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Self-stabilization is a versatile approach to fault-tolerance since itpermits a distributed system to recover from any transient fault thatarbitrarily corrupts the contents of all memories in the system. Byzantinetolerance is an attractive feature of distributed systems that permits to copewith arbitrary malicious behaviors. We consider the well known problem ofconstructing a breadth-first spanning tree in this context. Combining these twoproperties proves difficult: we demonstrate that it is impossible to containthe impact of Byzantine nodes in a strictly or strongly stabilizing manner. Wethen adopt the weaker scheme of topology-aware strict stabilization and wepresent a similar weakening of strong stabilization. We prove that theclassical $min+1$ protocol has optimal Byzantine containment properties withrespect to these criteria.

On Byzantine Containment Properties of the $min+1$ Protocol