Let Formula omitted. and Formula omitted. be Formula omitted. real numbers that satisfy the strict second-order Cauchy interlacing inequalities Formula omitted. for Formula omitted. and the nondegeneracy conditions Formula omitted. for Formula omitted. . Given a connected graph Formula omitted. on Formula omitted. vertices with adjacent vertices Formula omitted. and Formula omitted. , it is proven that there is a real symmetric matrix Formula omitted. whose graph is Formula omitted. such that Formula omitted. has eigenvalues Formula omitted. and Formula omitted. has eigenvalues Formula omitted. , provided some necessary combinatorial conditions on Formula omitted. are satisfied. We also provide generalizations when the two deleted vertices are not adjacent, as well as interpretation of the results in terms of perturbing one or two diagonal entries.