On “Movement-Assisted Connectivity Restoration in Wireless Sensor and Actor Networks”

Abstract

In wireless sensor and actor networks (WSANs), a set of static sensor nodes and a set of (mobile) actor nodes form a network that performs distributed sensing and actuation tasks. In [1], Abbasi et al. presented DARA, a Distributed Actor Recovery Algorithm, which restores the connectivity of the interactor network by efficiently relocating some mobile actors when failure of an actor happens. To restore 1 and 2-connectivity of the network, two algorithms are developed in [1]. Their basic idea is to find the smallest set of actors that needs to be repositioned to restore the required level of connectivity, with the objective to minimize the movement overhead of relocation. Here, we show that the algorithms proposed in [1] will not work smoothly in all scenarios as claimed and give counterexamples for some algorithms and theorems proposed in [1]. We then present a general actor relocation problem and propose methods that will work correctly for several subsets of the problems. Specifically, our method does result in an optimum movement strategy with minimum movement overhead for the problems studied in [1].

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On “Movement-Assisted Connectivity
Restoration in Wireless Sensor and Actor
Networks”
ShiGuang Wang⋆, XuFei Mao⋆, Shao-Jie Tang⋆, XiangYang Li†⋆‡, JiZhong Zhao‡, GuoJun Dai†
†Institute of Computer Application Technology, Hangzhou Dianzi University, Hangzhou, PRC.
⋆Department of Computer Science, Illinois Institute of Technology, Chicago, USA.
‡Department of Computer Science and Technology, Xi’an Jiaotong University, PRC.
Abstract—In wireless sensor and actor networks (WSANs), a set of static sensor nodes and a set of (mobile) actor nodes form
a network that performs distributed sensing and actuation tasks. In [1], Abbasi et al. presented DARA, a Distributed Actor Recovery
Algorithm, which restores the connectivity of the inter-actor network by efficiently relocating some mobile actors when failure of an actor
happens. To restore 1- and 2-connectivity requirements, two algorithms are developed in [1]. Their basic idea is to find the smallest
set of actors that needs to be repositioned to restore the required level of connectivity, with the objective to minimize the movement
overhead of relocation. Here, we show that the algorithms proposed in [1] will not work smoothly in all scenarios and we give counter-
examples for some algorithms and theorems proposed in [1]. We then present a general actor relocation problem and propose methods
that will work correctly for several subsets of the problems. Specifically, our method does result in an optimum movement strategy with
minimum movement overhead for the problems studied in [1].
Index Terms—Connectivity restoration, controlled node mobility, fault tolerance, wireless sensor and actor networks
!
1INTRODUCTION
In recent years, we have witnessed a growing research in-
terest and practical applications of wireless sensor networks
(WSNs). In a WSN, a group of sensors are deployed in a
region to gather information about the physical world. The
data could be collected and then aggregated up toward some
sink nodes for further processing. To further improve the
efficiency and effectiveness of wireless sensor networks, actor
nodes have been introduced and integrated in the wireless
sensor networks to form wireless sensor and actor networks
(WSANs). Unlike small wireless sensor nodes, actors are
usually resource-rich devices equipped with better processing
capabilities, stronger transmission powers and longer battery
life. In WSANs, (potentially mobile) actor nodes are deployed
to perform appropriate actions upon the environment data
collected by the sensors, which allows remote, automated
interaction with the environment. Unlike traditional control
system, in a WSAN network, an actor typically not only is
capable of perform individual tasks (such as deactivating a
landmine, extinguishinga fire, and rescuing a trapped survivor)
Emails: {swang44,xmao3,stang7}@iit.edu, xli@cs.iit.edu,
zjz@mail.xjtu.edu.cn, daigj@hdu.edu.cn.
The research of authors are partially supported by NSF CNS-0832120,
National Natural Science Foundation of China under Grant No. 60828003,
No.60773042, the Natural Science Foundation of Zhejiang Province under
Grant No.Z1080979, program for Zhejiang Provincial Key Innovative Re-
search Team, program for Zhejiang Provincial Overseas High-Level Talents
(One-hundred Talents Program), National Basic Research Program of China
(973 Program) under grant No. 2010CB328100, and the National High
Technology Research and Development Program of China (863 Program)
under grant No. 2007AA01Z180.
in response to the environment, it also is able to collaborate
with other actors in a networked system. Applications of
wireless sensor and actor networks thus may include a group
of mobile robots that will optimize their actions based on
data collected from the network. A WSAN will perceive the
environment from multiple disparate viewpoints based on the
data gathered by a sensor network. Since the actor nodes will
perform certain tasks immediately after collecting data from
sensor nodes, the issue of real-time communication is very
important in WSANs.
