Wireless Communications and Mobile Computing

Wireless Communications and Mobile Computing / 2021 / Article

Research Article | Open Access

Volume 2021 |Article ID 5977781 | https://doi.org/10.1155/2021/5977781

Xinyu Liang, Jiarong Liang, Weiguang Zhang, "Construction of Quality Virtual Backbones with Link Fault Tolerance in Wireless Sensor Networks", Wireless Communications and Mobile Computing, vol. 2021, Article ID 5977781, 20 pages, 2021. https://doi.org/10.1155/2021/5977781

Construction of Quality Virtual Backbones with Link Fault Tolerance in Wireless Sensor Networks

Academic Editor: Ghufran Ahmed
Received27 Jul 2021
Revised02 Nov 2021
Accepted06 Nov 2021
Published08 Dec 2021

Abstract

Wireless sensor networks (WSNs) are extensively utilized in various circumstances. For applications, the construction of the virtual backbones (VBs) of WSNs has attracted considerable attention in this field. Generally, a homogeneous WSN is formulated as a unit disk graph (UDG), and the VB of the corresponding WSN is modeled as a connected dominating set (CDS) in the UDG. In certain applications, communication between sensors in a network may fail for various reasons, such as sensor movement, signal interference, and the appearance of obstacles. Consequently, a CDS in a UDG should possess fault tolerance on the edges. In this paper, we introduce a new concept called the 2 edge-connected 2 edge-dominating set (-ECDS); then, we design an approximation algorithm for computing -ECDSs in UDGs, the performance ratio of which is 30.51. By means of simulations, we compare our algorithm and existing algorithms in terms of the CDS size, running time, success rate, and network lifetime. The simulation results indicate that our algorithm exhibits better performance and is more suitable for constructing a VB with edge fault tolerance in a WSN.

1. Introduction

Wireless sensor networks (WSNs) are multihop self-organizing network systems that contain numerous sensor nodes. Because no predefined infrastructure is required, WSNs can be flexibly deployed in various circumstances, such as the Internet of Things, ocean monitoring, and traffic surveillance [15]. However, the wireless sensors in WSNs are usually powered by batteries, which means that minimizing sensor energy consumption is a critical challenge [6, 7]. Generally, the energy of sensor nodes in a WSN is mainly consumed in the signal transmission process. Specifically, when a signal is transmitted from a source node to a target node, the required energy increases exponentially with increasing distance [8]. Thus, most WSNs use a multihop communication strategy rather than long-range direct communication to enhance the energy efficiency of the sensor nodes [914].

In multihop communication, a message from a source node to a target node may be retransmitted many times, which can result in substantial wireless signal interference and collision and waste a large amount of energy. Hence, a high-quality routing path for information transmission is critical. To address this issue in WSN communication, the concept of the VB, inspired by the backbone of wired networks, is supposed to reduce information overload and improve the efficiency and lifespan of the network [1].

In theory, a VB is a subset of sensor nodes in a WSN that meets the following requirements [15]. First, each node in a WSN is adjacent to a node in this subset. Second, any two nodes in the subset have a routing path between them in the VB. To investigate VBs in homogeneous WSNs, the WSN is typically modeled as a UDG , where a vertex in is a simplification of a sensor node and an edge in stands for the message transmission between two sensors; a VB of a WSN is modeled as a CDS in the corresponding graph . In the research of WSNs, since a smaller VB entails less routing overhead and wireless signal collision, the size of the VB is an important index in measuring the quality of the VB. Thus, it is natural to find a minimum CDS (MCDS) of when constructing a CDS. However, Garey and Johnson [16] have proven that calculating MCDSs is NP-hard. Consequently, many efforts have been made to obtain an approximate solution to this problem by designing approximation algorithms.

In practical applications, in a WSN, a few sensors and the links between them may be out of operation for various reasons, such as sensor movement, signal interference, and the appearance of obstacles. Thus, the VB for a WSN should be fault tolerant such that it continues to work when some nodes and links fail. Fortunately, in recent years, progress has been made in designing node fault-tolerant VBs for WSNs. In fact, in 2005, Dai and Wu [17] proposed a special CDS, the connected dominating set, for a node fault-tolerant VB in a WSN against node failure. Later, Thai et al. [18] extended this concept to the famous connected dominating set (CDS), where . Since the construction of the minimum CDS is NP-hard, the construction of the minimum CDS is also NP-hard. As a result, numerous researchers have focused on developing approximation algorithms to calculate a CDS with a worst-case performance guarantee for any given pair of integers and () [1927].

