Inside the moduli space of curves $\overline{\mathcal{M}}_{g,n}$ one can distinguish two classes of $F$curves isomorphic to $\mathbb{P}^1$: those of type $\overline{\mathcal{M}}_{0,4}$, and those of type $\overline{\mathcal{M}}_{1,1}$. Are there divisor classes that are trivial once restricted to one or the other (or both) types of $F$curves?

1$\begingroup$ This is bad form  I should really doublecheck this first. However, my vague recollection is that inside $\overline{\mathcal{M}}_{g,0}$, the following divisor class, $$ D = \sum_{h=1}^{[g/2]} h(gh) \Delta_h,$$ is zero on the $F$curves of the form $\overline{\mathcal{M}}_{0,4}$. Clearly it is not zero on some of the $F$curves of the form $\overline{\mathcal{M}}_{1,1}$. I will try to doublecheck this. If it is correct, then I will post it as an answer. $\endgroup$– Jason StarrJan 7 '14 at 20:45
The boundary divisor $\Delta_0$ has intersection number $0$ with every $F$curve of the form $\overline{\mathcal{M}}_{0,4}$, yet has nonzero intersection number with every $F$curve of the form $\overline{\mathcal{M}}_{1,1}$.
Edit. Also, since $\lambda$ is the pullback of a divisor class from the Satake compactification of $\mathcal{A}_g$, also $\lambda$ has intersection number $0$ with every $F$curve of the form $\overline{\mathcal{M}}_{0,4}$, yet has nonzero intersection number with every $F$curve of the form $\overline{\mathcal{M}}_{1,1}$.