# \(k\)-Circle Formation and \(k\)-epf by Asynchronous Robots
## 1 Introduction
### Related Works  
The arbitrary pattern formation problem asks the robots to form an arbitrary geometric pattern given as an input. [PERSON] et al. [10] studied the formation of geometric patterns in the plane by the robots. They characterized the class of geometric patterns that can be formed by the robots in terms of their initial configurations. [PERSON] et al. [11] investigated the solvability of the arbitrary pattern formation problem by asynchronous robots. [PERSON] et al. [12] proposed an algorithm that forms a given pattern \(P\) starting from any initial configuration \(I\), if \(\rho(I)\) divides \(\rho(P)\), where \(\rho(.)\) denotes the geometric symmetricity, provided that both \(I\) and \(P\) do not contain multiplicities. [PERSON] et al. [13] studied the arbitrary pattern formation problem without assuming common chirality. [PERSON] et al. [25] investigated the arbitrary pattern formation problem for asynchronous robots with lights. The presence of lights serves both as a medium of weak explicit communication and also as a form of memory. This work investigates the problem in a setting where the view of a robot is obstructed by the presence of other robots. In the discrete domain, [PERSON] et al. [26] investigated the arbitrary pattern formation problem on infinite grid for asynchronous robots.  
[PERSON] et al. [27] studied the circle formation problem for oblivious anonymous mobile robots without chirality. [PERSON] et al. [18] proposed a distributed algorithm, that ensures that the robots deterministically form a non-uniform circle in a finite number of steps and converge towards a solution to the uniform circle formation. The circle formation problem for asynchronous fat robots has been studied in [17; 19]. [PERSON] et al. [20] proved that the uniform circle formation problem is solvable for any initial configuration in which the robots are in distinct locations. [PERSON] et al. [16] proposed a distributed algorithm to solve the circle formation problem by asynchronous robots having one bit of persistent memory, which minimizes the maximum distance traveled by any robot. [PERSON] et al. [21] proposed a distributed algorithm for fat robots having limited visibility to form a uniform circle. [PERSON] et al. [28] studied the uniform circle formation for opaque robots with lights.  
The landmarks covering problem studied by [PERSON] et al. [14], asks the robots to reach a configuration where at each landmark point (which is a pre-fixed point in the plane), there is precisely one robot. They proposed an algorithm, assuming common chirality among the robots, which minimizes the total distance traveled by all the robots. [PERSON] et al. [15] studied the embedded pattern formation problem without assuming common chirality. [PERSON] et al. in [22] solved a variant of the gathering problem,where the robots need to gather at one of the pre-determined points, referred to as meeting points. They also studied this problem w.r.t. two objective functions, by minimizing the total distance traveled by all robots and by minimizing the maximum distance traveled by a single robot. [PERSON] et al. [23] studied the gathering over meeting nodes problem in an infinite grid.  
[PERSON] et al. studied the effect of the common orientation on the solvability of the partitioning problem [24] for homogeneous robots. [PERSON] et al. [29] studied the team assembling problem by heterogeneous robots. Given robots with \(k\) different colors, the robots are required to partition themselves into teams satisfying a given specification \(A=(a_{1},a_{2},\ldots,a_{k})\), where \(a_{i}\) is the number of robots with color \(i\) in one team.