Whitney proved long ago that every maximal planar graph with no separating triangles is Hamiltonian. However, the converse is not necessarily true, that is, there exist maximal planar graphs that are hamiltonian and have separating triangles. So what happens when a maximal planar graph has separating triangles? Is there still some condition under which we can say with certainty that it will be Hamiltonian as well?
One theorem of this kind was shown by Jackson and Yu.
Consider a maximal planar graph 
This way of decomposing maximal planar graphs has been popular. It was first used by William Cunningham and Jack Edmonds in 1980. They proved several interesting results about the decomposition graph
Things 
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