*3.5. Morse-Based Decomposition*

Another cellular decomposition method proposed by Acar et al. [42] is based on Morse functions [43]. The Morse-based decomposition method has the advantage of different cells shapes such as circular and can be applied in any dimensional space, such as concave, polygon, and irregular space. The cell decomposition is succeeded with a slice that sweeps through the area of interest. A slice is discontinued at the critical point of the Morse function, which is restricted from the obstacle boundaries, as shown in Figure 5. This method uses information concerning the area during motion planning. For this reason, the method can be classified as online [44,45].

**Figure 5.** Morse-based decomposition.

#### *3.6. Online Topological Coverage Algorithm*

Wong [46] presented an algorithm that finds the cell boundaries online using slice decomposition. Slice decomposition is a method for determining the cell boundaries using a sweeping line over the area of interest. As the line sweeps over the area, it separates the obstacles and free space in two regions or more, as shown in Figure 6. The algorithm constructs a topological map using the slice decomposition on the area of interest [47].

**Figure 6.** Slice decomposition.

#### *3.7. Contact Sensor-Based Coverage of Rectilinear Environments*

Butler et al. [48] present an exact cell decomposition algorithm for contact sensor-based robots for online coverage of the rectilinear environment. In contact sensor-based coverage, the robot's path is cycling with retracing, while at the same time it repeatedly constructs a cellular decomposition of the area of interest. When a robot's full-cycle path is unsuccessful, it chooses a new path based on its position and environment. The robot's motion depends on the area's cell decomposition state, updated as the CPP progresses.
