  Treewidth is a parameter that emerged from the study of minor closed classes
of graphs (i.e. classes closed under vertex and edge deletion, and edge
contraction). It in some sense describes the global structure of a graph.
Roughly, a graph has treewidth $k$ if it can be decomposed by a sequence of
noncrossing cutsets of size at most $k$ into pieces of size at most $k+1$. The
study of hereditary graph classes (i.e. those closed under vertex deletion
only) reveals a different picture, where cutsets that are not necessarily
bounded in size (such as star cutsets, 2-joins and their generalization) are
required to decompose the graph into simpler pieces that are structured but not
necessarily bounded in size. A number of such decomposition theorems are known
for complex hereditary graph classes, including even-hole-free graphs, perfect
graphs and others. These theorems do not describe the global structure in the
sense that a tree decomposition does, since the cutsets guaranteed by them are
far from being noncrossing. They are also of limited use in algorithmic
applications.
  We show that in the case of even-hole-free graphs of bounded degree the
cutsets described in the previous paragraph can be partitioned into a bounded
number of well-behaved collections. This allows us to prove that even-hole-free
graphs with bounded degree have bounded treewidth, resolving a conjecture of
Aboulker, Adler, Kim, Sintiari and Trotignon [arXiv:2008.05504]. As a
consequence, it follows that many algorithmic problems can be solved in
polynomial time for this class, and that even-hole-freeness is testable in the
bounded degree graph model of property testing. In fact we prove our results
for a larger class of graphs, namely the class of $C_4$-free odd-signable
graphs with bounded degree.
