Heegaard splittings with (disk, essential surface) pairs that intersect in one point

We consider a Heegaard splitting M = H₁ ∪_S H ₂ of a 3-manifold M having an essential disk D in H₁ and an essential surface F in H₂ with |D ∩ F| = 1. From H ₁∪_S H₂, we obtain another Heegaard splitting H′₁ ∪_S′ H′₂ by removing a neighborhood of F from H₂ and attaching it to H ₁. As an application, by using a theorem due to Casson and Gordon, we give examples of 3-manifolds admitting two Heegaard splittings of distinct genera, where one of them is a strongly irreducible non-minimal genus splitting and it is obtained from the other by the above construction. We also show that all Heegaard splittings of a Seifert fibered space are related via the above construction.