Extremality and Sharp Bounds for the $k$-edge-connectivity of Graphs

TL;DR

This paper investigates the $k$-edge-connectivity of graphs, providing exact values, sharp bounds, and algorithms to understand its extremality and relationships with other connectivities.

Contribution

It introduces new sharp bounds and exact values for $k$-edge-connectivity, along with an efficient algorithm for upper bounds based on maximum degree.

Findings

Computed exact values and bounds for $ ext{lambda}_k(G)$

Established relationships between $ ext{lambda}_k(G)$ and other connectivities

Developed an $ ext{O}(n^2)$ algorithm for sharp upper bounds

Abstract

Boesch and Chen (SIAM J. Appl. Math., 1978) introduced the cut-version of the generalized edge-connectivity, named $k$-edge-connectivity. For any integer $k$ with $2\leq k\leq n$, the {\em $k$-edge-connectivity} of a graph $G$, denoted by $\lambda_k(G)$, is defined as the smallest number of edges whose removal from $G$ produces a graph with at least $k$ components. In this paper, we first compute some exact values and sharp bounds for $\lambda_k(G)$ in terms of $n$ and $k$. We then discuss the relationships between $\lambda_k(G)$ and other generalized connectivities. An algorithm in $\mathcal{O}(n^2)$ time will be provided such that we can get a sharp upper bound in terms of the maximum degree. Among our results, we also compute some exact values and sharp bounds for the function $f(n,k,t)$ which is defined as the minimum size of a connected graph $G$ with order $n$ and $\lambda_k(G)=t$.

Equations

$${\kappa }_k^{\prime }(G)=\min \{{\kappa }_G^{\prime }(S)\mid S\subseteq V(G),|S|=k\}.$$

$${\lambda }_k^{\prime }(G)=\min \{{\lambda }_G^{\prime }(S)\mid S\subseteq V(G),|S|=k\}.$$

$$k-1\le {\lambda }_k(G)\le \left(\genfrac{}{}{0ex}{}{n}{2}\right)-\left(\genfrac{}{}{0ex}{}{n-k+1}{2}\right).$$

$$(k-1){\lambda }_k^{\prime }(G)\le {\lambda }_k(G)\le \Delta (G){\kappa }_{k-1}(G).$$

$${\lambda }_k(G)\ge (k-1){\kappa }_k^{\prime }(G).$$

$${\lambda }_k(G)\ge {\kappa }_k(L(G)).$$

$${\lambda }_k(G)\le (\Delta (G)-1)(k-1)+1.$$

$${\lambda }_k(G)\le (r-1)(k-1)+1.$$

$$n-k+t\le f(n,k,t)\le \left(\genfrac{}{}{0ex}{}{n-k+1}{2}\right)+t.$$

$$f(n,k,t)=\left\{\begin{array}[]{ll}n-k+t,&if~{}t\in\{k-1,k\}\\ {{n-k+1}\choose 2}+t,&if~{}t\in\{{n\choose 2}-{{n-k+1}\choose 2}-1,{n\choose 2}-{{n-k+1}\choose 2}\}.\end{array}\right.$$

Full text

In this paper, we first compute some exact values and sharp bounds for $\lambda_{k}(G)$ in terms of $n$ and $k$. We then discuss the relationships between $\lambda_{k}(G)$ and other generalized connectivities. An algorithm in $\mathcal{O}(n^{2})$ time will be provided such that we can get a sharp upper bound in terms of the maximum degree. Among our results, we also compute some exact values and sharp bounds for the function $f(n,k,t)$ which is defined as the minimum size of a connected graph $G$ with order $n$ and $\lambda_{k}(G)=t$.

Key Words: $k$-edge-connectivity, $k$-connectivity, generalized connectivity, $S$-Steiner tree, $k$-way cut.

1 Introduction

We refer to [3] for graph theoretical notation and terminology not described here. For a graph $G$, let $V(G)$, $E(G)$ be the set of vertices, the set of edges of $G$, respectively. For $X\subseteq V(G)$, we denote by $G-X$ the subgraph obtained by deleting from $G$ the vertices of $X$ together with the edges incident with them. For $Y\subseteq E(G)$, we denote by $G-Y$ the subgraph obtained by deleting from $G$ the edges of $Y$. For a set $S$, we use $|S|$ to denote its size. We use $P_{n}$, $C_{m}$ and $K_{\ell}$ to denote a path of order $n$, a cycle of order $m$ and a complete graph of order $\ell$, respectively.

Connectivity is one of the most basic concepts in graph theory, both in combinatorial sense and in algorithmic sense, see[3, 7, 11, 28]. The edge-connectivity of $G$, written by $\lambda(G)$, is the minimum size of an edge set $Y\subseteq E(G)$ such that $G-Y$ is disconnected. This definition is called the cut-version definition of the edge-connectivity. A well-known theorem of Menger provides an equivalent definition, which can be called the path-version definition of the edge-connectivity. For any two distinct vertices $x$ and $y$ in $G$, the local edge-connectivity $\lambda_{G}(x,y)$ is the maximum number of edge-disjoint paths connecting $x$ and $y$. Then $\lambda(G)=\min\{\lambda_{G}(x,y)\mid x,y\in V(G),x\neq y\}$ is defined to be the edge-connectivity of $G$. Similarly, there are cut-version and path-version definitions for the connectivity of graphs.

