  We refine the construction of quasi-homomorphisms on mapping class groups. It
is useful to know that there are unbounded quasi-homomorphisms which are
bounded when restricted to particular subgroups since then one deduces that the
mapping class group is not boundedly generated by these subgroups. In this note
we enlarge the class of such subgroups. The generalization is motivated by
considerations in first order theory of free groups.
