Gathering by oblivious robots on graphs

\(k \ge 2\) anonymous, oblivious robots stand on the nodes of a connected anonymous graph \(G\) with \(n\) nodes, which every robot sees whole. A robot runs Look–Compute–Move cycles: in a Look it takes a snapshot of the graph with its occupied nodes, it computes from that snapshot only, and it stays or moves to a neighbour; moves are instantaneous, so robots are always seen on nodes. A node with two or more robots is a tower. With multiplicity detection a snapshot shows which nodes are towers (weak) or how many robots each node holds (strong); without it a tower looks like one robot. The robots must reach a configuration in which all of them are on one node and none moves again. A setting fixes the scheduler, the multiplicity detection, whether towers may exist at the start, and the graph class; an algorithm must gather from every start of the setting except the starts listed as unsolvable. Under round robin the scheduler activates one robot at a time in a fixed cyclic order. Time is counted in epochs: an epoch is an interval in which every robot completes at least one cycle (under round robin, \(k\) consecutive activations).

Measures: Time, epochs until all robots are on one node and no robot moves again.