  Encounters between walkers performing a random motion on an appropriate
structure can describe a wide variety of natural phenomena ranging from
pharmacokinetics to foraging. On homogeneous structures the asymptotic
encounter probability between two walkers is (qualitatively) independent of
whether both walkers are moving or one is kept fixed. On infinite comb-like
structures this is no longer the case and here we deepen the mechanisms
underlying the emergence of a finite probability that two random walkers will
never meet, while one single random walker is certain to visit any site. In
particular, we introduce an analytical approach to address this problem and
even more general problems such as the case of two walkers with different
diffusivity, particles walking on a finite comb and on arbitrary bundled
structures, possibly in the presence of loops. Our investigations are both
analytical and numerical and highlight that, in general, the outcome of a
reaction involving two reactants on a comb-like architecture can be strongly
different according to whether both reactants are moving (no matter their
relative diffusivities) or only one, and according to the density of short-cuts
among the branches.