In [2], Boesch and Chen generalized the cut-version definition of the edge-connectivity, which has many applications in practice. However, they did not give a specific name for such a generalized edge-connectivity. Here we will use the name “$k$-edge-connectivity” from [23]. For any integer $k$ with $2\leq k\leq n$, the $k$-edge-connectivity of a graph $G$, denoted by $\lambda_{k}(G)$, is defined as the smallest number of edges whose removal from $G$ produces a graph with at least $k$ components. By definition, we clearly have $\lambda_{2}(G)=\lambda(G)$. Boesch and Chen [2] gave some properties of $\lambda_{k}(G)$ and obtained some bounds for $\lambda_{k}(G)$ in terms the minimum degree and the degree-sequence of $G$. They also studied some special cases, such as complete graphs.

The problem of $k$-edge-connectivity is also called the $k$-WAY CUT problem which is defined as follows: given an undirected graph $G$ and integers $k$ and $s$, remove at most $s$ edges from $G$ to obtain a graph with at least $k$ connected components. This problem has applications in numerous areas of computer science, such as finding cutting planes for the traveling salesman problem, clustering-related settings (e.g., VLSI design), or network reliability [4]. In [8, 9, 12, 17, 18], the authors considered the algorithms and computational complexity of this problem. In general, $k$-WAY CUT is NP-complete [12] but solvable in polynomial time for fixed $k$ [12, 17]. From the parameterized perspective, the $k$-WAY CUT problem parameterized by $k$ is W[1]-hard [9]. Kawarabayashi and Thorup [18] presented a fixed-parameter algorithm for $k$-WAY CUT parameterized by $s$. In [8], Cygan et al. showed that it is OR-compositional and, therefore, a polynomial kernelization algorithm is unlikely to exist.

In this paper, we continue to compute the exact values and sharp bounds of $k$-edge-connectivity for a graph $G$, and investigate the extremality for the $\lambda_{k}(G)$ of graphs. Some concepts and preliminary results will be introduced in the next section. In Section 3, we will characterize those graphs $G$ with $\lambda_{k}(G)=t$, where $t\in\{k-1,{n\choose 2}-{{n-k+1}\choose 2}-1,{n\choose 2}-{{n-k+1}\choose 2}\}$. For any connected graph $G$, we will obtain a sharp lower and a sharp upper bounds of $\lambda_{k}(G)$ in terms of $n$ and $k$, and we will give necessary and sufficient conditions for equalities to hold.

Relationships between $\lambda_{k}(G)$ and other generalized connectivities, including $\lambda^{\prime}_{k}(G),\kappa^{\prime}_{k}(G)$ and $\kappa_{k}(G)$, will also be discussed in Section 3. Note that definitions of these generalized connectivities will be introduced in Section 2. We will first compute a sharp lower bound which is about the relationship between $\lambda_{k}(G)$ and $\lambda^{\prime}_{k}(G)$ ($\kappa^{\prime}_{k}(G)$), and a sharp upper bound which concerns the relationship between $\lambda_{k}(G)$ and $\kappa_{k-1}(G)$. Moreover, a sharp bound that $\lambda_{k}(G)\geq\kappa_{k}(L(G))$ will also be deduced, where $L(G)$ is the line graph of $G$.

An algorithm in $\mathcal{O}(n^{2})$ time will be provided such that we can compute a sharp upper bound in terms of the maximum degree $\Delta(G)$ of a graph $G$.

For $2\leq k\leq n$ and $k-1\leq t\leq{n\choose 2}-{{n-k+1}\choose 2}$, the function $f(n,k,t)$ is defined as the minimum size of a connected graph $G$ with order $n$ and $\lambda_{k}(G)=t$. Bounds and some exact values for this function will be computed.

2 Preliminaries

We now introduce concepts of three generalized (edge-) connectivities which will be useful in our argument. Chartrand et al. [6] generalized the cut-version definition of the connectivity as follows: For an integer $k\geq 2$ and a graph $G$ of order $n\geq k$, the $k$-connectivity $\kappa_{k}(G)$ is the smallest number of vertices whose removal from $G$ produces a graph with at least $k$ components or a graph with fewer than $k$ vertices. By definition, we clearly have $\kappa_{2}(G)=\kappa(G)$. For more details about this topic, we refer to [6, 25, 29, 30].

