  We consider the free Schr\"odinger group $e^{-it \frac{d^2}{dx^2}}$ on a
tadpole graph ${\mathcal R}$. We first show that the time decay estimates $L^1
({\mathcal R}) \rightarrow L^\infty ({\mathcal R})$ is in $|t|^{-\frac12}$ with
a constant independent of the length of the circle. Our proof is based on an
appropriate decomposition of the kernel of the resolvent. Further we derive a
dispersive perturbation estimate, which proves that the solution on the queue
of the tadpole converges uniformly, after compensation of the underlying time
decay, to the solution of the Neumann half-line problem, as the circle shrinks
to a point. To obtain this result, we suppose that the initial condition
fulfills a high frequency cutoff.
