  A commonly accepted feature of an excitable medium is that a local excitation
leads to a propagation of circular or spiral excitation wave-fronts. This is
indeed the case in fully excitable medium. However, with a decrease of an
excitability localised wave-fragments emerge and propagate ballistically. Using
FitzhHugh-Nagumo model we numerically study how excitation wave-fronts behave
in a geometrically constrained medium and how the wave-fronts explore a random
planar graph. We uncover how excitability controls propagation of excitation in
angled branches, influences arrest of excitation entering a sudden expansion,
and determines patterns of traversing of a random planar graph by an excitation
waves.
