  Self-stabilization is a versatile approach to fault-tolerance since it
permits a distributed system to recover from any transient fault that
arbitrarily corrupts the contents of all memories in the system. Byzantine
tolerance is an attractive feature of distributed systems that permits to cope
with arbitrary malicious behaviors. We consider the well known problem of
constructing a breadth-first spanning tree in this context. Combining these two
properties proves difficult: we demonstrate that it is impossible to contain
the impact of Byzantine nodes in a strictly or strongly stabilizing manner. We
then adopt the weaker scheme of topology-aware strict stabilization and we
present a similar weakening of strong stabilization. We prove that the
classical $min+1$ protocol has optimal Byzantine containment properties with
respect to these criteria.