Note that the above literature considered only the problem of a fault-tolerant VB against node failure and neglected the impact of edge failure. In practical applications, edge failure may impact the operation of a VB in the considered WSN; in other words, edge failure may lead to failure of the VB even if all nodes in the VB are fault-free. For example, in a WSN with 9 nodes, which is modeled by the UDG shown in Figure 1, the VB consists of nodes , , and . When edge fails, some aspects of the VB, such as domination, will fail, even though each node is fault-free. In practice, edge failure in a WSN may be inevitable due to various factors, such as node movement, signal interference, and obstacles. For example, in the WSN shown in Figure 1, when node is moved to a position outside of the transmission range of (see Figure 1(b)), edge fails, which results in subset no longer being a VB.

Note that in a UDG, if a CDS is a node fault-tolerant CDS, such as -CDS with , it must be an edge fault-tolerant CDS. Formally, in a UDG , a CDS is a node fault-tolerant CDS if for any , is still a CDS in . Similarly, in a UDG , a CDS is an edge fault-tolerant CDS if for any , is still a CDS in . Compared to that of constructing an edge fault-tolerant CDS in a UDG, the cost of constructing a node fault-tolerant CDS may be higher. On the other hand, in some UDGs, there may not exist a node fault-tolerant CDS, while there is an edge fault-tolerant CDS. For example, in the UDG shown in Figure 2, there are no node fault-tolerant CDSs; however, the CDS is an edge fault-tolerant CDS because when any edge in fails, is still a CDS. Hence, it is necessary to explore how to construct an edge fault-tolerant CDS in a UDG. Unfortunately, few efforts have been made in this direction in the existing literature.

In this paper, we investigate the construction of an edge fault-tolerant CDS in a UDG. Specifically, this paper introduces a new concept called the 2 edge-connected 2 edge-dominating set (in short, -ECDS) and develops a constant factor approximation algorithm to calculate a -ECDS in a given UDG.

The main work of our paper is described as follows: (i)To obtain an edge fault-tolerant VB in a WSN, we introduce a new concept called the -ECDS(ii)A centralized approximation algorithm is designed for the construction of a -ECDS in a given UDG, whose performance ratio is proven to be 30.51(iii)The designed algorithms are supported by strict theoretical analysis and simulation experiments. In the theoretical analysis, we conduct strict mathematical proof of the correctness of these algorithms. By means of simulation experiments, we illustrate that the CDSs output by our algorithms outperforms those obtained with the state-of-the-art algorithms presented in [25, 26]

The rest of this paper is organized as follows. In Section 2, related research is introduced. In Section 3, an introduction of the definitions, terms, and related results used in the remaining sections is provided. Section 4 presents an approximation algorithm for constructing a -ECDS. Section 5 reports simulation results on the comparison between our algorithm and other related algorithms. Finally, in Section 6, we summarize our work and introduce potential directions for future work.

A WSN can be abstracted as a graph ; correspondingly, the VB of the WSN can be abstracted as a CDS of . In recent years, numerous efforts have been made to construct MCDSs. In a general graph, Guha and Khuller [28] proved that for the MCDS problem, it is impossible to obtain a polynomial time approximation algorithm with a performance ratio for any unless , where is the number of nodes in the graph. They introduced a -approximation algorithm and a -approximation algorithm based on a greedy strategy, where is the maximum degree of nodes in the corresponding network graph and is the harmonic function. In [29], Du et al. improved this result and presented an -approximation algorithm , which is the best result in general graphs thus far.

For UDGs, the MCDS problem has been widely investigated, and a variety of results have been obtained [26, 3039], in which different approximation algorithms have been proposed for this problem. These algorithms usually consist of two phases: (1) a maximal independent set (MIS), which serves as a dominating set in a given UDG, is determined, and (2) some nodes are selected into the MIS such that it becomes connected. Researchers analyzed the performance ratio of the resulting CDS from two aspects: the size of the MIS obtained in the first phase and the number of newly added nodes in the second phase. For the MIS size, in [33], Wan et al. demonstrated that the upper bound of an MIS is in a UDG, where represents the size of the MCDS in the UDG. Later, this upper bound was improved by [30, 3437], where [30] obtained an upper bound of , which is the best result thus far. For the size of the newly added nodes, in [33], Wan et al. showed that the size bound generated by their algorithms in the second phase is , where is the MIS obtained in the first phase. This result was used by [34, 35] to evaluate the performance ratios of their algorithms. In [31, 38], the Steiner tree strategy was used to design approximation algorithms for the MCDS problem, and the size of the newly added nodes in the second phrase was no more than , where is the number of Steiner nodes in the minimum Steiner tree containing the MIS obtained in the first phase. To our knowledge, this is the best result thus far.

