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We study tautological classes on the moduli space of stable $n$-pointedhyperelliptic curves of genus $g$ with rational tails. Our result gives acomplete description of tautological relations. The method is based on theapproach of Yin in comparing tautological classes on the moduli of curves andthe universal Jacobian. It is proven that all relations come from the Jacobianside. The intersection pairings are shown to be perfect in all degrees. We showthat the tautological algebra coincides with its image in cohomology via thecycle class map. The latter is identified with monodromy invariant classes incohomology. The connection with recent conjectures by Pixton is also discussed.

Tautological classes on the moduli space of hyperelliptic curves with rational tails