The generalized $k$-connectivity $\kappa^{\prime}_{k}(G)$ of a graph $G$ which was introduced by Hager [13] in 1985 is a natural generalization of the path-version definition of the connectivity. For a graph $G=(V,E)$ and a set $S\subseteq V$ of at least two vertices, an $S$-Steiner tree or a Steiner tree connecting $S$ (or simply, an $S$-tree) is a such subgraph $T$ of $G$ that is a tree with $S\subseteq V(T)$. Two $S$-trees $T_{1}$ and $T_{2}$ are said to be internally disjoint if $E(T_{1})\cap E(T_{2})=\emptyset$ and $V(T_{1})\cap V(T_{2})=S$. The generalized local connectivity $\kappa^{\prime}_{G}(S)$ is the maximum number of internally disjoint $S$-trees in $G$. For an integer $k$ with $2\leq k\leq n$, the generalized $k$-connectivity is defined as

[TABLE]

Thus, $\kappa^{\prime}_{k}(G)$ is the minimum value of $\kappa^{\prime}_{G}(S)$ when $S$ runs over all the $k$-subsets of $V(G)$. By definition, we clearly have $\kappa^{\prime}_{2}(G)=\kappa(G)$. By convention, for a connected graph $G$ with less than $k$ vertices, we set $\kappa^{\prime}_{k}(G)=1$, and $\kappa^{\prime}_{k}(G)=0$ when $G$ is disconnected. For more details about this topic, the reader can see [13, 20, 24, 21, 30].

As a natural counterpart of the generalized $k$-connectivity, Li, Mao and Sun [24] introduced the following concept of generalized edge-connectivity which is a generalization of the path-version definition of the edge-connectivity. Two $S$-trees $T_{1}$ and $T_{2}$ are said to be edge-disjoint if $E(T_{1})\cap E(T_{2})=\emptyset$. The generalized local edge-connectivity $\lambda^{\prime}_{G}(S)$ is the maximum number of edge-disjoint $S$-trees in $G$. For an integer $k$ with $2\leq k\leq n$, the generalized $k$-edge-connectivity is defined as

[TABLE]

Thus, $\lambda^{\prime}_{k}(G)$ is the minimum value of $\lambda^{\prime}_{G}(S)$ when $S$ runs over all the $k$-subsets of $V(G)$. Hence, we have $\lambda^{\prime}_{2}(G)=\lambda(G)$. By definitions of $\kappa^{\prime}_{k}(G)$ and $\lambda^{\prime}_{k}(G)$, $\kappa^{\prime}_{k}(G)\leq\lambda^{\prime}_{k}(G)$ holds. By definitions, the generalized local edge-connectivity is the famous Steiner Packing Problem, see [10, 19, 31].

Nowadays, more and more researchers are working in the topic of generalized connectivity with applications. There are many results on this type of generalized edge-connectivity, such as [22, 24]. The reader is also referred to a new book [23] for a detailed introduction of this field.

The following two observations can be obtained straightforwardly from the definition of $\lambda_{k}(G)$.

Observation 2.1**.**

Let $H$ be a connected spanning subgraph of a graph $G$, we have $\lambda_{k}(H)\leq\lambda_{k}(G)$.

Observation 2.2**.**

For any integer $k$ with $2\leq k\leq n-1$, we have $\lambda_{k}(G)\leq\lambda_{k+1}(G)$.

In the rest of this section, we will present exact values of $\lambda_{k}(G)$ for some special graph classes which will be used in our argument of the main results for general graphs given in the next section. A wheel graph $W_{n}$ of order $n$ is a graph that contains a cycle of order $n-1$, and every graph vertex in the cycle is connected to one other graph vertex, which is known as the hub.

Lemma 2.3**.**

*The following assertions hold:

$(i)$[2] $\lambda_{k}(T)=k-1$, where $T$ is a tree;

$(ii)$ $\lambda_{k}(C_{n})=k$;

$(iii)$ \lambda_{k}(W_{n})=\left\{\begin{array}[]{ll}2k-1,&if~{}2\leq k\leq n-1\\ 2k-2,&if~{}k=n.\end{array}\right.*

Proof: The assertion$(i)$ is from [2]. The assertion $(ii)$ is not hard, so we omit the details. We now prove $(iii)$ and assume that $2\leq k\leq n-1$ in the following argument since the special case that $k=n$ is clear. Let $G\cong W_{n}$ such that $C:v_{1},v_{2},\cdots,v_{n-1}$ is the cycle of order $n-1$ and $u$ is the hub. Let $Y_{0}$ be the set of edges incident to elements of $\{v_{i}\mid 1\leq i\leq k-1\}$. Clearly, $|Y_{0}|=2k-1$ and $G-Y_{0}$ contains $k$ components, then $\lambda_{k}(W_{n})\leq 2k-1$.

Let $Y\subseteq E(G)$ with $|Y|\leq 2k-2$. Suppose that $G^{\prime}=G-Y$ contains $\ell$ components: $G_{1},G_{2},\cdots,G_{\ell}$, where $\ell\geq k$. Without loss of generality, we assume that $u\in V(G_{1})$ and will consider the following two cases:

Case 1. $V(G_{1})=\{u\}$. In this case all edges incident to $u$ must belong to $Y$ and then $Y\cap E(C)\leq(2k-2)-(n-1)\leq k-2$, so $G^{\prime}[V(C)]=C-(Y\cap E(C))$ contains at most $k-2$ components. Since $G^{\prime}=G-Y$ contains at least $k$ components and $G_{1}$ is a trivial one with $\{u\}=V(G_{1})$, we have that $G^{\prime}[V(C)]$ has at least $k-1$ components, a contradiction.

