We study the boundary $L_{t}$ of the Milnor fiber for the
non-isolated singularities in $\mathbf{C}^{3}$ with equation
$z^{m} - g(x,y) = 0$ where $m \geq 2$ and $g(x,y)=0$ is a
non-reduced plane curve germ. We give a complete proof that
$L_{t}$ is a Waldhausen graph manifold and we provide the
tools to construct its plumbing graph. As an example, we give
the plumbing graph associated to the germs $z^{2} - (x^{2}
- y^{3})y^{l} = 0$ with $l$ odd and $l \geq 3$. We prove that
the boundary of the Milnor fiber is a Waldhausen manifold
new in complex geometry, as it cannot be the boundary of a
normal surface singularity.