On 3-Restricted Edge Connectivity of Replacement Product Graphs

Abstract

A 3-restricted edge cut is an edge cut of a connected graph that separates this graph into components, each having an order of at least 3. The minimum size of all 3-restricted edge cuts of a graph is called its 3-restricted edge connectivity. This work determines the upper and lower bounds on the 3-restricted edge connectivity of replacement product graphs and presents sufficient conditions for replacement product graphs to be maximally 3-restricted edge connected. As a result, the 3-restricted edge connectivity of replacement product graphs of some special graphs are determined.

1. Introduction

When modeling the reliability of a telecommunication network, it is reasonable to assume that its nodes (vertices) never fail, but links (edges) fail independently with an equally small probability. Such network models are now known as Moore–Shannon ones [1,2,3]. Let M be a Moore–Shannon network model that has size $\epsilon$ (the number of links or edges) and edge failure probability p, where $0<p<1$. If $C_h$ denotes the number of edge cuts of size h, then the reliability of M, namely the probability it remains connected, can be expressed as

$$R(M,p)=1-\sum _{h=1}^{\epsilon }{C}_h{p}^h{(1-p)}^{\epsilon -h}.$$

The right hand side is called the reliability polynomial of network M. To determine the reliability $R(M,p)$, one needs only to calculate all the coefficients $C_h$’s. However, Provan proves in [4] that it is NP-hard to calculate all these coefficients. Bauer presents a general expression of those coefficients $C_h$’s with $h\le \lambda \left(M\right)$ in [5], where $\lambda \left(M\right)$ denotes the edge connectivity of M. In order to estimate the reliability more accurately, Esfahanian introduces the concepts of restricted edge cut and restricted edge connectivity in [6], which can be defined as follows.

Definition 1.

A restricted edge cut is an edge cut of a connected graph which disconnects this graph without isolated vertices. The size of a minimum restricted edge cut of graph G is called its restricted edge connectivity.

In this work, we denote by ${\lambda }_2\left(G\right)$, or simply ${\lambda }_2$, the restricted edge connectivity of graph G. With the properties of restricted edge cuts and restricted edge connectivity, Li determines in [7] the first ${\lambda }_2-1$ coefficients of the reliability polynomial of networks with topologies being circulant graphs. His results show that for Moore–Shannon models with the same number of nodes and links, those that have a larger restricted edge connectivity and fewer minimum restricted edge cuts are locally more reliable, where a network M is locally more reliable than another network N if there is an integer $p_0<1$ such that $R(M,p)<R(N,p)$ holds for any positive real number $p\le p_0$. For accurate estimation and comparison of the reliability, the concepts of m-restricted edge cut and m-restricted edge connectivity are introduced in [8,9] as follows.

Definition 2.

An m-restricted edge cut is an edge cut of a connected graph which disconnects this graph with each component having order at least m. The size of a minimum m-restricted edge cut of graph G is called its m-restricted edge connectivity.

We denote by ${\lambda }_m\left(G\right)$, or simply ${\lambda }_m$, the m-restricted edge connectivity of graph G. Let F be a subgraph of graph G or a subset of the vertex set $V\left(G\right)$ of G. Denote by $G\backslash F$ the graph obtained by removing all the vertices of F from G, that is, the subgraph of G induced by the vertex set $V\left(G\right)\backslash V\left(F\right)$. Let $I\left(F\right)$ be the set of edges with one end in F and the other in $G\backslash F$ (the coboundary of F). Write $\partial \left(F\right)=\left|I\right(F\left)\right|$ and ${\xi }_m\left(G\right)$ or simply ${\xi }_m=$ min $\{\partial (F):F$ is a connected vertex-induced subgraph of order m of graph $G\}$. It is proven in [6,8] that when $m\le 3$, ${\lambda }_m\left(G\right)\le {\xi }_m\left(G\right)$ holds with a few trivial exceptions. So, a graph G is called maximally m-restricted edge connected if the equality holds in the previous inequality.

Let M and N be two Moore–Shannon network models that have the same number of nodes and links, respectively. From the results obtained in [5,7,10], one can easily concludes that if M and N are both maximally m-restricted edge connected and have least equal minimum m-restricted edge cuts for $m=1,2$, then N is locally more reliable than M when ${\lambda }_3\left(M\right)<{\lambda }_3\left(N\right)$. So, maximizing 3-restricted edge connectivity of graphs plays an important role in the design of the locally most reliable networks. We prove in [11] that undirected binary Kautz networks are maximally 3-restricted edge connected. Some sufficient conditions for graphs to be maximally 3-restricted edge connected are obtained in [10,12,13] and elsewhere. Fruitful results on maximizing m-restricted edge connectivity are obtained; the readers are suggested to refer to [5,6,7,8,9,14,15,16,17] and their references.

Before proceeding, let us introduce some more symbols and terminologies. Graphs indicated in this work are all simple and connected. They are k-regular if every vertex has degree k. The length of a shortest cycle of a graph G that contains cycles, denoted by $g\left(G\right)$, is called its girth. The connectivity and edge connectivity of graph G are denoted by $\kappa \left(G\right)$ and $\lambda \left(G\right)$, respectively. The minimum degree of G is denoted by $\delta \left(G\right)$. Suppose $X\subset V\left(G\right)$, let $G\left[X\right]$ denote the subgraph of G induced by X.

Definition 3

([18]). Let $G_1$ be a $k_1$-regular graph with vertex set $V_1$ and edge-set $E_1$, $G_2$ be a $k_2$-regular graph with vertex set $V_2=\{1,2,\dots ,k_1\}$ and edge-set $E_2$. For every vertex $x\in G_1$, label the edges of $G_1$ incident with x with $e_x^1,e_x^2,\dots ,e_x^{k_1}$. The replacement product graph $G_1\circledR G_2$ has vertex set $V_1\times V_2$, two of its different vertices $(x,i)$ and $(y,j)$, where $x,y\in V_1$ and $i,j\in V_2$, are adjacent if and only if either $x=y$ and $ij\in E_2$, or $xy\in E_1$ and $e_x^i=xy=e_y^j$.

