We consider
the problem of covering an
input graph class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003775&_mathId=si1.gif&_user=111111111&_pii=S0012365X15003775&_rdoc=1&_issn=0012365X&md5=070f5746f02d51bcdbdd31fd878fcd63" title="Click to view the MathML source">Hclass="mathContainer hidden">class="mathCode"> with graphs from a fixed
covering class class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003775&_mathId=si7.gif&_user=111111111&_pii=S0012365X15003775&_rdoc=1&_issn=0012365X&md5=6b9e94bf07e12a9a9c4f663b142aa237" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode">. The
classical covering number of
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003775&_mathId=si1.gif&_user=111111111&_pii=S0012365X15003775&_rdoc=1&_issn=0012365X&md5=070f5746f02d51bcdbdd31fd878fcd63" title="Click to view the MathML source">Hclass="mathContainer hidden">class="mathCode"> with respect to
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003775&_mathId=si7.gif&_user=111111111&_pii=S0012365X15003775&_rdoc=1&_issn=0012365X&md5=6b9e94bf07e12a9a9c4f663b142aa237" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode"> is
the minimum number of graphs from
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003775&_mathId=si7.gif&_user=111111111&_pii=S0012365X15003775&_rdoc=1&_issn=0012365X&md5=6b9e94bf07e12a9a9c4f663b142aa237" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode"> needed to cover
the edges of
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003775&_mathId=si1.gif&_user=111111111&_pii=S0012365X15003775&_rdoc=1&_issn=0012365X&md5=070f5746f02d51bcdbdd31fd878fcd63" title="Click to view the MathML source">Hclass="mathContainer hidden">class="mathCode"> without covering non-edges of
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003775&_mathId=si1.gif&_user=111111111&_pii=S0012365X15003775&_rdoc=1&_issn=0012365X&md5=070f5746f02d51bcdbdd31fd878fcd63" title="Click to view the MathML source">Hclass="mathContainer hidden">class="mathCode">. We introduce a unifying notion of three covering parameters with respect to
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003775&_mathId=si7.gif&_user=111111111&_pii=S0012365X15003775&_rdoc=1&_issn=0012365X&md5=6b9e94bf07e12a9a9c4f663b142aa237" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode">, two of which are novel concepts only considered in special cases before:
the local and
the folded covering number. Each parameter measures “how far”
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003775&_mathId=si1.gif&_user=111111111&_pii=S0012365X15003775&_rdoc=1&_issn=0012365X&md5=070f5746f02d51bcdbdd31fd878fcd63" title="Click to view the MathML source">Hclass="mathContainer hidden">class="mathCode"> is from
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003775&_mathId=si7.gif&_user=111111111&_pii=S0012365X15003775&_rdoc=1&_issn=0012365X&md5=6b9e94bf07e12a9a9c4f663b142aa237" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode"> 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.