## 4 Coverage-Preserving Control Scheduling Scheme
### Fault-Tolerant Backbone  
The main objective of topology management in wireless sensor network management is to ensure the network's connectivity and optimize the path for data transmission. Moreover, once failed nodes appears in the network, the practical topology control can adjust the communication links dynamically as far as possible to ensure the normal operation of the network. The topology management can be achieved through the deployment of nodes. For the unattended network with random deployment of nodes, topology management protocols or algorithms should be designed to improve the network performance. After constructing the maximal connected set of active nodes in each grid, the high coverage and connectivity between nodes can be guaranteed. Next, the virtual backbone should be constructed to forward the data collected in each grid to the sink with respect of ensuring the connectivity of the global network. Thus, it can effectively reduce the energy consumption and prolong the network lifetime. The most conventional technique applied of constructing virtual backbone network is to obtain a Connected Dominating Set (CDS). Most of the data transmission or forwarding occurs between the dominating nodes in multi-hop mode. If there are multiple paths between any source node and the destination, a node can change its choice of next hop once a path failure occurs. In this way, it is necessary to construct a virtual backbone network with fault tolerance. Actually, the centralized algorithms often require global information of the sensor nodes, thus it is hard to be achieved in most cases. In contrast, the distributed algorithms can be extended to the whole network only by local information. In this section, the two-connected dominating set algorithm is proposed to implement the architecture of distributed backbone network, which can ensure the node information collected in different regions is transmitted to the sink node through multi-hop mode. **Definition 7:**_Given a graph \(G=\) (V,E), and suppose a subset of vertexes \(D\subseteq V.\) If each vertex in \(V-D\) is dominated by at least \(k\) directed vertex in \(D\), \(D\) is denoted as a distance \(k\)-dominating set of graph \(G\). The vertex in \(D\) is described as dominator node, and the vertex in \(V-D\) is a dominated node correspondingly._  
**Definition 8**:: _Given a graph \(G=(V,E)\), and suppose a dominating set of vertex \(C\) and \(C\subseteq V\). If there exist at least two different paths between each pair of nodes in \(C\), and meanwhile \(C\) is a distance \(k\)-dominating set of graph \(G\), then \(C\) can be denoted as a two-connected \(k\)-dominating set of graph._  
The main steps for constructing two-connected \(k\)-dominating sets for a fault-tolerant backbone is as follows: (1) for network graph \(G\), the \(k\)-dominating set \(D\) is defined, which can be obtained by a distributed manner based on localized management; (2) The minimum spanning tree of the \(k\)-dominating set \(D\) is constructed, and the nodes in the path is added to the set \(D\); (3) the nodes of \(V-D\) can be labeled as backbone nodes, which satisfy two-connectivity for all dominated nodes. During the neighbor discovery phase, each node needs to broadcast HELLO messages to its neighbors for exchanging information. Each node creates a local neighbor list, which includes all one-hop neighboring nodes with respect to their distance. After the exchange of information, each node can obtain the comprehensive information, such as, neighbors ID, estimated distance, color status, etc. Next, the graph coloring algorithm is put to use to color all nodes, which can determine the node's state being dominating or dominated. Here, black represents the dominating node; gray represents the dominated node and all nodes initiate as white nodes. Each node contains two parameters of \(SUM\) and \(dom\). For node \(u\), the \(u.dom\) is set to \(k\) initially. In every round, a node will be selected to the dominating set.