Case 2. $V(G_{1})\setminus\{u\}\neq\emptyset$. Let $E_{u}$ be the set of edges incident to $u$ in $G$. We know that in $G^{\prime}$ there is no edge between $u$ and $V(G_{i})$ for $2\leq i\leq\ell$. Then $|Y\cap E_{u}|\geq\ell-1\geq k-1$ and so $|Y\cap E(C)|\leq(2k-2)-(k-1)=k-1$. Hence, $G^{\prime}[V(C)]=C-(Y\cap E(C))$ contains at most $k-1$ components. Since $G^{\prime}=G-Y$ contains at least $k$ components and $V(G_{1})\setminus\{u\}\neq\emptyset$, we know that $G^{\prime}[V(C)]$ has at least $k$ components, a contradiction.

By the above argument, we have $\lambda_{k}(W_{n})\geq 2k-1$, and furthermore $\lambda_{k}(W_{n})=2k-1$. $\Box$

Lemma 2.4**.**

[29]** Let $G$ be a graph with order $n$ and size $m$. If $G$ contains at least $k$ components, then $m\leq{{n-k+1}\choose 2}$; the equality holds if and only if $G$ has exactly $k$ components such that $k-1$ of them are trivial, the remaining one is a clique of order $n-k+1$.

Note that Boesch and Chen [2] have determined the precise value for $\lambda_{k}(K_{n})$. Here, we restate their result with a different argument which will be useful in the following discussion.

Lemma 2.5**.**

[2]** $\lambda_{k}(K_{n})={n\choose 2}-{{n-k+1}\choose 2}.$

Proof: Let $G\cong K_{n}$ with vertex set $V(G)=\{u_{i}\mid 1\leq i\leq n\}$. Let $Y_{0}$ be a set of edges which are incident to any member of $\{u_{i}\mid 1\leq i\leq k-1\}$. Clearly, $|Y_{0}|={n\choose 2}-{{n-k+1}\choose 2}$, and the graph $G-Y_{0}$ contains exactly $k$ components: a clique with order $n-k+1$ and $k-1$ trivial components. Hence, $\lambda_{k}(K_{n})\leq|Y_{0}|={n\choose 2}-{{n-k+1}\choose 2}.$

Let $Y$ be any edge set of $G$ such that $G-Y$ contains $\ell$ components, $G_{1},G_{2},\cdots,G_{\ell}$, where $\ell\geq k$. Without loss of generality, we can assume that $n_{1}\leq n_{2}\leq\cdots\leq n_{\ell}$, where $n_{i}=n(G_{i})$. Since $\sum_{i=1}^{k}{n_{i}}=n$ and $1\leq n_{i}\leq n-k+1$, by Lemma 2.4, it is not hard to show that $\sum_{i=1}^{\ell}{E(G_{i})}$ attains the maximum value if and only if $\ell=k,n_{1}=n_{2}=\cdots=n_{k-1}=1,n_{k}=n-k+1$, and each $G_{i}$ is a clique, that is, $\sum_{i=1}^{\ell}{E(G_{i})}\leq\frac{k-1}{2}+\frac{(n-k+1)^{2}}{2}-\frac{n}{2}={{n-k+1}\choose 2}$ with the equality holds only if $n_{1}=n_{2}=\cdots=n_{k-1}=1,n_{k}=n-k+1$, and each $G_{i}$ is a clique. Then $|Y|={n\choose 2}-\sum_{i=1}^{k}{E(G_{i})}\geq{n\choose 2}-{{n-k+1}\choose 2}$ and so $\lambda_{k}(K_{n})\geq{n\choose 2}-{{n-k+1}\choose 2}.$ This completes the proof. $\Box$

We use $K_{n}-e$ to denote a graph obtained from a complete graph $K_{n}$ by deleting any edge $e$. Then, we have the following lemma.

Lemma 2.6**.**

$\lambda_{k}(K_{n}-e)={n\choose 2}-{{n-k+1}\choose 2}-1.$**

Proof: Let $G\cong K_{n}-e$ with vertex set $V(G)=\{u_{i}\mid 1\leq i\leq n\}$ such that $u_{1}v_{1}\not\in E(G)$. Let $Y_{0}$ be a set of edges which are incident to any member of $\{u_{i}\mid 1\leq i\leq k-1\}$. Clearly, $|Y_{0}|={n\choose 2}-{{n-k+1}\choose 2}-1$, and the graph $G-Y_{0}$ contains exactly $k$ components: a clique with order $n-k+1$ and $k-1$ trivial components. Hence, $\lambda_{k}(K_{n})\leq|Y_{0}|={n\choose 2}-{{n-k+1}\choose 2}-1.$

Let $Y$ be any edge set of $G$ such that $G-Y$ contains $\ell$ components, $G_{1},G_{2},\cdots,G_{\ell}$, where $\ell\geq k$. With a similar argument to that of Lemma 2.5, we have $\sum_{i=1}^{\ell}{E(G_{i})}\leq{{n-k+1}\choose 2}$. Then $|Y|={n\choose 2}-1-\sum_{i=1}^{\ell}{E(G_{i})}\geq{n\choose 2}-{{n-k+1}\choose 2}-1$ and so $\lambda_{k}(K_{n})\geq{n\choose 2}-{{n-k+1}\choose 2}-1.$ This completes the proof. $\Box$

We still need the following lemma.

