  A spanning 2-forest separating vertices $u$ and $v$ of an undirected
connected graph is a spanning forest with 2 components such that $u$ and $v$
are in distinct components. Aside from their combinatorial significance,
spanning 2-forests have an important application to the calculation of
resistance distance or effective resistance. The resistance distance between
vertices $u$ and $v$ in a graph representing an electrical circuit with unit
resistance on each edge is the number of spanning 2-forests separating $u$ and
$v$ divided by the number of spanning trees in the graph. There are also
well-known matrix theoretic methods for calculating resistance distance, but
the way in which the structure of the underlying graph determines resistance
distance via these methods is not well understood.
  For any connected graph $G$ with a 2-separator separating vertices $u$ and
$v$, we show that the number of spanning trees and spanning 2-forests
separating $u$ and $v$ can be expressed in terms of these same quantities for
the smaller separated graphs, which makes computation significantly more
tractable. An important special case is the preservation of the number of
spanning 2-forests if $u$ and $v$ are in the same smaller graph. In this paper
we demonstrate that this method of calculating resistance distance is more
suitable for certain structured families of graphs than the more standard
methods. We apply our results to count the number of spanning 2-forests and
calculate the resistance distance in a family of Sierpinski triangles and in
the family of linear 2-trees with a single bend.