Note that the replacement product graph $G_1\circledR G_2$ replaces every vertex of $G_1$ by a copy of $G_2$ and $xy\in E\left(G_1\right)$ if and only if there is exactly one edge between $V\left(x{G}_2\right)$ and $V\left(y{G}_2\right)$ in $G_1\circledR G_2$. For an earlier version of the definition of $G_1\circledR G_2$ please refer to [19]. An illustration of the replacement product graph of $K_4$ and $C_3$ is provided in Figure 1.

For other symbols and terminologies not specified we follow those of [20].

2. Bounds on 3-Restricted Edge Connectivity

In what follows, we assume that $G_1$ is a $k_1$-regular connected graph on n vertices and $G_2$ is a $k_2$-regular connected graph with vertex set $\{1,2,\dots ,k_1\}$, where $k_1\ge 2$, $k_2\le k_1-1$ and $n\ge 3$. Under this assumption, $G_1\circledR G_2$ is $(k_2+1)$-regular and has $n{k}_1$ vertices. Moreover, the vertex set of $G_1\circledR G_2$ can be partitioned into $\{X_1,X_2,\dots ,X_n\}$ such that $G\left[X_i\right]\cong G_2$ for each $i\in I_n$, where $I_n=\{1,2,\dots ,n\}$. It is clear that $G_1\circledR G_2$ has 3-restricted edge cuts when $k_1\ge 3$. In this section, we investigate the bounds of the 3-restricted edge connectivity of these replacement product graphs.

Lemma 1

([21]). Let G be a connected k-regular graph of order at least $2m$ that contains an m-restricted edge cut. If $g\left(G\right)>m/2+1$, them ${\lambda }_m\left(G\right)\le {\xi }_m\left(G\right)$.

Note that if a connected regular graph contains 3-restricted edge cuts, then it has cycles, so the following lemma follows directly from Lemma 1.

Lemma 2.

Let G be a k-regular connected graph of order at least 6. If G contains a 3-restricted edge cut, then ${\lambda }_3\left(G\right)\le {\xi }_3\left(G\right)$.

Lemma 3

([18]). ${\lambda }_2(G\circledR K_n)=\lambda \left(G\right)$ for any n-regular connected graph G.

Lemma 4.

If $G_1$ is a 2-regular connected graph (a cycle), then ${\lambda }_3(G_1\circledR K_2)=\lambda \left(G_1\right)$.

Proof. 

Since $G_1$ is a cycle, by the definition of replacement product graph, it follows that $G_1\circledR K_2$ is also a cycle. Thus, the lemma follows. □

For $X,\overline{X}\subset V\left(G\right)$ and $X\cup \overline{X}=V\left(G\right)$, denote the set of edges of G with one end in X and the other end in $\overline{X}$ by $[X,\overline{X}]$.

Lemma 5.

If $G_i$ is a $k_i$-regular connected graph with $k_1\ge 3$, where $i=1,2$, then ${\lambda }_3(G_1\circledR G_2)\le \lambda \left(G_1\right)$.

Proof. 

Write $G=G_1\circledR G_2$. Let $X\subset V\left(G_1\right)$ such that $[X,\overline{X}]$ is a minimum edge-cut of $G_1$. Then both $G_1\left[X\right]$ and $G_1\left[\overline{X}\right]$ are connected. Therefore, $G\left[Y\right]$ and $G\left[\overline{Y}\right]$ are both connected, where $Y=\{(x,i):x\in X,i\in V\left(G_2\right)\}$, $\overline{Y}=\{(y,i):y\in \overline{X},i\in V\left(G_2\right)\}$. Since $\left|Y\right|\ge \left|V\right(G_2\left)\right|\ge 3$ and $|\overline{Y}|\ge |V\left(G_2\right)|\ge 3$, it follows that $[Y,\overline{Y}]$ is a 3-restricted edge cut of G. By the definition of replacement product graph, there is an edge $xy$ in $G_1$ if and only if there is exactly one edge between $V\left(x{G}_2\right)$ and $V\left(y{G}_2\right)$ in G, where $x{G}_2$ is the subgraph of G induced by the vertex set $\{(x,i):i\in V\left(G_2\right)\}$, and $y{G}_2$ is defined similarly. So, if $xy\in [X,\overline{X}]$, there exists $i,j\in V\left(G_2\right)$ such that $((x,i),(y,j))\in [Y,\overline{Y}]$. Hence, $|[Y,\overline{Y}]|=|[X,\overline{X}]|=\lambda \left(G_1\right)$ and

$${\lambda }_3\left(G\right)\le |[Y,\overline{Y}]|=|[X,\overline{X}]|=\lambda \left(G_1\right).$$

(1)

□

Lemma 6.

If $G_i$ is a $k_i$-regular connected graph with $k_1\ge 3$ and $g\left(G_2\right)=3$, $i=1,2$, then ${\lambda }_3(G_1\circledR G_2)\le min\{\lambda \left(G_1\right),3k_2-3\}$.

Proof. 

Write $G=G_1\circledR G_2$. Since $G_2$ is a connected subgraph of G with $g\left(G_2\right)=3$, it follows that $g\left(G\right)=3$. Recalling that G is $(k_2+1)$-regular, by Lemma 2 we have,

$${\lambda }_3\left(G\right)\le {\xi }_3\left(G\right)=3(k_2+1)-6=3k_2-3.$$

(2)

Together with Lemma 5, we have ${\lambda }_3\left(G\right)\le min\{\lambda \left(G_1\right),3k_2-3\}$. □

Lemma 7.

If graph $G_i$ is $k_i$-regular connected, $i=1,2$, and $k_1\ge 3$, $g\left(G_2\right)\ge 4$, then ${\lambda }_3(G_1\circledR G_2)\le min\{\lambda \left(G_1\right),3k_2-1\}$.

Proof. 

