Three ways to cover a graph

We consider the problem of covering a host graph G with several graphs from a fixed template class T . The classical covering number of G with respect to T is the minimum number of template graphs needed to cover the edges of G. We introduce two new parameters: the local and the folded covering number. Each parameter measures how far G is from the template class in a different way. Whereas the folded covering number has been investigated thoroughly for some template classes, e.g., interval graphs and planar graphs, the local covering number was given only little attention. We provide new bounds on each covering number w.r.t. the following template classes: linear forests, star forests, caterpillar forests, and interval graphs. The classical graph parameters turning up this way are interval-number, track-number, and linear-, star-, and caterpillar arboricity. As host graphs we consider graphs of bounded degeneracy, bounded degree, or bounded (simple) tree-width, as well as, outerplanar, planar bipartite and planar graphs. For several pairs of a host class and a template class we determine the maximum (local, folded) covering number of a host graph w.r.t. that template class exactly.