  The problem of computing vertex and edge connectivity of a graph are
classical problems in algorithmic graph theory. The focus of this paper is on
computing these parameters on embedded graphs. A typical example of an embedded
graph is a planar graph which can be drawn with no edge crossings. It has long
been known that vertex and edge connectivity of planar embedded graphs can be
computed in linear time. Very recently, Biedl and Murali extended the
techniques from planar graphs to 1-plane graphs without $\times$-crossings,
i.e., crossings whose endpoints induce a matching. While the tools used were
novel, they were highly tailored to 1-plane graphs, and do not provide much
leeway for further extension. In this paper, we develop alternate techniques
that are simpler, have wider applications to near-planar graphs, and can be
used to test both vertex and edge connectivity. Our technique works for all
those embedded graphs where any pair of crossing edges are connected by a path
that, roughly speaking, can be covered with few cells of the drawing. Important
examples of such graphs include optimal 2-planar and optimal 3-planar graphs,
$d$-map graphs, $d$-framed graphs, graphs with bounded crossing number, and
$k$-plane graphs with bounded number of $\times$-crossings.
