A Sufficient Condition for the Hamiltonicity of Vertex Envelopes of Plane Graphs

We explore a sufficient condition for the hamiltonicity of vertex envelopes of plane graphs. In particular we show that if a plane graph G contains an independent set of vertices [Formula: see text] such that [Formula: see text] is a tree, then its vertex envelope [Formula: see text] is Hamiltonian. Based on this criteria, we identify classes of plane graphs whose vertex envelopes are Hamiltonian. We also describe constructions that produce plane graphs whose vertex envelopes are Hamiltonian, and investigate the hamiltonicity of graphs obtained through edge subdivisions.