  In this paper, we consolidate and expand upon the current theory and
potential applications of the set of $k$ best \emph{cascading via-paths} (CVPs)
and the \emph{reciprocal pointer chain} (RPC) method for identifying them. CVPs
are a collection of up to $|V|$ paths between a source and a target node in a
graph $G = (V,E)$, computed using two shortest path trees, that have
distinctive properties relative to other path sets. They have been shown to be
particularly useful in geospatial applications, where they are an intuitive and
efficient means for identifying a set of spatially diverse alternatives to the
single shortest path between the source and target. However, spatial diversity
is not intrinsic to paths in a graph, and little theory has been developed
outside of application to describe the nature of these paths and the RPC method
in general. Here we divorce the RPC method from its typical geospatial
applications and develop a comprehensive theory of CVPs from an abstract
graph-theoretic perspective. Restricting ourselves to properties of the CVPs
and of the entire set of $k$-best CVPs that can be computed in $O(|E| + |V|
\log |V|)$, we are able to then propose, among other things, new and efficient
approaches to problems such as generating a diverse set of paths and to
computing the $k$ shortest loopless paths between two nodes in a graph. We
conclude by demonstrating the new theory in practice, first for a typical
application of finding alternative routes in road networks and then for a novel
application of identifying layer-boundaries in ground-penetrating radar (GPR)
data. It is our hope that by generalizing the RPC method, providing a sound
theoretical foundation, and demonstrating novel uses, we are able to broaden
its perceived applicability and stimulate new research in this area, both
applied and theoretical.