Lemma 2.7**.**

If $G\not\cong K_{n},K_{n}-e$, then $\lambda_{k}(G)\leq{n\choose 2}-{{n-k+1}\choose 2}-2.$

Proof: Let $G$ be a graph obtained from a complete graph $K_{n}$ by deleting two edges $e_{1},e_{2}$. Let $V(G)=\{u_{i}\mid 1\leq i\leq n\}$. In the following, we will show that $\lambda_{k}(G)\leq{n\choose 2}-{{n-k+1}\choose 2}-2$, and then our result clearly holds by Observation 2.1. We will consider two cases according to whether $e_{1}$ and $e_{2}$ are adjacent.

Case 1. $e_{1}$ and $e_{2}$ are adjacent. Without loss of generality, we assume that $e_{1}=u_{1}u_{2}$ and $e_{2}=u_{1}u_{3}$. Let $Y_{0}$ be a set of edges which are incident to any member of $\{u_{i}\mid 1\leq i\leq k-1\}$. Clearly, $|Y_{0}|={n\choose 2}-{{n-k+1}\choose 2}-2$, and the graph $G-Y_{0}$ contains exactly $k$ components: a clique with order $n-k+1$ and $k-1$ trivial components. Hence, $\lambda_{k}(K_{n})\leq|Y_{0}|={n\choose 2}-{{n-k+1}\choose 2}-2.$

Case 2. $e_{1}$ and $e_{2}$ are nonadjacent. Without loss of generality, we assume that $e_{1}=u_{1}u_{2}$ and $e_{2}=u_{3}u_{4}$.

We first consider the case that $k\geq 4$. Let $Y_{1}$ be a set of edges which are incident to any member of $\{u_{i}\mid 1\leq i\leq k-1\}$. Clearly, $|Y_{1}|={n\choose 2}-{{n-k+1}\choose 2}-2$, and the graph $G-Y_{1}$ contains exactly $k$ components: a clique with order $n-k+1$ and $k-1$ trivial components. Hence, $\lambda_{k}(K_{n})\leq|Y_{1}|={n\choose 2}-{{n-k+1}\choose 2}-2.$

We then consider the case that $k=3$. Let $Y_{2}=\{u_{1}u_{4},u_{1}u_{3},u_{2}u_{4}\}\cup E(\{u_{1},u_{4}\},A)$, where $A=V\setminus\{u_{i}\mid 1\leq i\leq 4\}$ and $E(\{u_{1},u_{4}\},A)$ denotes the set of edges between $\{u_{1},u_{4}\}$ and $A$. Clearly, $|Y_{2}|=2n-5={n\choose 2}-{{n-k+1}\choose 2}-2$, and the graph $G-Y_{2}$ contains exactly three components: a clique with order $n-2$ and two trivial components. Hence, $\lambda_{k}(K_{n})\leq|Y_{2}|={n\choose 2}-{{n-k+1}\choose 2}-2.$ $\Box$

3 Main results of computing exact values and sharp bounds

By Lemmas 2.5, 2.6 and 2.7, the following result clearly holds.

Proposition 3.1**.**

*The following assertions hold:

$(i)$ $\lambda_{k}(G)={n\choose 2}-{{n-k+1}\choose 2}-1$ if and only if $G\cong K_{n}-e$;

$(ii)$ $\lambda_{k}(G)={n\choose 2}-{{n-k+1}\choose 2}$ if and only if $G\cong K_{n}$.*

The following result concerns sharp bounds for $\lambda_{k}(G)$ of a general graph $G$.

Theorem 3.2**.**

For a connected graph $G$, we have

[TABLE]

Moreover, the lower bound can be attained if and only if $G$ contains at least $k-1$ cut edges, and the upper bound can be attained if and only if $G\cong K_{n}$.

Proof: The lower bound is clear by Observation 2.1 and $(i)$ of Lemma 2.3. If $G$ contains at least $k-1$ cut edges, then let $Y_{0}$ be a set of $k-1$ cut edges. Clearly, $G-Y_{0}$ contains $k$ components and so $\lambda_{k}(G)\leq k-1$. Hence, $\lambda_{k}(G)=k-1$ in this case. If $G$ contains at most $k-2$ cut edges, then let $Y=\{e_{1},e_{2},\cdots,e_{k-1}\}$ be a set of any $k-1$ edges of $G$. Without loss of generality, we assume that the former $k_{1}$ elements of $Y$ are cut edges. Then $G^{\prime}=G-\{e_{i}\mid 1\leq i\leq k_{1}\}$ has exactly $k_{1}+1$ components. We know that each element of $\{e_{i}|k_{1}+1\leq i\leq k-1\}$ is not a cut edge of $G^{\prime}$ and so the number of components in $G^{\prime\prime}=G-Y=G^{\prime}-\{e_{i}\mid 1\leq i\leq k_{1}\}$ will increase at most $(k-1-k_{1})-1$. Hence, the number of components in $G^{\prime\prime}$ is at most $(k-1-k_{1})-1+k_{1}+1=k-1$ components and so $\lambda_{k}(G)>|Y|=k-1$, a contradiction. Therefore, the lower bound can be attained if and only if $G$ contains at least $k-1$ cut edges.

