In this paper we fix 7 types of undirected graphs: paths, paths with
prescribed endvertices, circuits, forests, spanning trees, (not necessarily
spanning) trees and cuts. Given an undirected graph G=(V,E) and two "object
types" A and B chosen from the alternatives above, we
consider the following questions. \textbf{Packing problem:} can we find an
object of type A and one of type B in the edge set E of
G, so that they are edge-disjoint? \textbf{Partitioning problem:} can we
partition E into an object of type A and one of type B?
\textbf{Covering problem:} can we cover E with an object of type
A, and an object of type B? This framework includes 44
natural graph theoretic questions. Some of these problems were well-known
before, for example covering the edge-set of a graph with two spanning trees,
or finding an s-t path P and an s′-t′ path P′ that are
edge-disjoint. However, many others were not, for example can we find an
s-t path P⊆E and a spanning tree T⊆E that are
edge-disjoint? Most of these previously unknown problems turned out to be
NP-complete, many of them even in planar graphs. This paper determines the
status of these 44 problems. For the NP-complete problems we also investigate
the planar version, for the polynomial problems we consider the matroidal
generalization (wherever this makes sense)