A graph is 1-planar, if it admits a 1-planar embedding, where each edge
has at most one crossing. Unfortunately, testing 1-planarity of a graph
is known as NP-complete. This paper considers the problem of testing 2-
planarity of a graph, in particular,testing full outer-2-planarity of a
graph. A graph is fully-outer-2-planar, if it admits a fully-outer-2-
planar embedding such that every vertex is on the outer boundary, no
edge has more than two crossings, and no crossing appears along the
outer boundary. We present several structural properties of triconnected
outer-2-planar graphs and fully-outer-2-planar graphs, and prove that
triconnected fully-outer-2-planar graphs have constant number of fully-
outer-2-planar embeddings. Based on these properties, we present a
linear-time algorithm for testing fully outer-2-planarity of a graph G,
where G is triconnected, biconnected and oneconnected. The algorithm
also produce a fully outer-2-planar embedding of a graph, if it exists.
We also show that every fully-outer-2-planar embedding admits a straight
-line drawing.