On the Worst-Case Initial Configuration for Conservative Connectivity Preservation

We consider the system of robots with limited-visibility, where each robot can see only the robots within the unit visibility range a.k.a. the unit distance range). In this model, we focus on the inherent cost we have to pay for connectivity preservation in the conservative way (i.e., in any execution, no edge of the visibility graph is deleted). We present a bad configuration with the visibility graph of diameter D for which any conservative algorithm requires O(D2) rounds to make all robots movable, where D is the diameter of the initial visibility graph. This result implies that we inherently need edge-deletion mechanisms to solve many connectivity-preserving problems (as considered in AOSY99, FPSW05, SDY09) within o(D2) rounds.