In research on the construction of node fault-tolerant VBs, Dai and Wu [17] introduced a concept called the -CDS. For a -CDS in a given , after nodes are deleted from , the set of residual nodes in the -CDS is still a CDS for the subgraph induced by the residual nodes of . They presented three algorithms to compute a -CDS for a given UDG. Unfortunately, they did not present the performance ratios of these three algorithms. In [18], Thai et al. extended this concept to the famous -connected -dominating set, where . In their paper, a centralized approximation algorithm was provided for the minimum -CDS problem. Since then, numerous constant factor approximation algorithms for the construction of -CDSs have been proposed. In [40], Wu et al. presented a distributed algorithm with a performance ratio of . In [25], Shang et al. proposed another approximation algorithm, whose performance ratio is or 11 , for constructing a -CDS. In [26], Shi et al. noted mistakes in the proof process of the performance ratio of the algorithm in [25] and found that the ratio should be or 21 rather than or 11 . Moreover, Shi et al. [26] used a submodular function and greedy strategy to design a new approximation algorithm with a performance ratio of , where is the performance ratio of the algorithm for constructing the -CDS. In 2010, Kim et al. [23] introduced a approximation algorithm for computing a -CDS in a UDG, where is the performance ratio of the algorithm for the -CDS () problem. In [41], Liu et al. used Tutte decomposition to design a approximation algorithm to calculate a -CDS () in a UDG, where is the performance ratio of the algorithms for the -CDS () problem. When , they noted that the performance ratio of their algorithm is 62.30, which is better than the 580/3 performance ratio achieved by [23]. In [8], Wang et al. introduced the concepts of “good points” and “bad points” and proposed an algorithm to calculate a -CDS in a given UDG, where and . The main idea of their algorithm is to first obtain a -CDS by means of an existing algorithm and then convert all degree- bad points in the -CDS into good points by adding “-paths.” Second, they construct a simplified auxiliary graph for the CDS obtained in each iteration. Finally, the -CDS is constructed by means of the simplified auxiliary graph. They achieved a performance ratio of for the -CDS problem ( for the -CDS problem), where is the performance ratio of the algorithm proposed in [26] for -CDS problem .

In recent years, research on link failure in WSN communication has attracted considerable attention. In [42], Zonouz et al. proposed a model, called time-dependent lognormal shadowing radio propagation, to analyze and estimate the reliability of a wireless link under consideration of the power consumption in different modes and wireless channel conditions. In [43], Fu et al. noted that the capacity of links is significantly limited in most WSNs. They believed that when the load of a few links in the network exceeds their capacity, link failures may occur, which may cause paralysis of the global network. Later, in [44], Draz et al. provided a link failure detection algorithm to proactively diagnose faulty links in a WSN and introduced a recovery mechanism against link failures. In [45], Xu et al. used independent Bernoulli processes to describe link failures in a WSN and employed a Kalman-consensus filter and CI algorithm to estimate these link failures through a hierarchical fusion method. In [46], to improve the robustness of VBs, Liang et al. proposed a new concept, robust VB, and corresponding algorithms against link failures in WSNs with unstable transmission ranges. In their work, a more robust VB, named the -robust CDS, is obtained by their algorithms, and based on this VB, the communication in the corresponding WSN does not fail when , where is the effective transmission range of the nodes and is the initial transmission range of the nodes.

As described above, link failures can impact WSN communications. A natural idea is to investigate VBs with link fault tolerance, namely, the problem of edge fault-tolerant VBs against link failure. Unfortunately, research on the construction of edge fault-tolerant VBs in WSNs remains scarce. In this paper, we consider this problem and introduce the concept of the -ECDS for UDG, which is a kind of edge fault-tolerant VB in the corresponding homogeneous WSN, and propose an approximation algorithm for the construction problem of the -ECDS.