In many applications, the number of sensor nodes deployed
in studying a phenomenon may be in the order of hundreds or
even thousands. However, such a dense deployment is often
not necessary for actor nodes because of the different coverage
requirements and physical interaction methods of acting task.
In addition, actor nodes are often capable of moving from one
place to another place to perform a certain task in response to
the changing environment. Hence, in order to provide effective
sensing and acting, a distributed local coordination mechanism
is necessary among sensors and actors. In most applications,
it is often required that the network topology of a WSAN
should be a connected graph. Here each node (an actor or
a wireless sensor node) is mapped to a vertex in a graph,
and two vertices are connected if the corresponding nodes can
communicate directly with each other using wireless channels
without relying on relays by other nodes. A network is said
to be k-connected if for every pair of nodes, there are k
node disjoint paths connecting these two nodes. A k-connected
network, for k ≥ 2, is often said to be (k − 1)-fault-tolerant,
for fault-tolerant for simplicity. When the network topology

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is not fault-tolerant, an actor failure can cause the loss of
multiple inter-actor communication links and may partition
the network if alternate paths among the affected actors are
not available. So for a fault-tolerant network, bi-connectivity is
required. In a bi-connected network, the removal of any node
from the network will not partition the network into connected
components.
In the recent work [1], Abbasi et al. presented DARA, a
Distributed Actor Recovery Algorithm, which tries to effi-
ciently restore the connectivity of the inter-actor network that
has been affected by the failure of an actor. To address 1-
and 2-connectivity requirements, two variants of the algorithm
are developed by them (DARA-1C and DARA-2C). Due to
the fact that the energy is limited in WSANs and normally
movement of nodes consumes huge energy, compared to
communication and computation processes, their basic idea is
“ to identify the least set of actors that should be repositioned
in order to reestablish a particular level of connectivity”.
DARA strives to localize the scope of the recovery process
and minimize the movement overheadimposed on the involved
actors, although they did not explicitly prove and verify the
performances of their algorithms in terms of this metric.
Unfortunately, the paper [1] contains some significant discrep-
ancies that affect the correctness of the algorithms. Here we
demonstrate it by presenting some counter-examples to some
theorems and algorithms. We also present methods that can
fix some of the discrepancies in this article. We then formally
define the connectivity-restoration problem and show that it
can be solved in polynomial time if the number of candidate
positions of nodes is at most n+c for some constant c, for a
network of n nodes. Observe that, for all results presented in
[1], after an actor node fails, they will move some actor nodes
towards the old positions of some other actor nodes. Thus,
for a network of n actor nodes, after 1 actor node failed, the
number of candidate positions of actor nodes are still n. Thus,
the method proposed by us will result in an optimal solution
in the setting [1].
Designing wireless networks that is fault-tolerant, or k-
connected, and reconfiguring the network (by moving some
nodes or adding new relays) to be k-connected have been
studied extensively in the literature. Li et al. [14], [15] showed
how many nodes are needed to get a k-connected network with
high probability with random node placement and proposed
localized topology control method to retain k-connectivity. Yi
et al. [16] studied the asymptotic distribution of the number
of isolated nodes with Bernoulli failures in a random network.
Several results were proposed in the literature to get a new
k-connected network after some node failures, by moving
existing alive nodes or adding new nodes. Li et al. [13] proved
that it is NP-hard to add the minimum number of nodes
to improve the network connectivity. Almasaeid et al. [3]
proposed a method with objective of minimizing the number of
additional nodes needed to repair the connectivity. Ibrahim et
al. [11] also discussed network repair problem using minimum
number of additional relays. Ibrahim et al. [10] characterized
the network connectivity by Fiedler value, i.e., 2nd smallest
eigenvalue of the Laplacian matrix of the network. They
used the semi-definite-programming approach to find the best
locations of relay nodes to improve the connectivity. Given
a network, Kashyap et al. [12] proposed a polynomial time
method for placing some additional relay nodes for getting a
k-connected graph. The approximation ratio of their method
is 2DMST for nodes in d dimensions, where DMST is the
maximum degree of a minimum degree minimum spanning
tree of all nodes. Here DMST = 5 for two dimensional nodes.
