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

01/18/2019 ∙ by Yuefang Sun, et al. ∙ NetEase, Inc 0

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≤ k≤ n, the k-edge-connectivity of a graph G, denoted by λ_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 λ_k(G) in terms of n and k. We then discuss the relationships between λ_k(G) and other generalized connectivities. An algorithm in 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 λ_k(G)=t.

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1 Introduction

We refer to [3] for graph theoretical notation and terminology not described here. For a graph , let , be the set of vertices, the set of edges of , respectively. For , we denote by the subgraph obtained by deleting from the vertices of together with the edges incident with them. For , we denote by the subgraph obtained by deleting from the edges of . For a set , we use to denote its size. We use , and to denote a path of order , a cycle of order and a complete graph of order , 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 , written by , is the minimum size of an edge set such that 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 and in , the local edge-connectivity is the maximum number of edge-disjoint paths connecting and . Then is defined to be the edge-connectivity of . 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 “-edge-connectivity” from [23]. For any integer with , the -edge-connectivity of a graph , denoted by , is defined as the smallest number of edges whose removal from produces a graph with at least components. By definition, we clearly have . Boesch and Chen [2] gave some properties of and obtained some bounds for in terms the minimum degree and the degree-sequence of . They also studied some special cases, such as complete graphs.

The problem of -edge-connectivity is also called the -WAY CUT problem which is defined as follows: given an undirected graph and integers and , remove at most edges from to obtain a graph with at least 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, -WAY CUT is NP-complete [12] but solvable in polynomial time for fixed [12, 17]. From the parameterized perspective, the -WAY CUT problem parameterized by is W[1]-hard [9]. Kawarabayashi and Thorup [18] presented a fixed-parameter algorithm for -WAY CUT parameterized by . 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 -edge-connectivity for a graph , and investigate the extremality for the of graphs. Some concepts and preliminary results will be introduced in the next section. In Section 3, we will characterize those graphs with , where . For any connected graph , we will obtain a sharp lower and a sharp upper bounds of in terms of and , and we will give necessary and sufficient conditions for equalities to hold.

Relationships between and other generalized connectivities, including and , 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 and (), and a sharp upper bound which concerns the relationship between and . Moreover, a sharp bound that will also be deduced, where is the line graph of .

An algorithm in time will be provided such that we can compute a sharp upper bound in terms of the maximum degree of a graph .

For and , the function is defined as the minimum size of a connected graph with order and . 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 and a graph of order , the -connectivity is the smallest number of vertices whose removal from produces a graph with at least components or a graph with fewer than vertices. By definition, we clearly have . For more details about this topic, we refer to [6, 25, 29, 30].

The generalized -connectivity of a graph which was introduced by Hager [13] in 1985 is a natural generalization of the path-version definition of the connectivity. For a graph and a set of at least two vertices, an -Steiner tree or a Steiner tree connecting (or simply, an -tree) is a such subgraph of that is a tree with . Two -trees and are said to be internally disjoint if and . The generalized local connectivity is the maximum number of internally disjoint -trees in . For an integer with , the generalized -connectivity is defined as

Thus, is the minimum value of when runs over all the -subsets of . By definition, we clearly have . By convention, for a connected graph with less than vertices, we set , and when is disconnected. For more details about this topic, the reader can see [13, 20, 24, 21, 30].

As a natural counterpart of the generalized -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 -trees and are said to be edge-disjoint if . The generalized local edge-connectivity is the maximum number of edge-disjoint -trees in . For an integer with , the generalized -edge-connectivity is defined as

Thus, is the minimum value of when runs over all the -subsets of . Hence, we have . By definitions of and , 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 .

Observation 2.1.

Let be a connected spanning subgraph of a graph , we have .

Observation 2.2.

For any integer with , we have .

In the rest of this section, we will present exact values of 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 of order is a graph that contains a cycle of order , 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:
[2] , where is a tree;
 ;
 

Proof: The assertion is from [2]. The assertion is not hard, so we omit the details. We now prove and assume that in the following argument since the special case that is clear. Let such that is the cycle of order and is the hub. Let be the set of edges incident to elements of . Clearly, and contains components, then .

