  The celebrated Erd\H{o}s-P\'{o}sa Theorem, in one formulation, asserts that
for every $c\geq 1$, graphs with no subgraph (or equivalently, minor)
isomorphic to the disjoint union of $c$ cycles have bounded treewidth. What can
we say about the treewidth of graphs containing no induced subgraph isomorphic
to the disjoint union of $c$ cycles?
  Let us call these graphs $c$-perforated. While $1$-perforated graphs have
treewidth one, complete graphs and complete bipartite graphs are examples of
$2$-perforated graphs with arbitrarily large treewidth. But there are sparse
examples, too: Bonamy, Bonnet, D\'{e}pr\'{e}s, Esperet, Geniet, Hilaire,
Thomass\'{e} and Wesolek constructed $2$-perforated graphs with arbitrarily
large treewidth and no induced subgraph isomorphic to $K_3$ or $K_{3,3}$; we
call these graphs occultations. Indeed, it turns out that a mild (and
inevitable) adjustment of occultations provides examples of $2$-perforated
graphs with arbitrarily large treewidth and arbitrarily large girth, which we
refer to as full occultations.
  Our main result shows that the converse also holds: for every $c\geq 1$, a
$c$-perforated graph has large treewidth if and only if it contains, as an
induced subgraph, either a large complete graph, or a large complete bipartite
graph, or a large full occultation. This distinguishes $c$-perforated graphs,
among graph classes purely defined by forbidden induced subgraphs, as the first
to admit a grid-type theorem incorporating obstructions other than subdivided
walls and their line graphs.
  More generally, for all $c,o\geq 1$, we establish a full characterization of
induced subgraph obstructions to bounded treewidth in graphs containing no
induced subgraph isomorphic to the disjoint union of $c$ cycles, each of length
at least $o+2$.