Given the set of duty sensors in the plane and the upper
bound of the transmission range, Cheng et al. [5] proposed two
methods to compute the minimum number of relay sensors
such that the induced topology by all sensors is globally
connected. The first algorithm has a performance ratio of 3
and the second has a performance ratio of 21
a survey of various methods for constructing minimum cost
k-connected geometry networks.
Besides adding extra relay nodes, some approaches try to
move the existing nodes to improve the connectivity of net-
works. Das et al. [7] proposed a seminal localized movement
control algorithm for wireless mobile robot networks to estab-
lish/restore bi-connectivity using p-hop neighbor information.
Atay et al. [4] used k-redundancy to check and repair the
k-connectivity. Abbasi et al. [1] proposed several methods to
move nodes around to improve the network connectivity. They
assume that every actor node already has the information about
all nodes within two-hop.
This article is organized as follows. We review some def-
initions, and algorithms presented in [1] in Section 2, and
discuss some of the results of [1] in Section 3. We also
present our method that can optimally re-allocate nodes with
the minimum cost to restore the k-connectivity of the network
if it is possible. We conclude the article in Section 4.
2. See [6] for
2
In this section, we briefly review the paper [1]. The paper
[1] focused on the connectivity reestablishment problem in
Wireless Sensor and Actor Networks (WSANs) caused by
single node failure. In WSANs there are two kinds of nodes:
mobile/static sensor nodes and mobile actor nodes. Mobile
actor nodes act as the backbone of a WSAN; all the sensor
nodes transmit their data to the nearby actor nodes and actor
nodes forward the data to the sink node where the data is
processed. Due to a actor node’s failure, the connectivity of
actor nodes could be destroyed. So how to reestablish the
connectivity after one actor node damaged it naturally comes
forth. Note that [1] did not consider the coverage problem;
only connectivity reestablishment problem was focused.
In [1], the authors proposed two distributed algorithms
DARA-1C and DARA-2C to address the 1- and 2-connectivity
reestablishment problem, which we found not always valid.
Both DARA-1C and DARA-2C are real-time and perform
cascaded relocation of actor nodes. The input of DARA-1C
(resp. DATA-2C) is the 1-connected (resp. 2-connected) actor
network graph. In DARA-1C each actor node Aimaintains its
two-hop neighbors list, the set 2-hop-Neighbors(Ai), by using
periodic heartbeat messages. It is assumed that all nodes will
maintain an updated list of its two-hop neighbors. Missing
heartbeat messages can be used to detect the failure of actor
REVIEW OF RESULTS & DEFINITIONS IN [1]

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nodes. Note that not the failure of any actor node will destroy
the connectivity; only a cut vertex of the network graph of
the actor network will destroy it. Here we call an actor node
which is a cut vertex of the network graph simply as a cut
vertex. A vertex v in a connected graph G is called cut vertex
if the removal of v will partition G. However, the paper [1]
did not mention how to distinguish cut and non-cut vertices.
When a cut vertex Af fails, its heartbeat message is lost,
and thus its neighbors will discover the failure and a recovery
procedure happens. So it is needed to choose an actor node
ABC (called best candidate node) and relocate it to restore
connectivity. The best candidate ABCis chosen from one-hop
neighbors of Af, by the following criteria in order: Least node
degree, Closest proximity to the failed actor, Highest actor ID.
The relocation method is as follows:
1) Node ABCsends a “MOVING” message to its neighbors,
in the set Dependents(ABC,Af) , i.e. those one-hop
neighbors of ABCthat are not one-hop neighbors of Af,
about its movement and the time it will take to reach to
the new location.
2) Node ABCmoves to the exact location of the failed node
Af,
3) Node ABCbroadcasts “RECOVERED” message. Notice
that since other nodes are still in their previous positions,
this broadcast “RECOVERED” message can only be
received by nodes in the same connected component as
the node ABC at the position of failed node Af.
When relocation of ABC causes dis-connectivity of actor
network, another round of relocation happens using the same
method. The detection of network dis-connectivity caused by
the relocation of ABC is performed as follows:
1) The dependent neighbors (children) of ABCkeep waiting
until they receive the “RECOVERED” message indicat-
ing the restoration process has been completed;
2) If they did not hear the “RECOVERED” message within
the time period of 2T where T is the time needed by
the node ABC to travel for maximum possible distance
r, they assume dis-connectivity happens and do the
restoration process just as their parent ABC did.
3) Each node will move only once.
Note that different from the original relocation, here the new
best candidate in the children level is chosen among the
dependent neighbors of the previous best candidate ABC. In
paper [1], the authors also talked about an optimization of
DARA-1C for ring-like topologies; however, the it does not
increase the asymptotic bound of converge time.
