2 problems

Let $G=(V,E)$ be an involutive polyhedral graph: a self-dual polyhedral graph equipped with an involution $\tau$ satisfying $v\notin\tau(v)$ and $u\in\tau(v)\iff v\in\tau(u)$. A Re…

Let ${\mathcal{P}}_k^\text{irr}$ be the set of all irreducible polyhedral graphs with $k$ rigid vertices. Finiteness conjecture. Each set ${\mathcal{P}}_k^\text{irr}$ is finite. Th…