## 4 Coverage-Preserving Control Scheduling Scheme  
### Determination of Redundant Nodes  
How to prolong the network lifetime as long as possible under the condition of limited sensor node energy is the main challenge in designing ESNs. The reduction of the number of working sensors is an important and effective means to achieve the above purpose. That is, without impacting the performance of the system, some nodes are scheduled into sleep status to retain their residual energy, and others remain in an active state. In this way, we can reduce the node density of the network, lower the redundancy of monitored information, and cut down interference levels. Thus, the energy consumption of the whole system can be improved. In addition, when a node fails due to energy depletion, its neighboring nodes in sleep status can be activated to replace the failed node, and thus continue to maintain the network coverage. Therefore, density control can effectively prolong the lifetime of wireless sensor networks. Aiming to maintain the original performance of the network,density control must meet the following two conditions: (1) ensure coverage quality of the target area; (2) ensure the connectivity of the network. **Definition 5:**_Given a set of sensor nodes IS, \(G_{s}\) = (\(V_{s}\),\(E_{s}\)) is an undirected graph composed of all nodes in the set, in which \(V_{s}\) = \(S\), \(\forall N_{i}\),\(N_{j}\) \(\in\)\(S\) and \(\text{dis}(i\),\(j)\) \(\in\)\(E_{s}\). If and only if \(\text{dis}(i\),\(j)\)\(\leq\)\(\text{min}\{\text{C}_{i}\),\(C_{j}\}\), it can be denoted that the transmission graph \(G_{s}\) is constructed by all nodes from set \(S\). Moreover, a transmission path Path = {\(N_{1}\),\(N_{2}\),\(N_{3}\),...} derived from \(G_{s}\) consists of a series of nodes, in which every pair of adjacent nodes are communication neighbors. If there exists at least a communication path between any two nodes in the transmission graph \(G_{s}\), it is said that the graph is connected._  
**Definition 6:**_Given a sensor node set RS and target area \(R\), If every point within the \(R\) is covered by at least one of the nodes within the set \(S\), the \(S\) is denoted as the cover set of \(R\). Next, meanwhile if the transmission graph derived from the set \(S\) is connected, the set \(S\) can be denoted as a connected cover set of \(R\)._  
Considering that a sensor network deployed in the target area \(R\), the problem of the minimum connected cover set is to find out the minimum subset \(S^{*}\) (\(S^{*}\subseteq S\)), which \(R\subseteq\cup_{N_{i}\in S^{*}}\rho\left(N_{i},R_{i}\right)\) and the transmission graph derived from \(S^{*}\) is connected completely. In [52], the minimum connected cover set problem was proved to be a NP-hard problem. Since that the number of redundant nodes that can be hibernated at the same time determines the size of the final set, we can construct the minimal connected cover set according to multiple criteria, including node redundancy degree, the residual energy, node category, etc. In this paper, a greedy algorithm is used to find the redundant nodes set which can be scheduled to sleep simultaneously. Firstly, a node with the maximum redundancy degree in each grid should be selected and put into the temporary node set. Secondly, the temporary redundant nodes will be closed and at least one node of each type of its neighbors must be in a working state after being selected to the largest independent set. In order to make the final coverage set as small as possible, the above process iterates until no redundant nodes can be closed. At the end of the algorithm, all the redundant nodes are selected and form the maximum independent set. The rest of the nodes will remain in working state in the next round and form the coverage of the target area.