3. Preliminaries

For convenience, some concepts, notations, and knowledge related to graph theory are now introduced. In this paper, all considered graphs are connected unless otherwise specified.

3.1. System Model and Problem Definition

In this paper, we adopt the following assumptions: the sensors in the considered WSN are randomly placed in a given area and immobile thereafter; each sensor has the same transmission radius and initial energy; two sensors can communicate with each other if and only if one of them is within the transmission range of the other; and in the considered WSN, each link between sensors is bidirectional. We abstract each sensor of the WSN as a node of a graph and each communication link between two sensors as an edge between the corresponding nodes. Thus, a WSN is modeled as an undirected connected graph .

The problem that we address in this paper is called the minimum-ECDS problem. For the sake of understanding the minimum -ECDS problem, we first introduce some necessary concepts related to this problem.

Definition 1 (2 edge-connected graph). A graph is called a 2 edge-connected graph if for any , is connected, where is the spanning subgraph induced by edge set . In particular, a graph with a single node is defined as a 2 edge-connected graph.

Definition 2 (2 edge-dominating set). In a given graph , a subset is called a 2 edge-dominating set if for each node , .
In a given UDG , let and . If , then we say that is 2 edge-dominated by .

Definition 3 (2 edge-connected 2 edge-dominating set). In a given UDG , a CDS is a -ECDS if the following conditions hold: (1) is a 2 edge-dominating set of (2) is a 2 edge-connected graph

A -ECDS with the minimum cardinality is called a minimum -ECDS.

Definition 4 (minimum -ECDS problem). For a given graph , the minimum -ECDS problem is the problem of constructing a minimum -ECDS for .
Note that because the MCDS problem for a graph is NP-hard, the minimum -ECDS problem for a graph is also NP-hard. In this paper, we will design an approximation algorithm for the minimum -ECDS problem.

3.2. Notations

In this subsection, we use to denote a UDG. (i): the spanning subgraph induced by , namely, (ii): the subgraph induced by (iii): the edge subset, where each edge is incident to , namely, (iv): , where (v): the set of neighbor nodes of (vi): (vii): the ID of node (viii): the energy of node (ix): the set consisting of all edges of (x): the set consisting of all nodes of

3.3. Definitions

Definition 5 (weight function ). Given two 3-tuple variables and , for a weight function , if and only if at least one of the following conditions is true: (i)(ii) and (iii) and and

Definition 6 (bridge). In a given UDG , is called a bridge of if is disconnected.
According to Definitions 1 and 6, a given UDG is a 2 edge-connected graph if and only if each edge is not a bridge of .

Definition 7 (edge-disjoint). In a given UDG , let and be two different paths. Then, and are called edge-disjoint if .

Definition 8 (simple cycle). In a given UDG , let be a path of . The path is called a simple cycle in if and .
According to Whitney’s theorem [47], the proposition that for a given , is a bridge of is equal to any one of the following propositions. (1)In , such that all paths between and contain edge (2) has no simple cycle containing

Definition 9 (-block). In a given UDG , let . is a -block of if one of the following conditions holds: (1) consists of a single node, and for any nonempty subset , is not 2 edge-connected(2) consists of at least two nodes; is a 2 edge-connected graph; and for any nonempty subset , is not 2 edge-connected(3) and is a 2 edge-connected graph

Remark 10. According to the definitions of the -block and 2 edge-connected graphs, for a given graph UDG and a set , we have the following conclusions: (1)If is a -block of , then is 2 edge-connected blocks(2) is the unique -block of if and only if is 2 edge-connected

Definition 11 (path). In a given UDG , let and . A path from to with nodes is called a path if its inner nodes belong to .

Lemma 12. In a given UDG , for any , there exist two paths that are edge-disjoint between and if and only if is a 2 edge-connected graph.

According to Definition 7 and Whitney’s theorem, the proof of Lemma 12 is easy to obtain.

4. Algorithm and Analysis

In this section, we present an algorithm (called LLZ20) to construct a -ECDS in a 2 edge-connected UDG . LLZ20 includes three main parts. First, a CDS in is constructed with the two-stage approximation algorithm (called ACC11) in [32]. Next, a connected 2 edge-dominating set is calculated by adding some nodes to the CDS obtained in the first part; this can be implemented by calling Algorithm 2, whose flow chart is shown in Figure 3. Finally, paths are added to connect the nodes of the connected 2 edge-dominating set obtained in the second part to make it a ()-ECDS; this can be implemented by calling Algorithm 3, whose flow chart is shown in Figure 4. In Algorithm 3, to compute -blocks in a subgraph, we need to call another algorithm, Algorithm 4, whose flow chart is shown in Figure 5. LLZ20 is described formally in Algorithm 1.

