Degeneracies of Triangulated Graphs

A graph $G$ is $k$-degenerate if each subgraph has minimum degree at most $k$. The degeneracy\textbf{ }$D\left(G\right)$ is the smallest $k$ such that $G$ is $k$-degenerate. We determine the truth values of four statements (using different quantifiers) about when a planar graph $G$ with degeneracy $k$ has a triangulation with degeneracy $l$. We characterize which 3-connected planar graphs can only be triangulated to degeneracy 3. Then we consider analogous questions for maximal planar bipartite