Riemann-bench

A robust maximal independent set in a graph $G$ is a maximal independent set that remains maximal in all connected spanning subgraphs of $G$. How many connected graphs on $12$ vertices have the property that every maximal independent set is a robust maximal independent set, up to isomorphism?

Notation and definitions for background context:

Let $F$ be the field of order 2. Let $K$ be the field of Hahn series in indeterminate $t$ with value group $\mathbb{R}$ and residue field $F$. Let $A$ be the subring of $K$ consisting of those $a \in K$ with non-negative valuation. Consider $K$ as an $A$-module. For $q \in \mathbb{R}$, let $I_q = t^q A$ and $I_{>q} = \bigcup_{r>q} I_r$. Write $A/I_{>0}$ as $F$, since they are identical both as $A$-modules and as fields. Let $\Theta = K/I_{>0}$ and $\Phi = K/A$. We say that an $A$-module $M$ is 'basic' if it is isomorphic to $L/N$ for some $N < L \le K$, and that it is 'multibasic' if it is a direct sum of a (possibly empty) finite list of basic modules. For $A$-submodules $U, V$ of $K$, let $U + V = \{a + b : a \in U, b \in V\}$, $UV = \{ab : a \in U, b \in V\}$, and $c(V) = \{a \in K : aV \le I_{>0}\}$.

You may assume the following facts:

Fact 1: The decomposition of a multibasic $A$-module into basic submodules is unique up to the order of the summands.

Fact 2: If $M_i = L_i / N_i$ and $N_i < L_i \le K$ for $i = 0, 1$, then $M_0 \otimes M_1 = \frac{L_0 L_1}{L_0 N_1 + N_0 L_1}$ and $D(M_0) = c(N_0) / c(L_0)$.

Find the number of distinct isomorphism classes of multibasic $A$-modules $M$ satisfying the following conditions:

(i) $K \otimes \text{End}(M) = K$.

(ii) $F \otimes \text{End}(M) = F$.

(iii) Let $e_r = \dim_F(F \otimes I_r \text{Hom}(I_{>0}, M))$ for all real $r \ge 0$. Then $\lim_{p \to q^-} e_p = e_q$ for all real $q > 0$ except for integers $q$ with $29 \le q \le 328$.

If your answer is infinite, write -1.

Consider the Eynard Orantin Topological Recursion Formalism for the spectral curve $(\mathbb{C}\mathbb{P}^1, x, y, \omega_{0,2}(x, y))$, where $x = t + 1/t$ and $y = t^3 / 3$, and the fundamental bidifferential is given by $\omega_{0,2}(x_1, x_2) = \frac{dz_1 dz_2}{(z_1 - z_2)^2}$, with $z_1, z_2 \in \mathbb{C}\mathbb{P}^1$. Note that $x$ has two simple ramification points at $\pm 1$ of order $2$ with deck transformation $\theta(t) = 1/t$.

Please calculate the Free energies $F_2$ and return it as a rational fraction in the format $a/b$ for $a$ and $b$ coprime. Recall that the free energies $F_g$ can be computed as the following integral $F_g = \frac{1}{2g-2} \sum_{a \in \Delta} \text{Res}_{q=a} \Phi(q)\omega_{g,1}(q)$, where $\Phi(q) = \int_{o}^{q} y(t)dx(t)$, for an arbitrary base point $o$.