  Evolutionary games on graphs describe how strategic interactions and
population structure determine evolutionary success, quantified by the
probability that a single mutant takes over a population. Graph structures,
compared to the well-mixed case, can act as amplifiers or suppressors of
selection by increasing or decreasing the fixation probability of a beneficial
mutant. Properties of the associated mean fixation times can be more intricate,
especially when selection is strong. The intuition is that fixation of a
beneficial mutant happens fast (in a dominance game), that fixation takes very
long (in a coexistence game), and that strong selection eliminates demographic
noise. Here we show that these intuitions can be misleading in structured
populations. We analyze mean fixation times on the cycle graph under strong
frequency-dependent selection for two different microscopic evolutionary update
rules (death-birth and birth-death). We establish exact analytical results for
fixation times under strong selection, and show that there are coexistence
games in which fixation occurs in time polynomial in population size. Depending
on the underlying game, we observe inherence of demographic noise even under
strong selection, if the process is driven by random death before selection for
birth of an offspring (death-birth update). In contrast, if selection for an
offspring occurs before random removal (birth-death update), strong selection
can remove demographic noise almost entirely.