Write $G=G_1\circledR G_2$. By Lemma 5, we have ${\lambda }_3\left(G\right)\le \lambda \left(G_1\right)$. From the construction properties of replacement product $G_1\circledR G_2$, we deduce that for every vertex $(x,i)\in V\left(G\right)$, exactly one edge incident with it is not in $x{G}_2$. So, if a cycle C of G is not contained in some subgraph $y{G}_2$ of G with $y\in V\left(G_1\right)$ then C has a length of at least six. Since $G_2$ has a girth of at least four, it follows that G has a girth of at least four. Hence, G contains 3-restricted edge cuts. By Lemma 2, we have ${\lambda }_3\left(G\right)\le {\xi }_3\left(G\right)=3(k_2+1)-4=3k_2-1$. Thus, the lemma follows. □

Corollary 1.

Let $G_1$ be a $k_1$-regular connected graph. If $k_1\ge 2$, then ${\lambda }_3(G_1\circledR K_{k_1})=\lambda \left(G_1\right)$.

Proof. 

If $k_1=2$, by Lemma 4 the result follows. If $k_1\ge 3$, by Lemma 3, ${\lambda }_2(G_1\circledR K_{k_1})=\lambda \left(G_1\right)$. Since $\lambda \left(G_1\right)\le k_1$ and $3k_2-3=3k_1-6\ge k_1$. By Lemma 6, ${\lambda }_3(G_1\circledR K_{k_1})\le \lambda \left(G_1\right)$. Therefore,

$$\lambda \left(G_1\right)={\lambda }_2(G_1\circledR K_{k_1})\le {\lambda }_3(G_1\circledR K_{k_1})\le \lambda \left(G_1\right).$$

The corollary follows. □

Theorem 1.

Let $G_i$ be connected $k_i$-regular graphs, $i=1,2$. If $3\le k_1\le 5$ and $g\left(G_2\right)=3$, then

$${\lambda }_3(G_1\circledR G_2)=\lambda \left(G_1\right).$$

Proof. 

If $k_1=3$, since $G_2$ is regular it follows that $G_2$ is the complete graph $K_3$. If $k_1=4$ then $G_2=K_4$ since $g\left(G_2\right)=3$. Similarly, if $k_1=5$ then we can deduce that $G_2=K_5$. Hence, by Corollary 1 we have ${\lambda }_3(G_1\circledR G_2)=\lambda \left(G_1\right)$. □

Theorem 2.

Let $G_i$ be connected $k_i$-regular graphs, $i=1,2$. If $4\le k_1\le 5$ and $g\left(G_2\right)\ge 4$, then

$$min\{\lambda \left(G_1\right),\kappa \left(G_1\right)+1\}\le {\lambda }_3(G_1\circledR G_2)\le \lambda \left(G_1\right).$$

So, if $\kappa \left(G_1\right)\ge \lambda \left(G_1\right)-1$ then ${\lambda }_3(G_1\circledR G_2)=\lambda \left(G_1\right)$.

Proof. 

The second part of the theorem follows directly from the first one, and the second inequality of the first statement follows directly from Lemma 7. So, it suffices to prove $min\{\lambda \left(G_1\right),\kappa \left(G_1\right)+1\}\le {\lambda }_3(G_1\circledR G_2)$.

If $k_1=4$, since $G_2$ is a regular connected graph on $k_1$ vertices with girth $g\left(G_2\right)\ge 4$, then $G_2$ is the 4-cycle $C_4$ and $k_2=2$. Similarly, if $k_1=5$ then $G_2$ is the 5-cycle $C_5$. In any case, $G_2$ is a cycle. Write $G=G_1\circledR G_2$. Let $F=[X,\overline{X}]$ be a minimum 3-restricted edge cut (simply, a ${\lambda }_3$-cut) of G, where $\overline{X}$ and X are the vertex sets of two connected vertex-induced subgraphs of G with at least three vertices each and $|\overline{X}|\ge |X|$. Let $V\left(G_1\right)=\{x_1,x_2,\dots ,x_n\}$, and $X_i$ be the vertex set of $x_i{G}_2$. Then, $\{X_1,X_2,\dots ,X_n\}$ be a partition of $V\left(G\right)$ with $G\left[X_i\right]$ being isomorphic to $G_2$.

If every $G\left[X_i\right]$ is connected in $G-F$ of all $i\in I_n$, then either $X_i\subset X$ or $X_i\subset \overline{X}$ for any $i\in I_n$. Note that $xy\in E\left(G_1\right)$ if and only if G has exactly one edge between $V\left(x{G}_2\right)$ and $V\left(y{G}_2\right)$. Since F is a minimal 3-restricted edge cut of G, each edge in F connects a vertex of some $X_i$ to a vertex of some $X_j$ with $i\ne j$. Set $H=\left\{x_i{x}_j\right|x_i{x}_j\in E\left(G_1\right)$ and there is an edge in F between $X_i$ and $X_j\}$, then H is an edge cut of $G_1$, which disconnects vertex sets $\left\{x_i\right|x_i\in V\left(G_1\right),X_i\subset X\}$ and $\left\{x_j\right|x_j\in V\left(G_1\right),X_j\subset \overline{X}\}$ and $\left|H\right|=\left|F\right|$. It follows that ${\lambda }_3(G_1\circledR G_2)=\left|F\right|=\left|H\right|\ge \lambda \left(G_1\right)$. In what follows we continue to consider the case when at least one $G\left[X_i\right]$ is disconnected in $G-F$.

Firstly, we consider the case when there exists exactly one integer $i\in I_n$ such that $G\left[X_i\right]$ is disconnected in $G-F$.

