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.