  We consider a Heegaard splitting M=H_1 \cup_S H_2 of a 3-manifold M having an
essential disk D in H_1 and an essential surface F in H_2 with |D \cap F|=1.
(We require that boundary of F is in S when H_2 is a compressionbody with
non-empty "minus" boundary.) Let F be a genus g surface with n boundary
components. From S, we obtain a genus g(S)+2g+n-2 Heegaard splitting M=H'_1
\cup_S' H'_2 by cutting H_2 along F and attaching F \times [0,1] to H_1. As an
application, by using a theorem due to Casson and Gordon, we give examples of
3-manifolds having two Heegaard splittings of distinct genera where one of the
two Heegaard splittings is a strongly irreducible non-minimal genus splitting
and it is obtained from the other by the above construction.