Case 1. $X\subset X_i$. In this case, $\overline{X}=(V\left(G\right)\backslash X_i)\cup (X_i\backslash X)$. Thus,

$$\begin{array}{ccc}\hfill \left|F\right|& =& \left|\right[X,\overline{X}\left]\right|\hfill \\ & =& |[X,X_i\backslash X]|+|[X,V\left(G\right)\backslash X_i]|\hfill \\ & =& \left|\right[X,X_i\backslash X\left]\right|+\left|X\right|.\hfill \end{array}$$

The last equality holds since every vertex of $X_i$ is incident with exactly one edge not in $x_i{G}_1$. Since F is a 3-restricted edge cut, it follows that $3\le \left|X\right|\le |G_2|-1=k_1-1$. Noting that $[X,X_i\backslash X]$ is an edge-cut of $G\left[X_i\right]$, we have $\left|F\right|\ge \lambda \left(G\left[X_i\right]\right)+\left|X\right|\ge \lambda \left(G_2\right)+3=5\ge min\{\lambda \left(G_1\right),\kappa \left(G_1\right)+1\}$.

Case 2. $X\not\subset X_i$. Since $|\overline{X}|\ge |X|$, it follows that $\overline{X}\not\subset X_i$. Thus, $X_i\cap X\ne \varnothing$, $X_i\cap \overline{X}\ne \varnothing$ and there are at least two sets of $X_j$ and $X_k$ other than $X_i$ such that $X_j\subset X$ and $X_k\subset \overline{X}$. Since $\kappa (G_1-x_i)\ge \kappa \left(G_1\right)-1$, there exists at least $\kappa \left(G_1\right)-1$ internally vertex-disjoint paths between any two vertices u and v, where $u,v\in V(G_1-x_i)$. By the definition of G, there exists at least $\kappa \left(G_1\right)-1$ internally vertex-disjoint paths between $X_j$ and $X_k$ in $G-X_i$. Each of these paths has at least one edge of F. Hence

$$\left|F\right|\ge \kappa \left(G_1\right)-1+\left|[X_i\cap X,X_i\cap \overline{X}]\right|\ge \kappa \left(G_1\right)-1+\lambda \left(G_2\right)=\kappa \left(G_1\right)-1+2=\kappa \left(G_1\right)+1.$$

Therefore, $\left|F\right|\ge min\{\lambda \left(G_1\right),\kappa \left(G_1\right)+1\}$ in this case.

Secondly, we consider the case when there exists exactly two integers $i,j\in I_n$ such that $G\left[X_i\right]$ and $G\left[X_j\right]$ are disconnected in $G-F$.

Case 3.$X\subset \left(X_i\cup X_j\right)$ or $\overline{X}\subset \left(X_i\cup X_j\right)$, say $X\subset \left(X_i\cup X_j\right)$. In this case, $\overline{X}=(V\left(G\right)\backslash (X_i\cup X_j))\cup (X_j\backslash (X_j\cap X))\cup (X_i\backslash (X_i\cap X))$ and

$$\left|F\right|=|[X,\overline{X}]|=|[X_i\cap X,X_i\backslash (X_i\cap X)]|+|[X_j\cap X,X_j\backslash (X_j\cap X)]|+|[X,V\left(G\right)\backslash (X_i\cup X_j)]|.$$

Without loss of generality, we assume $|X_i\cap X|\le |X_j\cap X|$. If $|X_i\cap X|=1$, then $2\le |X_j\cap X|$ since otherwise X would be an isolated edge in G by the property that every vertex of $x_i{G}_1$ is incident with exactly one edge not contained in $x_i{G}_1$. Thus, $X_j$ contains at least one vertex which is incident with one edge between $x_j{G}_1$ and some $x_s{G}_1$ such that $s\ne i,j$. Therefore,

$$\left|F\right|\ge 2\lambda \left(G_2\right)+1=5\ge \lambda \left(G_1\right)\ge min\{\lambda \left(G_1\right),\kappa \left(G_1\right)+1\}.$$

If $2\le |X_i\cap X|$, we also have $2\le |X_j\cap X|$. Thus, the the above formula is also true.

Case 4. $X\not\subset \left(X_i\cup X_j\right)$ and $\overline{X}\not\subset \left(X_i\cup X_j\right)$. In this case, there are at least two sets of $X_k$ and $X_l$ other than $X_i$ and $X_j$ such that $X_k\subset X$ and $X_l\subset \overline{X}$. Since $\kappa (G_1-x_i-x_j)\ge \kappa \left(G_1\right)-2$, $G_1-\{x_i,x_j\}$ has at least $\kappa \left(G_1\right)-2$ internally vertex-disjoint paths between vertices $x_k$ and $x_l$. These paths corresponds to the same number of internally vertex-disjoint paths between $X_k$ and $X_l$ in $G-X_i-X_j$, each of which contains at least one edge of F. Therefore,

$$\begin{array}{ccc}\hfill \left|F\right|& =& \left|\right[X,\overline{X}\left]\right|\hfill \\ & \ge & \kappa \left(G_1\right)-2+\left|[(X_i\cup X_j)\cap X,(X_i\cup X_j)\cap \overline{X}]\right|\hfill \\ & \ge & \kappa \left(G_1\right)-2+2\lambda \left(G_2\right)\hfill \\ & =& \kappa \left(G_1\right)-2+4\hfill \\ & =& \kappa \left(G_1\right)+2>min\{\lambda \left(G_1\right),\kappa \left(G_1\right)+1\}.\hfill \end{array}$$

Finally, we consider the case when there exists at least three integers $i,j,k\in I_n$ such that $G\left[X_i\right],G\left[X_j\right]$ and $G\left[X_k\right]$ are disconnected in $G-F$. In this case, we have

$$\left|F\right|\ge \lambda \left(G\left[X_i\right]\right)+\lambda \left(G\left[X_j\right]\right)+\lambda \left(G\left[X_k\right]\right)=3\lambda \left(G_2\right)=6.$$

Thus, the theorem follows. □

Lemma 8.

Let $G_i$ be connected $k_i$-regular graphs, $i=1,2$. If $k_1\ge 6$, then

$$\begin{array}{ccc}\hfill {\lambda }_3(G_1\circledR G_2)& \ge & min\{\lambda \left(G_1\right),{\lambda }_3\left(G_2\right)+3,k_1+\lambda \left(G_2\right)-2,\hfill \\ & & \kappa \left(G_1\right)+\lambda \left(G_2\right)-1,\lambda \left(G_2\right)+{\lambda }_2\left(G_2\right)+1,3\lambda \left(G_2\right)\}\hfill \end{array}$$

Proof. 