We now prove the upper bound. By Observation 2.1 and Lemma 2.5, we have $\lambda_{k}(G)\leq\lambda_{k}(K_{n})={n\choose 2}-{{n-k+1}\choose 2}.$ By Proposition 3.1, the upper bound can be attained if and only if $G\cong K_{n}$. $\Box$

Note that by Proposition 3.1 and Theorem 3.2, we can characterize those graphs $G$ with $\lambda_{k}(G)=t$ for $t\in\{k-1,{n\choose 2}-{{n-k+1}\choose 2}-1,{n\choose 2}-{{n-k+1}\choose 2}\}$.

We now discuss the relationships between $\lambda_{k}(G)$ and other generalized connectivities, including $\lambda^{\prime}_{k}(G),\kappa^{\prime}_{k}(G)$ and $\kappa_{k}(G)$. We first give a lower bound which concerns the relationship between $\lambda_{k}(G)$ and $\lambda^{\prime}_{k}(G)$, and an upper bound which is about the relationship between $\lambda_{k}(G)$ and $\kappa_{k-1}(G)$.

Theorem 3.3**.**

For a connected graph $G$ with maximum degree $\Delta(G)$, we have

[TABLE]

Moreover, both bounds are sharp.

Proof: We first prove the lower bound and its sharpness. For the case $k=2$, the result clearly holds. In the following argument, we assume that $k\geq 3$. Let $Y$ be a set of edges of $G$ with $|Y|=\lambda_{k}(G)$ such that the graph $G-Y$ contains $\ell$ components, say $G_{1},G_{2},\cdots,G_{\ell}$, where $\ell\geq k$. Let $S=\{u_{i}\mid u_{i}\in V(G_{i}),1\leq i\leq k\}$. By the definition of $\lambda^{\prime}_{k}(G)$, there are at least $\lambda^{\prime}_{k}(G)$ edge-disjoint $S$-trees in $G$. For each such tree $T$, we have that $|E(T)\cap Y|\geq k-1$, then $\lambda_{k}(G)=|Y|\geq(k-1){\lambda^{\prime}_{k}(G)}.$

For the sharpness of the lower bound, we just consider the case that $G$ is a tree. In this case we have that $\lambda^{\prime}_{k}(G)=1$ and $\lambda_{k}(G)=k-1$, so $\lambda_{k}(G)=(k-1){\lambda^{\prime}_{k}(G)}.$

We now prove the upper bound and its sharpness. Let $X\subseteq V(G)$ such that $|X|=\kappa_{k-1}(G)$ and $G-X$ contains at least $k-1$ components. Let $E^{\prime}$ be the set of edges between $X$ and $V(G)\setminus X$ in $G$. Clearly, the graph $G-E^{\prime}$ contains at least $k$ components. Then $\lambda_{k}(G)\leq|E^{\prime}|\leq{\Delta(G)}{|X|}={\Delta(G)}{\kappa_{k-1}(G)}.$

For the sharpness of the upper bound, we just consider the following graph $G$: Let $G$ be obtained by identifying the center vertex, say $u_{k-1}$, of a star graph $S$ with an end vertex, say $v_{n-k+2}$, of a path $P$ such that $V(S)=\{u_{i}\mid 1\leq i\leq k-1\}$, $V(P)=\{v_{j}\mid 1\leq j\leq n-k+2\}$ and the new vertex of $G$ is denoted by $u$. Clearly, $G$ is a tree with maximum degree $\Delta(G)=deg_{G}(u)=k-1$ and so $\lambda_{k}(G)=k-1$. It is not hard to show that $\kappa_{k-1}(G)=1$. Hence, $\lambda_{k}(G)={\Delta(G)}{\kappa_{k-1}(G)}$ in this case. $\Box$

Note that for some graphs, the equalities of bounds in Theorem 3.3 may not hold. For the upper bound, let $G$ be a cycle with order $n\geq 2k$, where $k\geq 3$. In this case we have that $\kappa_{k-1}(G)=k-1$ and $\lambda_{k}(G)=k$, so $\lambda_{k}(G)<{\Delta(G)}{\kappa_{k-1}(G)}$. For the lower bound, let $G$ be a wheel graph with order $n\geq k+1$. It is not hard to show that $\lambda^{\prime}_{k}(G)\leq 2$. By Lemma 2.3, we have $\lambda^{\prime}_{k}(G)=2k-1$, so $(k-1){\lambda^{\prime}_{k}(G)}<\lambda_{k}(G).$

Recall the fact that $\kappa^{\prime}_{k}(G)\leq\lambda^{\prime}_{k}(G)$ and by Theorem 3.3, we have the following corollary. For the sharpness of this bound, we just let $G$ be a tree.