For restoring 2-connectivity after a failure of an actor node,
the authors proposed DARA-2C. First, the authors gave several
definitions of key terms which are recalled as follows:
Definition 1 [1]: The geometric convex hull [9] is the mini-
mal convex set of points containing a nonempty finite set
of points.
Definition 2 [1]: A network periphery is the convex hull of
the nodes of the network. Since the nodes are placed in a
plane, the network periphery is a simple closed polygonal
chain, unless the nodes are collinear.
Definition 3 [1]: A connectivity hole [8] is an area in which
the edges between nodes form a closed polygonal without
links between nodes that are not adjacent on the polyg-
onal chain.
Definition 4 [1]: A boundary node is one that is located at
the network periphery or on the closed polygonal chain
surrounding a connectivity hole.
The definitions are illustrated in Fig. 1, which is a similar
illustration as that in [1]. Observe that their original definitions
1 (on convex hull), 2 (on network periphery), 4 (on boundary
nodes) are clear and self-contained. However, definition 3
is not clear and confusing. For special network examples,
such as the one illustrated by Fig. 1, the definitions are
correct. Later, however, we will show that the definitions of
network periphery and connectivity hole are not correct in
all scenarios; some counter-examples will be given. Thus the
definition of boundary node is also incorrect. Their original
definition of network periphery is not correct, but could be
fixed. Unfortunately, we could not find a correct way to fix
their original definition of connectivity hole, and consequently
the definition of boundary nodes, to make their algorithms
correct.
Network Periphery
A1
A2
A3
A4
A5
A6
A7
8
A
A9
A10
A11
A12
A13
A14
A15
A16
A17
A18
A19
A20
A21
A22
A23
A24
A25
A26
A27
Connectivity Hole
Fig. 1.
A1,A2,...,A18 is the convex hull of the network graph
and it is also the network periphery by Definition 2 in
[1]. By Definition 3 in [1], the polygonal area constructed
by vertices A8,A9,...,A27is the connectivity hole of the
network (which is a similar illustration as Fig. 6 in [1]). The
network boundary node is all the vertex A1,A2,...,A27,
for that they are either located on the network periphery
or on the connectivity hole.
The convex polygon constructed by vertices
The following two key theorems are the foundation based on
which DARA-2C is proposed, which are recalled as follows:
Theorem 6 [1]: In a 2-connected network, a failure of a node
Af can cause another node AC to be a cut vertex only
if Af is a boundary node.
Theorem 7 [1]: In a 2-connected network, the biconnectivity
lost due to a failure of a node Af can be restored by
repositioning the neighbors of Af that are located at the
network periphery or on the boundary of a connectivity
hole.
In DARA-2C, each actor node knows its one-hop and
two-hop neighbors as in DARA-1C, which includes the IDs,
locations, and node degrees of these neighbors. The paper [1]

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employs the localized boundary detection algorithm proposed
by [17] to determine whether an actor node is a boundary node
or not. We point out here that the traditional network boundary
detection method will not work for the definition of network
boundary proposed in [1]. The reason is that traditional
methods typically only detect the periphery of the network.
We later will also point out that the definition of periphery [1]
is also problematic. [1] also employs the cut-vertex detection
method proposed by [2], which probabilistically determines
whether a node is a cut vertex or not.
Now we briefly review DARA-2C. As in DARA-1C, the
failure actor node is detected by heartbeat message. Based
on Theorem 7 in [1], the restoration process will be initiated
only on every actor node Aj ∈ Neighbors(Af), based on
the following criteria: (1) Af is a boundary node, (2) Aj
is a boundary node, and (3) the failure of Af introduces
a cut vertex in the network. Nodes satisfy these conditions
are referred to as candidates thereafter. DARA-2C restores
the 2-connectivity of a network by relocating one of these
candidates. Two problems are addressed in DARA-2C as that
in DARA-1C: (1) which candidate node ABC(Best Candidate)
is chosen to relocate, (2) how to relocate ABC. ABC is
chosen by the following three criteria in order: (1) Lowest
node degree, (2)Least distance, and (3)Highest actor ID. The
relocation of ABC works as follows:
1) If the cardinality of the candidate set is two, the selected
ABC will move towards the other candidate until it
becomes within its radio range.
2) If the cardinality of the candidate set is three or more,
the ABC will move to the position of Af.
3) Any node will move only once.