Write $G=G_1\circledR G_2$. Let $F=[X,\overline{X}]$ be a ${\lambda }_3$-cut of G, where $X,\overline{X}\subset V\left(G\right)$ with $|\overline{X}|\ge |X|\ge 3$. Let $\{X_1,X_2,\cdots ,X_n\}$ be the partition of $V\left(G\right)$ defined in the proof of Lemma 2. If $G\left[X_i\right]$ is connected in $G-F$ for any $i\in I_n$, then either $X_i\subset X$ or $X_i\subset \overline{X}$. Thus, $[Y,\overline{Y}]$ is an edge-cut of $G_1$, where $Y=\{x_i\in V\left(G_1\right):x_i{G}_1\subset X\}$. From the construction properties of replacement product we deduce that $\left|F\right|=|[Y,\overline{Y}]|\ge \lambda \left(G_1\right)$. So, the lemma follows in this case. In what follows we assume that there is some $i\in I_n$ such that $G\left[X_i\right]$ is disconnected in $G-F$.

Firstly, similarly to the proof of Theorem 2, we consider the case when there exists exactly one integer $i\in I_n$ such that $G\left[X_i\right]$ is disconnected in $G-F$.

Case 1. $X\subset X_i$. In this case, we have $\overline{X}=(V\left(G\right)\backslash X_i)\cup (X_i\backslash X)$. Since every vertex of X is incident with exactly one edge not contained in $x_i{G}_2$, it follows that

$$\begin{array}{ccc}\hfill \left|F\right|& =& \left|\right[X,\overline{X}\left]\right|\hfill \\ & =& |[X,V\left(G\right)\backslash X_i]|+|[X,X_i\backslash X]|\hfill \\ & =& \left|X\right|+\left|\right[X,X_i\backslash X\left]\right|.\hfill \end{array}$$

Note that $\left|X\right|\ge 3$ and $\left|X\right|\le |G_2|-1=k_1-1$ in this case. If $\left|X\right|=k_1-1$, then $|[X,X_i\backslash X]|=k_2$. Therefore,

$$\left|F\right|=(k_1-1)+k_2\ge \kappa \left(G_1\right)-1+\lambda \left(G_2\right).$$

If $\left|X\right|=k_1-2$, since F is a minimum 3-restricted edge cut it follows that $[X,X_i\backslash X]$ is a restricted edge cut of $G\left[X_i\right]\cong G_2$. Therefore,

$$\left|F\right|\ge k_1-2+{\lambda }_2\left(G_2\right).$$

If $3\le \left|X\right|\le k_1-3$ and $G[X_i\backslash X]$ is connected, then $[X,X_i\backslash X]$ is a 3-restricted edge cut of $G\left[X_i\right]\cong G_2$. Hence,

$$\left|F\right|\ge 3+{\lambda }_3\left(G_2\right).$$

If $3\le \left|X\right|\le k_1-3$ and $G[X_i\backslash X]$ has exactly two connected components, then $[X,X_i\backslash X]$ contains a restricted edge cut and an edge cut of $G\left[X_i\right]\cong G_2$. So,

$$\left|F\right|\ge 3+{\lambda }_2\left(G_2\right)+\lambda \left(G_2\right).$$

If $3\le \left|X\right|\le k_1-3$ and $G[X_i\backslash X]$ has at least three connected components, then $[X,X_i\backslash X]$ contains at least three edge cuts of $G\left[X_i\right]\cong G_2$. Thus,

$$\left|F\right|\ge 3\lambda \left(G_2\right)+3\ge 3\lambda \left(G_2\right).$$

Therefore, the lemma follows in this case.

Case 2. $X\not\subset X_i$. Since $|\overline{X}|\ge |X|$, it follows that $\overline{X}\not\subset X_i$. So, $X_i\cap X\ne \varnothing$, $X_i\cap \overline{X}\ne \varnothing$ and there are at least two sets of $X_j$ and $X_k$ other than $X_i$ such that $X_j\subset X$ and $X_k\subset \overline{X}$. Since $\kappa (G_1-x_i)\ge \kappa \left(G_1\right)-1$, there exists at least $\kappa \left(G_1\right)-1$ internally vertex-disjoint paths between any two vertices $x_k$ and $x_l$. Correspondingly, G has at least $\kappa \left(G_1\right)-1$ internally vertex-disjoint paths between $X_j$ and $X_k$ in $G-X_i$, and each of these paths contains at least one edge of F. Hence,

$$\left|F\right|=(\kappa \left(G_1\right)-1)+\left|[X_i\cap X,X_i\cap \overline{X}]\right|\ge \kappa \left(G_1\right)-1+\lambda \left(G_2\right).$$

Therefore, the lemma is also true in this case.

Secondly, we consider the case when there are exactly two integers $i,j\in I_n$ such that $G\left[X_i\right]$ and $G\left[X_j\right]$ are disconnected in $G-F$.

Case 3. $X\subset \left(X_i\cup X_j\right)$ or $\overline{X}\subset \left(X_i\cup X_j\right)$, say $X\subset \left(X_i\cup X_j\right)$. In this case, $\overline{X}=(V\left(G\right)\backslash (X_i\cup X_j))\cup (X_j\backslash (X_j\cap X))\cup (X_i\backslash (X_i\cap X))$. For simplicity, we assume without loss of generality that $|X_i\cap X|\le |X_j\cap X|$. Then,

$$\begin{array}{ccc}\hfill \left|F\right|& =& \left|\right[X,\overline{X}\left]\right|\hfill \\ & =& |[X_i\cap X,X_i\backslash (X_i\cap X)]|+|[X_j\cap X,X_j\backslash (X_j\cap X)]|+\hfill \\ & & \left|\right[X,V\left(G\right)\backslash (X_i\cup X_j)\left]\right|.\hfill \end{array}$$