Corollary 3.4**.**

For a connected graph $G$, we have

[TABLE]

Moreover, the bound is sharp.

The $line~{}graph$ $L(G)$ of a graph $G$ is the graph whose vertex set is $V(L(G))=E(G)$ and two vertices $e_{1}$, $e_{2}$ of $L(G)$ are adjacent if and only if these edges are adjacent in $G$. By using the particular properties of line graphs shown in [14] and [15], we can give the following lower bound for $\lambda_{k}(G)$ in terms of $k$-connectivity of the line graph of $G$.

Theorem 3.5**.**

Let $G$ be a connected graph with order $n$. For $2\leq k\leq n$, we have

[TABLE]

Moreover, the bound is sharp.

Proof: Let $Y=\{e_{1},e_{2},\cdots,e_{t}\}\subseteq E(G)$ with $t=\lambda_{k}(G)$ such that $G-Y$ contains the following $k$ components: $G_{1},G_{2},\cdots,G_{k}$. Let $X=\{v_{e_{1}},v_{e_{2}},\cdots,v_{e_{t}}\}\subseteq V(L(G))$, where $v_{e_{i}}$ denotes the vertex in $L(G)$ corresponding to the edge $e_{i}$ in $G$ for $1\leq i\leq t$. It is not hard to show that ${L(G)}-X$ contains the following $k$ components: $L(G_{1}),L(G_{2}),\cdots,L(G_{k})$, where $L(G_{j})$ denotes the line graph of $G_{j}$ for $1\leq j\leq k$. Hence, $\kappa_{k}(L(G))\leq|X|=|Y|=t=\lambda_{k}(G)$.

For the sharpness of the bound, we just let $G=C_{n}$ with $n\geq 2k$, then we have $\lambda_{k}(G)=k$. Since $L(G)\cong C_{n}$, we obtain $\kappa_{k}(L(G))=k$. Hence, $\lambda_{k}(G)=\kappa_{k}(L(G))$ in this case. $\Box$

The following result is a sharp upper bound for $\lambda_{k}(G)$ which is also a function of the maximum degree $\Delta(G)$.

Theorem 3.6**.**

Let $G$ be a connected graph with order $n$ and maximum degree $\Delta(G)$. For $2\leq k\leq n$, we have

[TABLE]

Moreover, the bound is sharp and can be obtained in $\mathcal{O}(kn)$ time.

Proof: We use Algorithm 1 to prove our bound. In our algorithm, let $Y_{i}$ be the set of edges incident with $v_{i}$ in $G_{i}$ for $1\leq i\leq k$. Note that in line 3 of our algorithm, we can choose the vertex $v$ in the maximum component of $G_{i}$ as $v_{i}$ such that $v$ is adjacent to some vertex of $\{v_{1},v_{2},\cdots,v_{i-1}\}$, so the vertex $v_{i}$ must exist. When the algorithm terminates, the final graph is denoted by $G^{\prime}=G_{k}$. It is not hard to show that $G^{\prime}$ contains at least $k$ components. Since the total number edges deleted during the algorithm is $\sum_{i=1}^{k-1}{|Y_{i}|}\leq\Delta(G)+(\Delta(G)-1)(k-2)=(\Delta(G)-1)(k-1)+1$, we have $\lambda_{k}(G)\leq(\Delta(G)-1)(k-1)+1.$

For the sharpness of the bound, we just let $G=C_{n}$. By Lemma 2.3, we have $\lambda_{k}(G)=k=(\Delta(G)-1)(k-1)+1$ in this case.

It remains to analyze the running time. In line 3 of Algorithm 1, it takes $\mathcal{O}(n)$ time to find the maximum component of $G_{i}$ and choose a vertex $v_{i}$ with degree at most $\Delta(G)-1$ in this component. Therefore the total running time is $\mathcal{O}(kn)$. $\Box$

Obviously, by the above argument, the total running time of Algorithm 1 is at most $\mathcal{O}(n^{2})$ since $k\leq n$.

Recall that we proved that $\lambda_{k}(G)\leq{\Delta(G)}{\kappa_{k-1}(G)}$ in Theorem 3.3 and $\lambda_{k}(G)\leq(\Delta(G)-1)(k-1)+1$ in Theorem 3.6. For some graphs, the inequality ${\Delta(G)}{\kappa_{k-1}(G)}<(\Delta(G)-1)(k-1)+1$ holds. For example, let $G$ be the second example in the proof of Theorem 3.3, we have ${\Delta(G)}{\kappa_{k-1}(G)}=k-1<(k-2)(k-1)+1=(\Delta(G)-1)(k-1)+1$ in this case. For some other graphs, the inequality ${\Delta(G)}{\kappa_{k-1}(G)}>(\Delta(G)-1)(k-1)+1$ holds. For example, let $G$ be a cycle with order $n\geq 2k$, where $k\geq 3$. We have ${\Delta(G)}{\kappa_{k-1}(G)}=2(k-1)>k=(\Delta(G)-1)(k-1)+1$ in this case. By Theorem 3.6, we clearly have the following corollary.

