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 graphs. We prove some structural results on these graphs, including results on decomposition of planar graphs into various types of bipartite graphs.