  We consider the problem of covering an input graph $H$ with graphs from a
fixed covering class $G$. The classical covering number of $H$ with respect to
$G$ is the minimum number of graphs from $G$ needed to cover the edges of $H$
without covering non-edges of $H$. We introduce a unifying notion of three
covering parameters with respect to $G$, two of which are novel concepts only
considered in special cases before: the local and the folded covering number.
Each parameter measures "how far'' $H$ is from $G$ in a different way. Whereas
the folded covering number has been investigated thoroughly for some covering
classes, e.g., interval graphs and planar graphs, the local covering number has
received little attention.
  We provide new bounds on each covering number with respect to the following
covering classes: linear forests, star forests, caterpillar forests, and
interval graphs. The classical graph parameters that result this way are
interval number, track number, linear arboricity, star arboricity, and
caterpillar arboricity. As input graphs we consider graphs of bounded
degeneracy, bounded degree, bounded tree-width or bounded simple tree-width, as
well as outerplanar, planar bipartite, and planar graphs. For several pairs of
an input class and a covering class we determine exactly the maximum ordinary,
local, and folded covering number of an input graph with respect to that
covering class.
