The paper studies the cyclic pursuit problem in presence of connectivity constraints among single-integrator agents. The robots, each one pursuing its leading neighbor along the line of sight rotated by a common offset angle, are supposed to have a communication set described by a disk of constant radius. Given the initial position of the agents, we determine the communication radii that preserve the connectivity of the robots while they rendezvous at a point or converge to an evenly spaced circle formation. The special case that the initial condition is a linear combination of the eigenvectors of the dynamic matrix of the system, is studied in detail. On the other hand, given the communication radii, we find the set of initial conditions that guarantee the robots remain always connected. As a final contribution, once assigned a ldquonon-optimalrdquo radius, we study the stability of the hybrid system describing the dynamics of the robotic network under variable connectivity levels.