When $|X_i\cap X|=1$, as shown in the proof of Theorem 2 (case 3), we have $|X_j\cap X|\ge 2$. If $|X_j\cap X|\le k_1-2$, then $[X_i\cap X,X_i\backslash (X_i\cap X)]$ is an edge-cut of $G\left[X_i\right]$, and $[X_j\cap X,X_j\backslash (X_j\cap X)]$ is a restricted edge-cut of $G\left[X_j\right]$. Noticing $\left|\right[X,V\left(G\right)\backslash (X_i\cup X_j)\left]\right|\ge 1$, we conclude that

$$\left|F\right|\ge \lambda \left(G_2\right)+{\lambda }_2\left(G_2\right)+1.$$

If $|X_i\cap X|=1$ and $|X_j\cap X|=k_1-1$, noticing that the vertices of $X\cap X_j$ are incident with $k_1-2+k_2$ edges of F and the vertices of $X\cap X_i$ are incident with $k_2$ edges of F, thus, we deduce that $\left|F\right|=2k_2+(k_1-2)>k_1+\lambda \left(G_2\right)-2$.

When $2\le |X_i\cap X|\le k_1-2$, $[X_i\cap X,X_i\backslash (X_i\cap X)]$ is an restricted edge cut of $G\left[X_i\right]$, and $[X_j\cap X,X_j\backslash (X_j\cap X)]$ is a ${\lambda }_2$-cut of $G\left[X_j\right]$. Note that $X_i$ is incident with at least one edge of F which is not contained in $G\left[X_i\right]$, and so does $G\left[X_j\right]$. It follows that

$$\left|F\right|\ge \lambda \left(G_2\right)+{\lambda }_2\left(G_2\right)+2.$$

When $|X_i\cap X|=k_1-1$, by the assumption that $|X_j\cap X|\ge |X_i\cap X|$, we have $|X_j\cap X|=k_1-1$. So,

$$\left|F\right|\ge 2k_2+2(k_1-2)\ge \kappa \left(G_1\right)+\lambda \left(G_2\right)-1.$$

The inequality holds since $k_1\ge 6$. So, the lemma follows in this case.

Case 4. $X\not\subset \left(X_i\cup X_j\right)$ and $\overline{X}\not\subset \left(X_i\cup X_j\right)$. So, there are two sets of $X_k$ and $X_l$ other than $X_i$ and $X_j$ such that $X_k\subset X$ and $X_l\subset \overline{X}$. Since $\kappa (G_1-x_i-x_j)\ge \kappa \left(G_1\right)-2$, $G_1$ has at least $\kappa \left(G_1\right)-2$ internally vertex-disjoint paths between vertices $x_k$ and $x_l$. Correspondingly, G has at least $\kappa \left(G_1\right)-2$ internally vertex-disjoint paths between $X_k$ and $X_l$ in $G-X_i-X_j$, and each of these paths contains at least one edge of F. Hence,

$$\begin{array}{ccc}\hfill \left|F\right|& =& \left|\right[X,\overline{X}\left]\right|\hfill \\ & =& \kappa \left(G_1\right)-2+\left|[(X_i\cup X_j)\cap X,(X_i\cup X_j)\cap \overline{X}]\right|\hfill \\ & \ge & \kappa \left(G_1\right)-2+2\lambda \left(G_2\right)\hfill \\ & \ge & \kappa \left(G_1\right)-1+\lambda \left(G_2\right).\hfill \end{array}$$

Finally, we consider the case when there exist at least three integers $i,j,k\in I_n$ such that $G\left[X_i\right],G\left[X_j\right]$ and $G\left[X_k\right]$ are disconnected in $G-F$. In this case,

$$\left|F\right|\ge \lambda \left(G\left[X_i\right]\right)+\lambda \left(G\left[X_j\right]\right)+\lambda \left(G\left[X_k\right]\right)=3\lambda \left(G_2\right).$$

Therefore, the lemma follows in all cases. □

Theorem 3.

Let $G_i$ be $k_i$-regular connected graphs, $i=1,2$. If $k_1\ge 6$ and $g\left(G_2\right)=3$, then

$$\begin{array}{ccc}& & min\{\lambda \left(G_1\right),{\lambda }_3\left(G_2\right)+3,k_1+\lambda \left(G_2\right)-2,\hfill \\ & & \kappa \left(G_1\right)+\lambda \left(G_2\right)-1,\lambda \left(G_2\right)+{\lambda }_2\left(G_2\right)+1,3\lambda \left(G_2\right)\}\hfill \\ & & \le {\lambda }_3(G_1\circledR G_2)\le min\{\lambda \left(G_1\right),3k_2-3\}.\hfill \end{array}$$

Proof. 

The theorem follows directly from Lemmas 6 and 8. □

Theorem 4.

Let $G_i$ be $k_i$-regular connected graphs, $i=1,2$. If $k_1\ge 6$ and $g\left(G_2\right)\ge 4$, then

$$\begin{array}{ccc}& & min\{\lambda \left(G_1\right),{\lambda }_3\left(G_2\right)+3,k_1+\lambda \left(G_2\right)-2,\hfill \\ & & \kappa \left(G_1\right)+\lambda \left(G_2\right)-1,\lambda \left(G_2\right)+{\lambda }_2\left(G_2\right)+1,3\lambda \left(G_2\right)\}\hfill \\ & & \le {\lambda }_3(G_1\circledR G_2)\le min\{\lambda \left(G_1\right),3k_2-1\}.\hfill \end{array}$$

Proof. 

The theorem follows directly from Lemmas 7 and 8. □

Note that a connected regular graph $G_2$ with $g\left(G_2\right)\ge 4$ and $|V\left(G_2\right)|=k_1=3$ does not exist. The results of Theorems 1–4 can be summarized in

Table 1. A summary of the results of Theorems 1–4.