Input: A 2 edge-connected UDG .
Output: A -ECDS .
1 Construct a CDS with ACC11 in .
2 Construct a connected 2 edge-dominating set and a pseudocontrol node set with Algorithm 2 based on .
3 Construct a -ECDS with Algorithm 3 based on and .
4 return.
Input: A connected graph and a CDS of .
Output: A connected 2 edge-dominating set and a pseudocontrol node set .
1 
2 for each node do
3  Ifthen
4  
5  end if
6 end for
7, ,
8 whiledo
9  Find a node with the highest .
10  ifthen
11  
12  end if
13  and .
14 end while
15
16 return and .
Input: A 2 edge-connected graph , a connected 2 edge-dominating set in and the corresponding pseudocontrol node set .
Output: A -ECDS of .
1 Color all nodes white and assign each node a ID (called Eblock ID, denoted by ). Initially, let for each node .
2 Let , and calculate all 2E-blocks with Algorithm 4 in .
3 while the number of 2E-blocks in is more than 1 do.
4 if there exists an path connecting at least two 2E-blocks, then
5  Find an path whose inner node connects the most 2E-blocks.
6  . Recompute the 2E-blocks in with Algorithm 4.
7 else
8  Find an path such that the number of 2E-blocks connected by its inner nodes is the most.
9  , where and are the inner nodes of . Recompute the 2E-blocks in with Algorithm 4.
10 end if
11 end while
12 and
13 for each node in do
14 if is 2-edge connected then
15  
16 end if
17 end for
18 return
Input: A 2 edge-connected graph , a colored subgraph of and , where .
Output: The set consisting of all 2E-blocks in .
1 Construct a BFS tree in , and let be the set of edges in . Let , .
2 for to do
3  Select a node with the smallest in .
4  ifthen
5  if is a white node then
6  Color blue, and .
7  end if
8  else
9  Let
10  for each edge do
11  Find a simple circle , whose edges are all in the edge set , and let be the set of all nodes in .
12  if there are no red nodes in then.
13  Let be the minimum ID of nodes in .
14  For each , let .
15  Color all nodes in red.
16  else
17  Let . Let and .
18  For each , .
19  Color all nodes in red.
20  for each do
21  For each node with , .
22  end for
23  end if
24  end for
25  .
26 end if
27 .
28 end for
29 Let and . Let
30 return.

For readers to understand LLZ20 more intuitively, an example network graph is presented in Figure 6 to illustrate the steps of LLZ20. For convenience, let the initial energy of each node be the same. (1)For a given WSN, which is abstracted as the network graph (shown in Figure 6(a)), LLZ20 calls ACC11 to construct a CDS in (line 1). In this process, , , and are selected consecutively to form a CDS, and then, pseudocontrol node is deleted because does not affect the connectivity of the CDS. When this step is complete, (blue nodes shown in Figure 6(b)) is the resulting CDS (2)In line 2 of LLZ20, Algorithm 2 is called to calculate a dominating set (green nodes shown in Figure 6(c)) in . Before Algorithm 2 is executed, nodes , , , , and are not 2 edge-dominated by the CDS . When Algorithm 2 is executed, and are consecutively selected to obtain a connected 2 edge-dominating set , and the pseudocontrol node set is an empty set (see Figure 6(c) for details)(3)In line 3 of LLZ20, Algorithm 3 is called to calculate the set , which consists of the inner nodes of the selected path. Specifically, before Algorithm 2 is executed, is not 2 edge-connected. When Algorithm 2 is executed, the path and the path are consecutively selected; in other words, and are consecutively selected into the set . After this step is finished, is the resulting -ECDS (see Figure 6(d) for details)

4.1. Analysis of Algorithm Correctness

In this subsection, we provide proof of the correctness of our algorithms and show that a -ECDS must be obtained after executing LLZ20. For this purpose, we need to show only that the following three results are true: (1) ACC11 in [32] can generate a CDS, (2) Algorithm 2 can generate a connected 2 edge-dominating set, and (3) Algorithm 3 can generate a -ECDS.

