  Distributing entangled pairs among multiple users is a fundamental problem in
quantum networks. Existing protocols like $X$ protocol introduced in (npj
Quantum Information 5, 76 (2019)) use graph theoretic tools like local
complementation to optimize the number of measurements required to extract any
Bell pair among the network users. However, such a protocol relies on finding
the shortest path between the users. Here, the existing results are extended to
establish a counter-intuitive notion that, in general, the most optimal path to
perform the $X$ protocol is not along the shortest path. Specific examples of
this advantage are provided on networks of size as small as 12 qubits.
Bottlenecks in establishing simultaneous Bell pairs in nearest-neighbor
architectures are also explored. Recent results suggesting the unsuitability of
the line and ring networks for the implementation of quantum networks due to
the existence of bottlenecks are revisited, and using local equivalency
relations from graph theory, it is hinted at the possibility that even grid
graphs are not exempt from bottleneck issues. Further, it's noted that the
results obtained here would be of use in analyzing the advantages of
measurement-based quantum network coding.
