An outerplanar graph is a graph that can be embedded in the plane such that all vertices lie on the outer face. Outerplanar graphs are planar and, by their definition, connected graphs.
The complete graph 
and complete bipartite
graph 
are planar, but not outerplanar (West 2000, p. 240).
In fact, a graph is outerplanar iff it contains no subgraph
homeomorphic to the complete graph 
or complete bipartite
graph 
(Skiena 1990, p. 251, with 
corrected to 
).
A graph is outerplanar iff the graph formed by adding a new vertex connected to each of the original vertices is a planar graph (Felsner 2004).
An outerplanar graph on 
vertices is Hamiltonian iff it is biconnected,
in which case the vertices of outer face comprise a unique Hamiltonian
cycle.
Every outerplanar graph is 3-colorable.