According to [32], the first result is true. Next, we show the correctness of the second result that Algorithm 2 can generate a connected 2 edge-dominating set.

Lemma 13. After executing Algorithm 2, the output is a connected 2 edge-dominating set.

Proof. We need to prove that is connected and that is a 2 edge-dominating set. First, since is a CDS and , is connected. Next, we prove that is a 2 edge-dominating set. Let , where (, respectively) is (, respectively) obtained after executing line 15. Then, . Let ; then, . We claim that there are two edges with a common endpoint such that their other endpoints are in . If , then according to lines 2-7, , our claim is true. If , then according to lines 8-14, after executing line 14, is an MIS in , which means there is an edge with one endpoint and another endpoint belonging to . On the other hand, since is a CDS, there is an edge with endpoint and another endpoint belonging to . Hence, our claim is true. According to the above analysis, is a 2 edge-dominating set.

According to the above lemma and lines 10-12 of Algorithm 2, it is easy to obtain the following corollary.

Corollary 14. After Algorithm 2 is executed, for any , is a 2 edge-dominating set in .

Now, we show that (the result of Algorithm 3) is a -ECDS. Before we present the proof, let us first introduce some definitions and lemmas used in the following discussions. The following preliminaries, which serve as proof of the correctness of the results, are necessary.

For node chosen in line 3, let (line 9), be a set of red nodes with the same after executing the th iteration of the for-loop (lines 10-24) in Algorithm 4, which is the iteration for (line 10). Let be the number of of the red nodes in (line 11) before executing the th iteration of the for-loop (lines 10-24), and let be the subset of that consists of all nonred nodes in (line 11) before executing the th iteration of the for-loop (lines 10-24) in Algorithm 4. When , let be the nonempty subset of , which consists of all red nodes with the same before executing the th iteration of the for-loop (lines 10-24) in Algorithm 4. Then, . In this case, is called the TC decomposition of , is called a red TC decomposition piece of , and is called a nonred TC decomposition piece of . Clearly, if there is no TC decomposition in , then .

For a better understanding of TC decomposition, let us consider the network graph shown in Figure 7, which is one part of the network obtained before executing the th iteration of the for-loop (lines 10-24) in Algorithm 4, where the integer is determined as follows. Assume that in network graph , is the node chosen by line 3 in Algorithm 4 and , the th iteration of the for-loop (lines 10-24) is referred to as the iteration, in which the edge is considered (line 10). Assume that is the simple circle determined by line 11 in the th iteration, where and before the th iteration of the for-loop (lines 10-24). Then, , , , , . Finally, is a TC decomposition of .

Lemma 15. If there is no TC decomposition in , then contains a simple circle with all nodes of .

Proof. According to the above discussion, it is easy to see that the result is true.

Lemma 16. Suppose that and is the TC decomposition of , where . If is 2 edge-connected, then is 2 edge-connected.

Proof. According to the definition of and line 19, , where is the node set of the simple circle selected in line 11 of Algorithm 4. Clearly, . We claim that . Otherwise, after executing the th iteration of the for-loop (lines 10-24) in Algorithm 4, the s of the nodes in are not equal to , which implies that is not contained in , a contradiction. Let and be the edge sets in and , respectively. Construct a new graph , where , . Clearly, is connected: we claim that is 2 edge-connected. We need to prove only that for any edge , is connected.

Case 1. . Since is 2 edge-connected, is still connected. According to and the fact that is connected, we have that is connected.

Case 2. . Since is a simple circle, is still connected. Note that and are 2 edge-connected ; thus, we have that is connected.

Hence, is 2 edge-connected. Since is a spanning subgraph of , is 2 edge-connected.

Lemma 17. After executing line 29 in Algorithm 4, is a 2-block in .

Proof. According to line 6, lines 14-15, and lines 18-22, consists of red nodes with the same or a single blue node. Consider the following cases.

Case 1. consists of a single blue node. Let be this node. Clearly, is not in any simple circle in (otherwise, according to lines 10-24, would be colored red). We claim that for any nonempty subset , is not 2 edge-connected. In contrast, assume that there exists a nonempty subset such that is 2 edge-connected. Let ; then, there exist two edge-disjoint paths between and in that form a simple circle and contain node , a contradiction. Thus, this claim is true. According to the definition of a -block, is a -block.

Case 2.