  Cyclic pursuit frameworks provide an efficient way to create useful global
behaviors out of pairwise interactions in a collective of autonomous robots.
Earlier work studied cyclic pursuit with a constant bearing (CB) pursuit law,
and has demonstrated the existence of a variety of interesting behaviors for
the corresponding dynamics. In this work, by attaching multiple branches to a
single cycle, we introduce a modified version of this framework which allows us
to consider any weakly connected pursuit graph where each node has an outdegree
of 1. This provides a further generalization of the cyclic pursuit setting.
Then, after showing existence of relative equilibria (rectilinear or circling
motion), pure shape equilibria (spiraling motion) and periodic orbits, we also
derive necessary conditions for stability of a 3-agent collective. By paving a
way for individual agents to join or leave a collective without perturbing the
motion of others, our approach leads to improved reliability of the overall
system.
