  A closed formula is obtained for the integral
$\int_{\mathcal{\bar{H}}_g^1}\kappa_{1}\psi^{2g-2}$ of tautological classes
over the locus of hyperelliptic Weierstra\ss{} points in the moduli space of
curves. As a corollary, a relation between Hodge integrals is obtained.
  The calculation utilizes the homeomorphism between the moduli space of curves
$\mathcal{M}_{g,1}$ and the combinatorial moduli space
$\mathcal{M}^{comb}_{g,1}$, a PL-orbifold whose cells are enumerated by
fatgraphs. This cell decomposition can be used to naturally construct
combinatorial PL-cycles $W_a\subset\mathcal{M}^{comb}_{g,1}$ whose homology
classes are essentially the Poincar\'e duals of the Mumford-Morita-Miller
classes $\kappa_a$. In this paper we construct another PL-cycle
$\mathcal{H}^{comb}_g \subset \mathcal{M}^{comb}_{g,1}$ representing the locus
of hyperelliptic Weierstra\ss{} points and explicitly describe the chain level
intersection of this cycle with $W_1$. Using this description of
$\mathcal{H}^{comb}_g\cap W_1$, the duality between Witten cycles $W_a$ and the
$\kappa_a$ classes, and Kontsevich's scheme of integrating $\psi$ classes, the
integral $\int_{\mathcal{\bar{H}}_g^1}\kappa_{1}\psi^{2g-2}$ is reduced to a
weighted sum over graphs and is evaluated by the enumeration of trees.