Similar to DARA-1C, when ABC stops, it broadcasts a
“RECOVERED” message indicating the completion of the
restoration process. The neighbors of ABC will keep waiting
for the “RECOVERED” message; if they received it, they
conclude that they are still connected, otherwise they will
assume that they got disconnected and apply Theorem 7 in [1]
again as if ABCstopped functioning. This means that a node,
Aj ∈ Neighbors(ABC) will move only if it is a boundary
node. The recovery process will be applied recursively to
trigger the cascaded relocation of affected actors. Each of these
boundary nodes will individually move toward ABCand non-
boundary neighbors of ABC do not need to move.
3
METHOD
In this section, we will briefly discuss some discrepancies
found in some methods presented in [1]. We will then fix
some discrepancies in [1] and later formally propose the
connectivity-restoration problem.
SOME DISCREPANCIESIN [1] AND NEW
3.1
The basic idea of the algorithm proposed in [1], called DARA-
1C, for 1-connectivity case is reviewed in Section 2. We found
that there is some problem when apply this algorithm. We
describe one network example in Fig. 2. In [1], the authors
stated the method to choose a new best candidate in the child
Discrepancies in Algorithm DARA-1C
Af
ABC
A5
A6
A4
A1
A2
A3
Fig. 2. The dependents set of ABCis {A1,A2}, and the
detached dependents set of ABCafter ABCrelocated is
{A1,A2}, but A1 and A2 themselves do not know this
set. The dependents set, denoted by Dependents(Ai,Af)
[1], is the set of actors that are neighbors of Ai but not
neighbors of Af. The detached dependents set of ABCis
the subset of Dependents(ABC,Af) which becomes not
connected to the ABCafter ABCrelocated to the position
of Af.
level of the old best candidate ABCas “Thus, these detached
dependents identify a BC at the children level to relocate to
the position of their parent.”, which means that the new best
candidate is chosen from the set of all the detached dependents
of the old best candidate ABC. But how can each detached
dependent node know this detached dependents set? First, the
old best candidate ABC cannot tell which dependent will be
detached before its relocation, which is because that ABConly
maintains its 2-hop neighbors’ information. In Fig. 2, ABC
does not know that A3is connected with Af by another path,
because that A6 is not in the 2-hop neighbor table of ABC.
Second, the detached dependents could not be connected to the
network after ABC relocated and possibly they are not con-
nected with one another either, which means that they cannot
communicate with each other under some possible situation
as A1 and A2 in Fig. 2. Thus, this detached dependents set
cannot be learned by these detached dependents under some
situation, which leads to the case that the selection of a new
best candidate after ABC relocated cannot be processed. In
other words, detached dependents of ABC cannot make a
uniform decision on the new best candidate node. Thus the
goal of [1] (Section 4.3.3 of [1]) “...preventing conflicts in
the selection of the BC” could not be achieved under some
situation.
We next make a brief summary: Any best candidate node
like ABC can not be simply treated as a failure node. The
main reason is for a real failure node Af, all of its neighbors
must have the same observation regarding the failure of Af.
On the other hand, since any node like ABC does not really
fail but just change its location, it will cause the inconsistency
among the observation among its original neighbors. As shown
in previous example, the detached dependents set cannot be
computed exactly among all the detached dependents.
3.2
Here we propose a new approach in order to fix this problem.
Before the node ABCmove to some new location, it should
unicast Dependents(ABC,Af) (as {A1,A2,A3} in Fig. 2) to
all its dependents. And based on the criteria of selecting the
Our method for restoring 1-connectivity

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best candidate, each dependent can get a priority number, for
example, if A3should be the best candidate after ABCmoved,
then A3got the number 1, and A2is the next candidate, then
A2 got the number 2. We use Wi to denote the number Ai
got. Let T denote the time for traveling the distance 2r. Our
refined method, after ABC moved to the new location, is as
follows:
• Each node Ai∈ Dependents(ABC,Af) waits to receive
the “RECOVERED” message in time T.
• If such a message is received, then this node does not
move.
• If such a message is not received within time T, then
wait for another T · Witime.
– If within the T · Wi time, it received the “RECOV-
ERED” message, then it does not move.
– Otherwise, at the end of T · Witime, it moves to the
location to ABCand then broadcast a“RECOVERED”
message.
It is not difficult to show that the neighboring nodes of the
best candidate node will have the same view on what to do
next and after the algorithm terminates, the final structure is
indeed a connected network. We mainly let the candidate node
ABC to decide the priority number of each of its neighbors,
and send the priority number information to all its neighbors
before node ABC moves. Consequently, they will have the
same decision.
