  In many cases the multi-robot systems are desired to execute simultaneously
multiple behaviors with different controllers, and sequences of behaviors in
real time, which we call \textit{behavior mixing}. Behavior mixing is
accomplished when different subgroups of the overall robot team change their
controllers to collectively achieve given tasks while maintaining connectivity
within and across subgroups in one connected communication graph. In this
paper, we present a provably minimum connectivity maintenance framework to
ensure the subgroups and overall robot team stay connected at all times while
providing the highest freedom for behavior mixing. In particular, we propose a
real-time distributed Minimum Connectivity Constraint Spanning Tree (MCCST)
algorithm to select the minimum inter-robot connectivity constraints preserving
subgroup and global connectivity that are \textit{least likely to be violated}
by the original controllers. With the employed safety and connectivity barrier
certificates for the activated connectivity constraints and collision
avoidance, the behavior mixing controllers are thus minimally modified from the
original controllers. We demonstrate the effectiveness and scalability of our
approach via simulations of up to 100 robots with multiple behaviors.
