## 2 Background
### Coverage and Connectivity in WSN  
Two essential issues in WSNs are coverage and connectivity. If the sensor node is active and can detect an object in the monitored region, then we can say that this object is covered by the WSN. Coverage can be divided into three types, i.e., area, point, and barrier coverage [28; 29]. Area coverage can be full or partial coverage; full coverage means that each point in the monitoring area should be covered by at least one sensor node or K (where K \(\geq\) 1) node, according to the application requirements, such as a battlefield, where it is necessary to have precise information about the observed area. Other applications, such as environment applications, may require partial coverage. Partial coverage can be useful when the number of sensor nodes is not enough for full coverage. This can maximize network lifetime and save energy. Figure 5: Deterministic deployment. Figure 6: Random deployment. In point coverage, a specific point or target should be monitored, which may be static or mobile, such as observing the behavior of an animal in the region of interest or monitoring some important points on the enemy battlefield. Barrier coverage means monitoring international borders and trying to detect any illegal behavior. Additionally, barrier coverage can be full or partial coverage; sensors are deployed based on the Poisson point access model in full and partial coverage and used with a limited number of sensors. Concerning connectivity, the network is said to be fully connected if each sensor node has at least one path to the sink node to transmit and receive data. If there is full coverage without connectivity, the WSN quality of service will be degraded. So, these two issues, i.e., coverage and connectivity, should be always considered simultaneously in the deployment of WSNs. In some applications that required full coverage, full connectivity is also required to achieve data gathering and transmitting to the sink node. Two types of connectivity are available, i.e., one-connectivity when there is a single path between the sensor and sink nodes and k-connectivity when there are multiple paths from the sensor node to the sink node. There is always a relation between coverage and connectivity for each application. It is almost assumed that the range of connectivity (Rc) is greater than, or twice, the sensing range (Rs), as shown in Figure 8[25; 30]. Figure 8: Sensing range Rs and communication range Rc. Figure 7: Deployment methods. If we assume that there is a point P located at the (x, y) coordinates and sensor node Si located at (xi, yi) coordinates, then the Euclidean distance between point P and the sensor node Si can be expressed as follows [31; 32]:  
\[\mathrm{d}\left(\mathrm{Si},\mathrm{P}\right)=\sqrt{\left(\mathrm{xi-x}\right)^ {2}+\left(\mathrm{yi-y}\right)^{2}} \tag{1}\]  
The general sensibility of a sensor node on point P can be expressed as follows:  
\[\left(\mathrm{Si},\mathrm{P}\right)=\
icefrac{{\delta}}{{\left(\mathrm{d} \left(\mathrm{Si},\mathrm{P}\right)\right)^{k}}} \tag{2}\]  
where \(\delta\) and \(\mathbf{k}\) represent the non-negative, sensor-dependent constants. There is an inverse relationship between the sensor sensitivity and Euclidean distance between the sensor node and monitored point in the region of interest--as the distance increased, the sensitivity decreased. **Tow sensing models are found**:  
**Binary Disk sensing model**: The simplest sensing model is the binary sensing model, which is represented in Figure 9a. When point P lies within the sensing radius of the sensor node Rs, then this point is covered by the sensor node--otherwise not. The coverage equation can be expressed as follows:  
\[\mathrm{C_{xy}}\left(\mathrm{Si}\right)= \begin{cases}1&\mathrm{if}\ \mathrm{d}\left(\mathrm{Si},\mathrm{P}\right)<\mathrm{Rs}\\ 0&\mathrm{other\ \mathrm{wise}}\end{cases}