	$\mathit{g}\left({\mathit{G}}_2\right)=3$	$\mathit{g}\left({\mathit{G}}_2\right)\ge 4$
$k_1=3$	${\lambda }_3(G_1\circledR G_2)=\lambda \left(G_1\right)$	$--$
$4\le k_1\le 5$	${\lambda }_3(G_1\circledR G_2)=\lambda \left(G_1\right)$	$\begin{array}{c}min\{\lambda \left(G_1\right),\kappa \left(G_1\right)+1\}\le \\ {\lambda }_3(G_1\circledR G_2)\le \lambda \left(G_1\right)\end{array}$
$k_1\ge 6$	$\begin{array}{c}min\{\lambda \left(G_1\right),{\lambda }_3\left(G_2\right)+3,k_1+\lambda \left(G_2\right)-\\ 2,\kappa \left(G_1\right)+\lambda \left(G_2\right)-1,\lambda \left(G_2\right)+{\lambda }_2\left(G_2\right)+\\ 1,3\lambda \left(G_2\right)\}\le {\lambda }_3(G_1\circledR G_2)\le \\ min\{\lambda \left(G_1\right),3k_2-3\}\end{array}$	$\begin{array}{c}min\{\lambda \left(G_1\right),{\lambda }_3\left(G_2\right)+3,k_1+\lambda \left(G_2\right)-\\ 2,\kappa \left(G_1\right)+\lambda \left(G_2\right)-1,\lambda \left(G_2\right)+{\lambda }_2\left(G_2\right)+\\ 1,3\lambda \left(G_2\right)\}\le {\lambda }_3(G_1\circledR G_2)\le \\ min\{\lambda \left(G_1\right),3k_2-1\}\end{array}$

.

3. Optimization of 3-Restricted Edge Connectivity

This section will present some sufficient conditions for replacement product graphs to be maximally 3-restricted edge connected, namely, sufficient conditions for 3-restricted edge connectivity to arrive to its upper bound.

Theorem 5.

Let $G_i$ be $k_i$-regular connected graphs, $i=1,2$, with $k_1\ge 6$ and $k_2\ge 4$. If $g\left(G_2\right)=3$, $\kappa \left(G_1\right)\ge min\{\lambda \left(G_1\right)-\lambda \left(G_2\right)+1,{\lambda }_2\left(G_2\right)+2\}$, and $G_2$ is maximally 3-restricted edge connected, then

$${\lambda }_3(G_1\circledR G_2)=min\{\lambda \left(G_1\right),3k_2-3\}.$$

Proof. 

Since $G_2$ is maximally 3-restricted edge connected with $k_2\ge 4$ and $g\left(G_2\right)=3$, it follows that

$${\lambda }_3\left(G_2\right)={\xi }_3\left(G_2\right)=3k_2-6,{\lambda }_2\left(G_2\right)=2k_2-2,\lambda \left(G_2\right)=k_2.$$

Therefore,

$${\lambda }_3\left(G_2\right)+3=3k_2-3,$$

$$k_1+\lambda \left(G_2\right)-2>\lambda \left(G_1\right),$$

$$\lambda \left(G_2\right)+{\lambda }_2\left(G_2\right)+1=3k_2-1>3k_2-3,$$

$$3\lambda \left(G_2\right)=3k_2>3k_2-3.$$

Since $\kappa \left(G_1\right)\ge min\{\lambda \left(G_1\right)-\lambda \left(G_2\right)+1,{\lambda }_2\left(G_2\right)+2\}$, it follows that $\kappa \left(G_1\right)+\lambda \left(G_2\right)-1\ge \lambda \left(G_1\right)$ or $\kappa \left(G_1\right)+\lambda \left(G_2\right)-1\ge \lambda \left(G_2\right)+{\lambda }_2\left(G_2\right)+1$. Therefore, $min\{\lambda \left(G_1\right),{\lambda }_3\left(G_2\right)+3,k_1+\lambda \left(G_2\right)-2,\kappa \left(G_1\right)+\lambda \left(G_2\right)-1,\lambda \left(G_2\right)+{\lambda }_2\left(G_2\right)+1,3\lambda \left(G_2\right)\}=min\{\lambda \left(G_1\right),3k_2-3\}$. By Theorem 3, ${\lambda }_3(G_1\circledR G_2)=min\{\lambda \left(G_1\right),3k_2-3\}$, and the theorem follows. □

Corollary 2.

Let $G_i$ be $k_i$-regular connected graphs, $i=1,2$, with $k_1\ge 6$, $k_2\ge 4$ and $g\left(G_2\right)=3$. If $\kappa \left(G_1\right)=\lambda \left(G_1\right)$ and $G_2$ is maximally 3-restricted edge connected, then

$${\lambda }_3(G_1\circledR G_2)=min\{\lambda \left(G_1\right),3k_2-3\}.$$

Proof. 

This corollary follows directly from Theorem 5. □

Theorem 6.

Let $G_i$ be $k_i$-regular connected graphs, $i=1,2$, with $k_1\ge 6$. If $g\left(G_2\right)\ge 4$, $\kappa \left(G_1\right)\ge min\{\lambda \left(G_1\right)-\lambda \left(G_2\right)+1,{\lambda }_2\left(G_2\right)+2\}$ and $G_2$ is maximally 3-restricted edge connected, then

$${\lambda }_3(G_1\circledR G_2)=min\{\lambda \left(G_1\right),3k_2-1\}.$$

Proof. 