Let with . Suppose that contains components: , where . Without loss of generality, we assume that and will consider the following two cases:

Case 1. . In this case all edges incident to must belong to and then , so contains at most components. Since contains at least components and is a trivial one with , we have that has at least components, a contradiction.

Case 2. . Let be the set of edges incident to in . We know that in there is no edge between and for . Then and so . Hence, contains at most components. Since contains at least components and , we know that has at least components, a contradiction.

By the above argument, we have , and furthermore .

Lemma 2.4.

[29] Let be a graph with order and size . If contains at least components, then ; the equality holds if and only if has exactly components such that of them are trivial, the remaining one is a clique of order .

Note that Boesch and Chen [2] have determined the precise value for . Here, we restate their result with a different argument which will be useful in the following discussion.

Lemma 2.5.

[2]

Proof: Let with vertex set . Let be a set of edges which are incident to any member of . Clearly, , and the graph contains exactly components: a clique with order and trivial components. Hence,

Let be any edge set of such that contains components, , where . Without loss of generality, we can assume that , where . Since and , by Lemma 2.4, it is not hard to show that attains the maximum value if and only if , and each is a clique, that is, with the equality holds only if , and each is a clique. Then and so This completes the proof.

We use to denote a graph obtained from a complete graph by deleting any edge . Then, we have the following lemma.

Lemma 2.6.

Proof: Let with vertex set such that . Let be a set of edges which are incident to any member of . Clearly, , and the graph contains exactly components: a clique with order and trivial components. Hence,

Let be any edge set of such that contains components, , where . With a similar argument to that of Lemma 2.5, we have . Then and so This completes the proof.

We still need the following lemma.

Lemma 2.7.

If , then

Proof: Let be a graph obtained from a complete graph by deleting two edges . Let . In the following, we will show that , and then our result clearly holds by Observation 2.1. We will consider two cases according to whether and are adjacent.

Case 1. and are adjacent. Without loss of generality, we assume that and . Let be a set of edges which are incident to any member of . Clearly, , and the graph contains exactly components: a clique with order and trivial components. Hence,

Case 2. and are nonadjacent. Without loss of generality, we assume that and .

We first consider the case that . Let be a set of edges which are incident to any member of . Clearly, , and the graph contains exactly components: a clique with order and trivial components. Hence,

We then consider the case that . Let , where and denotes the set of edges between and . Clearly, , and the graph contains exactly three components: a clique with order and two trivial components. Hence,

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:
  if and only if ;
  if and only if .

The following result concerns sharp bounds for of a general graph .

Theorem 3.2.

For a connected graph , we have

Moreover, the lower bound can be attained if and only if contains at least cut edges, and the upper bound can be attained if and only if .

Proof: The lower bound is clear by Observation 2.1 and of Lemma 2.3. If contains at least cut edges, then let be a set of cut edges. Clearly, contains components and so . Hence, in this case. If contains at most cut edges, then let be a set of any edges of . Without loss of generality, we assume that the former elements of are cut edges. Then has exactly components. We know that each element of is not a cut edge of and so the number of components in will increase at most . Hence, the number of components in is at most components and so , a contradiction. Therefore, the lower bound can be attained if and only if contains at least cut edges.

We now prove the upper bound. By Observation 2.1 and Lemma 2.5, we have By Proposition 3.1, the upper bound can be attained if and only if .

Note that by Proposition 3.1 and Theorem 3.2, we can characterize those graphs with for .

We now discuss the relationships between and other generalized connectivities, including and . We first give a lower bound which concerns the relationship between and , and an upper bound which is about the relationship between and .

Theorem 3.3.

For a connected graph with maximum degree , we have

Moreover, both bounds are sharp.

Proof: We first prove the lower bound and its sharpness. For the case , the result clearly holds. In the following argument, we assume that . Let be a set of edges of with such that the graph contains components, say , where . Let . By the definition of , there are at least edge-disjoint -trees in . For each such tree , we have that , then

For the sharpness of the lower bound, we just consider the case that is a tree. In this case we have that and , so

We now prove the upper bound and its sharpness. Let such that and contains at least components. Let be the set of edges between and in . Clearly, the graph contains at least components. Then

For the sharpness of the upper bound, we just consider the following graph : Let be obtained by identifying the center vertex, say , of a star graph with an end vertex, say , of a path such that , and the new vertex of is denoted by . Clearly, is a tree with maximum degree and so . It is not hard to show that . Hence, in this case.

