Assume that we have a bi-connected planar graph $G$ with $\Delta(G)>3$, and we want to find a Hamiltonian Path in $G$. As we know the st-order of a bi-connected planar graph can be computed in linear time (By this Reference). The st-order of a bi-connected planar graph makes a path that covers all vertices of $G$. So can we say Hamiltonian Path can be solved in linear time in bi-connected planar graphs with $\Delta(G)>3$ $?$
Note: I said $\Delta(G)>3$, because we know the following rule (Wikipedia Reference):
Hamiltonian Path problem remain NP-complete even for undirected planar graphs of maximum degree three.
The st-order of a 2-connected planar graph with $\Delta(G)>3$ may doesn't give a Hamiltonian Path for it. The following image is a counterexample that shows an st-order which isn't a Hamiltonian Path.
But there is an algorithm for 4-connected planar graphs that can solve Hamiltonian Path problem in linear time (The hamiltonian cycle problem is linear-time solvable for 4-connected planar graphs).