  Two mobile agents, starting at arbitrary, possibly different times from
arbitrary nodes of an unknown network, have to meet at some node. Agents move
in synchronous rounds: in each round an agent can either stay at the current
node or move to one of its neighbors. Agents have different labels which are
positive integers. Each agent knows its own label, but not the label of the
other agent. In traditional formulations of the rendezvous problem, meeting is
accomplished when the agents get to the same node in the same round. We want to
achieve a more demanding goal, called rendezvous with detection: agents must
become aware that the meeting is accomplished, simultaneously declare this and
stop. This awareness depends on how an agent can communicate to the other agent
its presence at a node. We use two variations of the arguably weakest model of
communication, called the beeping model, introduced in [8]. In each round an
agent can either listen or beep. In the local beeping model, an agent hears a
beep in a round if it listens in this round and if the other agent is at the
same node and beeps. In the global beeping model, an agent hears a loud beep in
a round if it listens in this round and if the other agent is at the same node
and beeps, and it hears a soft beep in a round if it listens in this round and
if the other agent is at some other node and beeps. We first present a
deterministic algorithm of rendezvous with detection working, even for the
local beeping model, in an arbitrary unknown network in time polynomial in the
size of the network and in the length of the smaller label (i.e., in the
logarithm of this label). However, in this algorithm, agents spend a lot of
energy: the number of moves that an agent must make, is proportional to the
time of rendezvous. It is thus natural to ask if bounded-energy agents can
always achieve rendezvous with detection as well...