Note that for some graphs, the equalities of bounds in Theorem 3.3 may not hold. For the upper bound, let be a cycle with order , where . In this case we have that and , so . For the lower bound, let be a wheel graph with order . It is not hard to show that . By Lemma 2.3, we have , so

Recall the fact that and by Theorem 3.3, we have the following corollary. For the sharpness of this bound, we just let be a tree.

Corollary 3.4.

For a connected graph , we have

Moreover, the bound is sharp.

The of a graph is the graph whose vertex set is and two vertices , of are adjacent if and only if these edges are adjacent in . By using the particular properties of line graphs shown in [14] and [15], we can give the following lower bound for in terms of -connectivity of the line graph of .

Theorem 3.5.

Let be a connected graph with order . For , we have

Moreover, the bound is sharp.

Proof: Let with such that contains the following components: . Let , where denotes the vertex in corresponding to the edge in for . It is not hard to show that contains the following components: , where denotes the line graph of for . Hence, .

For the sharpness of the bound, we just let with , then we have . Since , we obtain . Hence, in this case.

The following result is a sharp upper bound for which is also a function of the maximum degree .

Theorem 3.6.

Let be a connected graph with order and maximum degree . For , we have

Moreover, the bound is sharp and can be obtained in time.

Proof: We use Algorithm 1 to prove our bound. In our algorithm, let be the set of edges incident with in for . Note that in line 3 of our algorithm, we can choose the vertex in the maximum component of as such that is adjacent to some vertex of , so the vertex must exist. When the algorithm terminates, the final graph is denoted by . It is not hard to show that contains at least components. Since the total number edges deleted during the algorithm is , we have

For the sharpness of the bound, we just let . By Lemma 2.3, we have in this case.

It remains to analyze the running time. In line 3 of Algorithm 1, it takes time to find the maximum component of and choose a vertex with degree at most in this component. Therefore the total running time is .

Obviously, by the above argument, the total running time of Algorithm 1 is at most since .

Input: A connected graph with order and maximum degree .
Output: A subgraph of with at least components.

1:Choose any vertex as in , set .
2:for  do
3:     Choose a vertex with degree at most in the maximum component of ; ; .
4:end for
5:return
Algorithm 1

Recall that we proved that in Theorem 3.3 and in Theorem 3.6. For some graphs, the inequality holds. For example, let be the second example in the proof of Theorem 3.3, we have in this case. For some other graphs, the inequality holds. For example, let be a cycle with order , where . We have in this case. By Theorem 3.6, we clearly have the following corollary.

Corollary 3.7.

Let be a connected -regular graph with order . For , we have

Moreover, the bound is sharp.

Recall that is the minimum size of a connected graph with order and , where and . We now prove the following result.

Theorem 3.8.

For a connected graph , we have

Moreover, we have

Proof: Let be a connected graph of order with . Let with such that contains at least components, say , where .

We claim that . If not, then and there is an edge with and . Without loss of generality, we assume that and . Let . It is not hard to show that contains the following components: . This means that , a contradiction. Hence, we have and so there are exactly components in .

For the lower bound, we have . Hence,

For the upper bound, we have that by Lemma 2.4. Hence,

By Lemma 2.3 and Propositions 3.1, we have


Note that in Theorem 3.8, we give an upper bound and a lower bound for the function . The upper bound can be attained when , and the lower bound can be attained when .

4 Concluding remarks

In this paper, we investigate the -edge-connectivity of a graph and compute some exact values and sharp bounds for in terms of some other graph parameters, such as and , where . Specially, we prove that and characterize the graphs with , where . Then the following problem is interesting.

Problem 4.1.

 Determine the graphs with for a small integer , that is, is close to .
 Determine the graphs with for a large integer , that is, is close to