Since $G_2$ is maximally 3-restricted edge connected and $g\left(G_2\right)\ge 4$, it follows that

$${\lambda }_3\left(G_2\right)={\xi }_3\left(G_2\right)=3k_2-4,{\lambda }_2\left(G_2\right)=2k_2-2,\lambda \left(G_2\right)=k_2.$$

Therefore,

$${\lambda }_3\left(G_2\right)+3=3k_2-1,$$

$$k_1+\lambda \left(G_2\right)-2\ge \lambda \left(G_1\right),$$

$$\lambda \left(G_2\right)+{\lambda }_2\left(G_2\right)+1=3k_2-1,$$

$$3\lambda \left(G_2\right)=3k_2>3k_2-1.$$

Since $\kappa \left(G_1\right)\ge min\{\lambda \left(G_1\right)-\lambda \left(G_2\right)+1,{\lambda }_2\left(G_2\right)+2\}$, it follows that either $\kappa \left(G_1\right)+\lambda \left(G_2\right)-1>\lambda \left(G_1\right)$ or $\kappa \left(G_1\right)+\lambda \left(G_2\right)-1\ge \lambda \left(G_2\right)+{\lambda }_2\left(G_2\right)+1)$. So, $min\{\lambda \left(G_1\right),{\lambda }_3\left(G_2\right)+3,k_1+\lambda \left(G_2\right)-2,\kappa \left(G_1\right)+\lambda \left(G_2\right)-1,\lambda \left(G_2\right)+{\lambda }_2\left(G_2\right)+1,3\lambda \left(G_2\right)\}=min\{\lambda \left(G_1\right),3k_2-1\}$. By Theorem 4, we have ${\lambda }_3(G_1\circledR G_2)=min\{\lambda \left(G_1\right),3k_2-1\}$. Hence the theorem follows. □

The following corollary follows directly from Theorem 6.

Corollary 3.

Let $G_i$ be $k_i$-regular connected graphs, $i=1,2$, with $k_1\ge 6$ and $g\left(G_2\right)\ge 4$. If $\kappa \left(G_1\right)=\lambda \left(G_1\right)$ and $G_3$ is maximally 3-restricted edge connected, then

$${\lambda }_3(G_1\circledR G_2)=min\{\lambda \left(G_1\right),3k_2-1\}.$$

Corollary 4.

If G is an n-regular connected graph with $n\ge 3$ and $\kappa \left(G\right)\ge 4$, then ${\lambda }_3(G\circledR C_n)=min\{\lambda \left(G\right),5\}$.

Proof. 

If $n=3$, then the corollary follows from Corollary 1. If $4\le n\le 5$, by Theorem2, we have $min\{\lambda \left(G\right),\kappa \left(G\right)+1\}\le {\lambda }_3(G\circledR C_n)\le \lambda \left(G\right)$. Since $\kappa \left(G\right)\ge 4$ and $\lambda \left(G\right)\le n\le 5$, it follows that $min\left\{\lambda \right(G),\kappa (G)+1\}=\lambda \left(G\right)$. So, ${\lambda }_3(G\circledR C_n)=\lambda \left(G\right)$. If $n\ge 6$, then $C_n$ is maximally 3-restricted edge connected. Since $\kappa \left(G\right)\ge 4$ and $3k_2-1=5$, by Theorem 6, we have that ${\lambda }_3(G\circledR C_n)=min\{\lambda \left(G\right),5\}$. Hence, The corollary follows. □

Theorem 7.

Let $G_i$ be $k_i$-regular connected graphs, $i=1,2$, with $k_1\ge 6$, $k_2\ge 4$, $g\left(G_2\right)=3$ and $\kappa \left(G_1\right)\ge min\{\lambda \left(G_1\right)-\lambda \left(G_2\right)+1,{\lambda }_2\left(G_2\right)+2\}$. If $G_2$ is maximally 3-restricted edge connected, then $G_1\circledR G_2$ is maximally 3-restricted edge connected if and only if $\lambda \left(G_1\right)\ge 3k_2-3$.

Proof. 

Since $g\left(G_2\right)=3$, it follows that $G_1\circledR G_2$ also has a girth of three. So, ${\xi }_3(G_1\circledR G_2)=3k_2-3$. By Theorem 5, we have ${\lambda }_3\left(G\right)=min\{\lambda \left(G_1\right),3\delta \left(G_2\right)-3\}$. Therefore, $G_1\circledR G_2$ is maximally 3-restricted edge connected if and only if $\lambda \left(G_1\right)\ge 3k_2-3$. □

Theorem 8.

Let $G_i$ be $k_i$-regular connected graphs, $i=1,2$, with $k_1\ge 6$, $g\left(G_2\right)\ge 4$ and $\kappa \left(G_1\right)\ge min\{\lambda \left(G_1\right)-\lambda \left(G_2\right)+1,{\lambda }_2\left(G_2\right)+2\}$. If $G_2$ is maximally 3-restricted edge connected, then $G_1\circledR G_2$ is maximally 3-restricted edge connected if and only if $\lambda \left(G_1\right)\ge 3k_2-1$.

Proof. 

Since $g\left(G_2\right)\ge 4$, it follows that $G_1\circledR G_2$ also has a girth of at least four. Then ${\xi }_3(G_1\circledR G_2)=3k_2-1$. By Theorem 6, we have ${\lambda }_3\left(G\right)=min\{\lambda \left(G_1\right),3k_2-1\}$. Therefore, $G_1\circledR G_2$ is maximally 3-restricted edge connected if and only if $\lambda \left(G_1\right)\ge 3k_2-1$. □

4. Conclusions

Two small networks can generate an extensive network by the replacing product operation. In this research, the upper and lower bounds on 3-restricted edge connectivity of a replacement product graphs on two small networks $G_1$ and $G_2$ are determined by the connectivities and degrees of $G_1$ and $G_2$. Furthermore, some sufficient conditions depicted with $\kappa \left(G_1\right)$ and $G_2$ are characterized for ${\lambda }_3(G_1\circledR G_2)$ to achieve the upper bound and for $G_1\circledR G_2$ to be maximally 3-restricted edge connected. However, the method in this research seems to not be effective for the case of ${\lambda }_m(G_1\circledR G_2)$ ($m\ge 4$). This is because there would be more cases when considering a connected subgraph with at least four vertices in $G_1\circledR G_2$, and more parameters shall be used to characterize the bounds of ${\lambda }_m(G_1\circledR G_2)$ ($m\ge 4$). We will explore new ways to investigate the bounds for ${\lambda }_m(G_1\circledR G_2)$ ($m\ge 4$) in our future research. Additionally, we will try to improve the bounds of ${\lambda }_3(G_1\circledR G_2)$ in our future research.