Subgraphs of maximal planar graph

Question

Does a maximal planar graph with $n$ vertices (and $3n - 6$ edges respectively, please correct me if I am wrong) contain, as a proper subgraph, any planar graph (up to isomorphism) with at most $n$ vertices? Otherwise, what is a counterexample? I define a maximum planar graph as a planar graph that, when a new edge is added to it, is impossible to draw on a plane without intersections.

Answer 1

A star graph on $n-1$ vertices (one in the middle, $n-2$ on the outside) is planar. But it is usually not contained within a maximal planar graph having $n$ vertices. For example, the Fritsch graph is a maximal planar graph on 9 vertices, and there is no 8-star to be found (as can be seen by inspecting the degrees of the nodes: the maximal degree here is 5, so there cannot be a proper subgraph where one of the nodes has degree 7). It also does not contain the 8-star as a minor, which is less immediate to see, but the basic observation is that there's no edge which you could contract, such that the two vertices on either end are between them connected to all of the remaining vertices.

This situation is generally applicable because maximal planar graphs (triangulated graphs) are under no obligation to have one of their vertices connected to $n-2$ of the others.

(image from https://commons.wikimedia.org/wiki/File:Fritsch_graph.svg)