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We define hyperbolic analogues of the graphs of free splittings, of cyclic splittings, and of maximally cyclic splittings of    F  N    for free products of groups. Given a countable group   G   which splits as    G  =   G  1   *  ⋯  *   G  k   *  F   , where   F   denotes a finitely generated free group, we identify the Gromov boundary of the graph of relative cyclic splittings with the space of equivalence classes of   Z  ‐averse trees in the boundary of the corresponding outer space. A tree is   Z   ‐averse  if it is not compatible with any tree    T  '    that is itself compatible with a relative cyclic splitting. Two   Z  ‐averse trees are  equivalent  if they are both compatible with a common tree in the boundary of the corresponding outer space. We give a similar description of the Gromov boundary of the graph of maximally cyclic splittings.

Hyperbolic graphs for free products, and the Gromov boundary of the graph of cyclic splittings