  We study the reachability problem for networks of timed communicating
processes. Each process is a timed automaton communicating with other processes
by exchanging messages over unbounded FIFO channels. Messages carry clocks
which are checked at the time of transmission and reception with suitable
timing constraints. Each automaton can only access its set of local clocks and
message clocks of sent/received messages. Time is dense and all clocks evolve
at the same rate. Our main contribution is a complete characterisation of
decidable and undecidable communication topologies generalising and unifying
previous work. From a technical point of view, we use quantifier elimination
and a reduction to counter automata with registers.