Corollary 3.7**.**

Let $G$ be a connected $r$-regular graph with order $n$. For $2\leq k\leq n$, we have

[TABLE]

Moreover, the bound is sharp.

Recall that $f(n,k,t)$ is the minimum size of a connected graph $G$ with order $n$ and $\lambda_{k}(G)=t$, where $2\leq k\leq n$ and $k-1\leq t\leq{n\choose 2}-{{n-k+1}\choose 2}$. We now prove the following result.

Theorem 3.8**.**

For a connected graph $G$, we have

[TABLE]

Moreover, we have

[TABLE]

Proof: Let $G$ be a connected graph of order $n$ with $\lambda_{k}(G)=t$. Let $Y_{0}\subseteq E(G)$ with $|Y_{0}|=\lambda_{k}(G)$ such that $G-Y_{0}$ contains at least $k$ components, say $G_{1},G_{2},\cdots,G_{\ell}$, where $\ell\geq k$.

We claim that $\ell=k$. If not, then $\ell\geq k+1$ and there is an edge $e_{0}=xy\in Y_{0}$ with $x\in\cup_{i=1}^{k}{V(G_{i})}$ and $y\in\cup_{j={k+1}}^{\ell}{V(G_{j})}$. Without loss of generality, we assume that $x\in V(G_{k})$ and $y\in V(G_{k+1})$. Let $Y_{1}=Y_{0}\setminus\{e_{0}\}$. It is not hard to show that $G-Y_{1}$ contains the following $k$ components: $G_{1},G_{2},\cdots,G_{k}\cup G_{k+1}$. This means that $\lambda_{k}(G)\leq|Y_{1}|=|Y_{0}|-1=\lambda_{k}(G)-1$, a contradiction. Hence, we have $\ell=k$ and so there are exactly $k$ components in $G-Y_{0}$.

For the lower bound, we have $m(G)=\sum_{i=1}^{k}{m(G_{i})}+t\geq\sum_{i=1}^{k}{(n(G_{i})-1)}+t=n-k+t$. Hence, $f(n,k,t)\geq n-k+t.$

For the upper bound, we have that $m(G)=\sum_{i=1}^{k}{m(G_{i})}+t\leq{{n-k+1}\choose 2}+t$ by Lemma 2.4. Hence, $f(n,k,t)\leq{{n-k+1}\choose 2}+t.$

By Lemma 2.3 and Propositions 3.1, we have

[TABLE]

$\Box$

Note that in Theorem 3.8, we give an upper bound and a lower bound for the function $f(n,k,t)$. The upper bound can be attained when $t\in\{{n\choose 2}-{{n-k+1}\choose 2}-1,{n\choose 2}-{{n-k+1}\choose 2}\}$, and the lower bound can be attained when $t\in\{k-1,k\}$.

4 Concluding remarks

In this paper, we investigate the $k$-edge-connectivity $\lambda_{k}(G)$ of a graph $G$ and compute some exact values and sharp bounds for $\lambda_{k}(G)$ in terms of some other graph parameters, such as $\lambda^{\prime}_{k}(G)$ and $\kappa^{\prime}_{k-1}(G)$, where $2\leq k\leq n$. Specially, we prove that $k-1\leq\lambda_{k}(G)\leq{n\choose 2}-{{n-k+1}\choose 2}$ and characterize the graphs with $\lambda_{k}(G)=t$, where $t\in\{k-1,{n\choose 2}-{{n-k+1}\choose 2}-1,{n\choose 2}-{{n-k+1}\choose 2}\}$. Then the following problem is interesting.

Problem 4.1**.**

$(i)$* Determine the graphs with $\lambda_{k}(G)=t$ for a small integer $t$, that is, $t$ is close to $k-1$.

$(ii)$ Determine the graphs with $\lambda_{k}(G)=t$ for a large integer $t$, that is, $t$ is close to ${n\choose 2}-{{n-k+1}\choose 2}$.*

Recall that we compute the precise values for the $k$-edge-connectivity of some graph classes which can be used in the results for general graphs. Products of graphs occur naturally in discrete mathematics as tools in combinatorial constructions, they give rise to important classes of graphs and deep structural problems, and they also play a key role in design and analysis of networks [5, 16]. Some researchers have investigated a generalized product of graphs [1] and the connectivity which can model and construct large reliable networks under optimal conditions in the past several decades [26, 27]. It is also interesting to obtain some sharp upper bounds for the $k$-edge-connectivity of generalized graph products in terms of some parameters of original graphs, such as the order and the minimum degree.

We further study the function $f(n,k,t)$ which is defined as the minimum size of a connected graph $G$ with order $n$ and $\lambda_{k}(G)=t$, and give bounds and some exact values for this function. Since it is quite difficult to determine the exact values of $f(n,k,t)$ for a general $t$, it is interesting to try the following problem.

Problem 4.2**.**

$(i)$* Determine the exact values of $f(n,k,t)$ for a small integer $t$, that is, $t$ is close to $k-1$.

$(ii)$ Determine the exact values of $f(n,k,t)$ for a large integer $t$, that is, $t$ is close to ${n\choose 2}-{{n-k+1}\choose 2}$.*