  Populations of flashing fireflies, claps of applauding audience, cells of
cardiac and circadian pacemakers reach synchrony via event-triggered
interactions, referred to as pulse couplings. Synchronization via pulse
coupling is widely used in wireless sensor networks, providing clock
synchronization with parsimonious packet exchanges. In spite of serious
attention paid to networks of pulse coupled oscillators, there is a lack of
mathematical results, addressing networks with general communication topologies
and general phase-response curves of the oscillators. The most general results
of this type (Wang et al., 2012, 2015) establish synchronization of oscillators
with a delay-advance phase-response curve over strongly connected networks. In
this paper we extend this result by relaxing the connectivity condition to the
existence of a root node (or a directed spanning tree) in the graph. This
condition is also necessary for synchronization.
