The concept of avoidable paths in graphs was introduced by Beisegel,
Chudnovsky, Gurvich, Milani\v{c}, and Servatius in 2019 as a common
generalization of avoidable vertices and simplicial paths. In 2020, Bonamy,
Defrain, Hatzel, and Thiebaut proved that every graph containing an induced
path of order k also contains an avoidable induced path of the same order.
They also asked whether one could generalize this result to other avoidable
structures, leaving the notion of avoidability up to interpretation. In this
paper we address this question: we specify the concept of avoidability for
arbitrary graphs equipped with two terminal vertices. We provide both positive
and negative results, some of which appear to be related to the recent work by
Chudnovsky, Norin, Seymour, and Turcotte [arXiv:2301.13175]