3.3
The most significant discrepancy happens at Section 5 of [1].
We found that based on the definitions of “boundary node”
in [1], Theorem 6 is not valid at all. We first use Fig. 5 in
paper [1] to illustrate the definition of “network periphery” and
“connectivity hole”. (Please refer to [1] to find Fig. 5.) In [9],
the geometric convex hull of Fig. 5 in [1] is the cycle defined
by vertexes sequence A1,A6,A4,A12,A3,A13,A7,A5,A10.
(Note that vertexes A2,A14are not included.) The paper [1]
uses the convex hull as the network periphery. The usage of
convex hull to define network periphery as in [1] will cause
some nodes that are actually on the network periphery be
excluded.
Cited [8], Abbasi et al. define the connectivity hole as
“the area in which the edges between nodes form a closed
polygonal without links between nodes that are adjacent on the
polygonal chain”, which is unclear to understand so we here
cite the original definition in [8]. In [8], connectivity hole is
defined as “an area where nodes can not be placed and edges
can not exist”, i.e. the connectivity hole is independent with
the network topology. Connectivity hole used in [8] reflects
a geographic region where we cannot physically put nodes
in. Based on the definition of connectivity hole in [8], in
Fig. 5 of [1], all the nodes are not on the boundary of a
connectivity hole. Abbasi et al. [1] give the definition of
boundary node as the one located at the network periphery
(defined as convex hull) or on the closed polygonal chain
surrounding a connectivity hole (cited [8]). With the definition
of boundary node in [1], we provide two counterexamples for
Theorem 6 [1] as shown in Fig. 3.
Discrepancies in Algorithm DARA-2C
5
A1
A3
A4
A2
A
A
1
A2
4
A
A3
A5
A
6
A7
A8
A9
A10
A11
(a) for network periphery,(b) for connectivity hole
Fig. 3. The counterexamples for boundary node.
In Fig. 3(a), the length of all the edge is 1 which is the
maximum transmission range, and the distance between A3
and A1(A2) is 1+ǫ, where ǫ > 0 is an arbitrarily small positive
constant. By the definition of network periphery in [1], we
know that in Fig. 3(a) the network periphery is A1,A2,A4,A5,
but the failure of A3, not on the network periphery, can destroy
the 2-connectivity of the network, which implies that Theorem
6 in [1] is incorrect.
It is clear the network periphery of Fig. 5 in [1] should
be defined by the vertexes sequence A1,A6,A2,A4, A12, A3,
A13, A7, A5, A14, A10, which is possibly what was proposed
by the authors in [1].
Correct the discrepancy in paper [1]: Here we give a
correct definition of “network periphery”.
Definition 1: The network periphery is a subgraph of the
network graph induced by the edges which can be connected
to a point infinitely far away with curves that do not cross any
edge or vertex of the network graph.
With the corrected definition of network periphery, all the
vertices in Fig. 3(a) are in the network periphery.
In Fig. 3(b), the length of all the edges are 1, and the
distance between A10 and A1(,A9,A8) is 1 + ǫ, and the
distance between A11and A6(, A7,A8) is 1+ǫ, where ǫ > 0
is an arbitrarily small positive constant. So by the definition
of connectivity hole [8], there is no connectivity hole in
Fig. 3(b). However, the failure of A10 (or A11) can destroy
the 2-connectivity of the network, which again illustrates the
incorrectness of Theorem 6 in [1].
However, we found that it is very hard to correct the
definition of connectivity hole in [1]. Maybe the authors of [1]
wanted to define the connectivity hole as the faces which have
more than 3 sides, for example A3,A4,A11,A10in Fig. 3(b).
However, by this definition almost all the vertices of the
network graph except the ones on the network periphery are on
the boundary of connectivity hole, which makes the definition
meaningless. For example, in Fig. 3(b), A10 and A11are on
the boundary of connectivity hole and they are the only two
vertices not on the network periphery. seems meaningless.
In Fig. 4, we give a series of input graphs in which
when any vertex fails there is some vertex that can be-
come a cut vertex. In the example, there are several circles
with the same center and a line segment crosses the center
and intersects with each circle at two intersection points.
On each intersection points we deploy a vertex (such as
A1,A2,A3,A4,A5,A6), and we deploy other vertices uni-
formly on the circles and the line segment to guarantee
connectivity (such as Ac1,Ac2,Ac3,Ac4,Ac5).