The closure of the hyperelliptic locus is a cycle whose points parameterize curves admitting a double cover of the projective line.
Although it's a classical locus in the moduli space of curves, relatively little is still known about its class in the tautological ring.
In this talk we will introduce and explore the notion of graph formula for a tautological class (a la Pixton).
We will then discuss recent ongoing work with Tarasca, and (time permitting) Clader and Ross on the class of the hyperelliptic locus and on the class of hyperelliptic curves with marked Weierstrass points.