  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" $\mathrm{A}$ and $\mathrm{B}$ chosen from the alternatives above, we
consider the following questions. \textbf{Packing problem:} can we find an
object of type $\mathrm{A}$ and one of type $\mathrm{B}$ 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 $\mathrm{A}$ and one of type $\mathrm{B}$?
\textbf{Covering problem:} can we cover $E$ with an object of type
$\mathrm{A}$, and an object of type $\mathrm{B}$? 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\subseteq E $ and a spanning tree $T\subseteq E$ 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).
