A subhamiltonian graph is a finite graph that can be extended by adding edges inside faces of a planar graph embedding to a planar Hamiltonian graph. For simple graphs with at least three vertices, this is equivalent to book thickness at most 2 (Bernhart and Kainen 1979).
Every planar graph on at least three vertices with maximum vertex degree 
is subhamiltonian (Bekos et al. 2014). Polyhedral graphs need not be subhamiltonian, as
the Goldner-Harary graph shows.
Classes for which all members are planar and Hamiltonian are subhamiltonian, including k-connected
graphs with 
that are planar (Nishizeki and Chiba 2008), Halin
graphs (Cornuéjols et al. 1983), and triangular
grid graphs with at least three vertices. Every
series-parallel graph and every outerplanar
graph (including every tree) on at least three vertices
is subhamiltonian. Under the convention in this work that 
is Hamiltonian but
is not, 
is also subhamiltonian, while 
and the empty graph on two
vertices are not. These small-order cases are exceptions
to the book thickness characterization: 
and the two-vertex empty graph
both have book thickness at most 1.
A subhamiltonian graph that is not outerplanar has book thickness exactly 2.
