A graph-based approach is going to result in the same equations, just expressed in a different form.
I'm assuming you are calculating admittance parameters by making one port an open or short circuit and measuring the current vs. voltage response at the other port. The admittance parameters (or Y-parameters) are a system of equations that allow you to describe the port currents \$ i_1 \$ and \$ i_2 \$ as functions of the port voltages \$ v_1 \$ and \$ v_2 \$:
$$
\begin{pmatrix}
i_1 \\
i_2
\end{pmatrix} =
\begin{bmatrix}
Y_{11} & Y_{12} \\
Y_{21} & Y_{22}
\end{bmatrix}
\begin{pmatrix}
v_1 \\
v_2
\end{pmatrix}
$$
You should end up going through a series of steps such as the following:
$$
Y_{11} = \frac{i_1}{v_1}\lvert_{V_2 = 0}
$$
$$
Y_{12} = \frac{i_1}{v_2}\lvert_{V_1 = 0}
$$
$$
Y_{21} = \frac{i_2}{v_1}\lvert_{V_2 = 0}
$$
$$
Y_{22} = \frac{i_2}{v_2}\lvert_{V_1 = 0}
$$
I would argue this itself is pretty algorithmic, since for each Y parameter, you end up calculating the input admittance of a one-port. Because you know how each of the components in the network behave, you will end up having to solve some node or mesh equations to obtain the current behavior from voltage.
Now, let's consider a graph-based approach. You either end up solving a system of \$ n \$ KCL differential equations for the \$ n \$ nodes in consideration or \$ l \$ KVL differential equations for the \$ l \$ meshes in your graph. In contrast with typical analysis however, one difference in your case is that you will have to keep track of some extra branch currents because \$ i_1 \$ and \$ i_2 \$ enter and exit the network.
Thus, instead of having all KCL equations that look as follows:
$$
i_a + i_b + ... = 0
$$
you instead end up having some KCL equations that look like:
$$
i_a + i_b + ... = i_1
$$
or
$$
i_a + i_b + ... = i_2
$$
Your next task would be to manipulate the system of equations you obtain to look like the system above, where all currents are expressed as transformations of voltages. Ultimately, you'll get the same solution.
One note:
The gyrator sitting in the middle of this two port makes analysis a little tricky. If your circuit is composed of just resistors, capacitors, and inductors, there are additional "shortcuts" for coming up with the matrix equations describing your circuit (such as modified nodal analysis). These can also be automated to create circuit analysis programs.
