, we can get the following observation.

Observation 4.6

Let $G$ be a connected graph of order $n$. If $n-\kappa(G)+1\leq k\leq n$, then $sdiam_{k}(G)=k-1$.

Proposition 4.7

$(1)$* For network $HP_{3}$ and $HL_{3}$,*

[TABLE]

$(2)$* For network $HL_{4}$,*

[TABLE]

$(3)$* For network $HP_{4}$,*

[TABLE]

Proof. $(1)$ Observe that $HL_{3}$ is just the Petersen graph. Set $G=HL_{3}$. Choose $S=\{v_{1},v_{3},v_{9}\}$. One can see that any $S$-Steiner tree must use at least $4$ edges of $G$, and hence $sdiam_{3}(G)\geq d_{G}(S)\geq 4$. One can check that $d_{G}(S)\leq 4$ for any $S\subseteq V(G)$ and $|S|=3$. Therefore, $sdiam_{3}(G)\leq 4$, and hence $sdiam_{3}(G)=sdiam_{3}(HL_{3})=4$. Since $HL_{3}=HP_{3}$, we have $sdiam_{3}(HP_{3})=sdiam_{3}(HL_{3})=4$. Since $\kappa(G)=3$, it follows from Observation 4.6 that $sdiam_{k}(G)=k-1$ if $8\leq k\leq 10$. If $k=4$, then we choose $S=\{v_{1},v_{4},v_{7},v_{8}\}$. One can see that any $S$-Steiner tree must use at least $5$ edges of $G$, and hence $sdiam_{4}(G)\geq d_{G}(S)\geq 5$. One can check that $d_{G}(S)\leq 5$ for any $S\subseteq V(G)$ and $|S|=3$. So, we have $sdiam_{4}(G)=5$. Similarly, we can prove that $sdiam_{k}(G)=k$ if $5\leq k\leq 7$.

$(2)$ For network $HL_{4}$, there are two copies of Petersen graphs, say $HL_{3}$ and $HL^{\prime}_{3}$. Set $G=HL_{4}$, $V(HL_{3})=\{v_{i}\,|\,1\leq i\leq 10\}$ and $V(HL^{\prime}_{3})=\{u_{i}\,|\,1\leq i\leq 10\}$. Choose $S=\{v_{1},v_{2},v_{9}\}$. One can see that any $S$-Steiner tree must use at least $3$ edges of $G$, and hence $sdiam_{3}(G)\geq d_{G}(S)\geq 3$. It suffices to show that $d_{G}(S)\leq 3$ for any $S\subseteq V(G)$ and $|S|=3$. Suppose $S\subseteq V(HL_{3})$ or $S\subseteq V(HL^{\prime}_{3})$. Without loss of generality, let $S=\{v_{1},v_{2},v_{3}\}\subseteq V(HL_{3})$. If $d_{HL_{3}}(S)=4$, then the tree induced by the edges in $\{u_{1}v_{1},u_{1}v_{2},u_{1}v_{3}\}$ is an $S$-Steiner tree, and hence $d_{G}(S)\leq 3$. Otherwise, $d_{G}(S)\leq d_{HL_{3}}(S)\leq 3$, as desired. Suppose $|S\cap V(HL_{3})|=2$ or $|S\cap V(HL^{\prime}_{3})|=2$. Without loss of generality, let $|S\cap V(HL_{3})|=2$ and $S=\{v_{1},v_{2},u_{1}\}$. Then the tree induced by the edges in $\{u_{1}v_{1},u_{1}v_{2}\}$ is an $S$-Steiner tree, and hence $d_{G}(S)\leq 2$, as desired. So $sdiam_{3}(HL_{4})=3$. Since $\kappa(G)=13$, it follows from Observation 4.6 that $sdiam_{k}(G)=k-1$ if $8\leq k\leq 20$. One can also prove that $sdiam_{k}(G)=k$ if $3\leq k\leq 7$.

$(3)$ For network $HP_{4}$, there are two copies of Petersen graphs, say $HP_{3}$ and $HP^{\prime}_{3}$. Set $G=HP_{4}$, $V(HP_{3})=\{v_{i}\,|\,1\leq i\leq 10\}$ and $V(HP^{\prime}_{3})=\{u_{i}\,|\,1\leq i\leq 10\}$. Choose $S=\{u_{1},u_{3},v_{10}\}$. One can see that any $S$-Steiner tree must use at least $5$ edges of $G$, and hence $sdiam_{3}(G)\geq d_{G}(S)\geq 5$. One can check that $d_{G}(S)\leq 5$ for any $S\subseteq V(G)$ and $|S|=3$. Then $sdiam_{3}(HP_{4})\leq 5$, and hence $sdiam_{3}(HP_{4})=5$. Since $\kappa(G)=4$, it follows from Observation 4.6 that $sdiam_{k}(G)=k-1$ if $17\leq k\leq 20$. For $4\leq k\leq 16$, we have $sdiam_{k}(HP_{4})\geq k-1$, and for any $S\subseteq V(G)$ with $|S|=k$, we let $S\cap V(HP_{3})=S_{1}$ and $S\cap V(HP^{\prime}_{3})=S_{2}$. Without loss of generality, let $|S_{1}|\geq\lceil\frac{k}{2}\rceil$. Let $S_{2}=S\cap V(HP^{\prime}_{3})=\{u_{1},u_{2},\ldots,u_{x}\}$, where $x\leq\left\lfloor k/2\right\rfloor$. Since $HP_{3}$ is connected, it follows that it contains a spanning tree $T$ of size $9$. Then the tree induced by the edges in $E(T)\cup\{u_{i}v_{i}\,|\,1\leq i\leq x\}$ is an $S$-Steiner tree in $G$, and hence $d_{G}(S)\leq x+9\leq\left\lfloor k/2\right\rfloor+9$. \qed

Acknowledgements.

The authors are very grateful to the referees’ comments and suggestions, which helped to improve the presentation of the paper.