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We consider a Heegaard splitting M=H_1 \cup_S H_2 of a 3-manifold M having anessential 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 withnon-empty "minus" boundary.) Let F be a genus g surface with n boundarycomponents. 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 anapplication, by using a theorem due to Casson and Gordon, we give examples of3-manifolds having two Heegaard splittings of distinct genera where one of thetwo Heegaard splittings is a strongly irreducible non-minimal genus splittingand it is obtained from the other by the above construction.

(Disk, Essential surface) pairs of Heegaard splittings that intersect in one point