  Many hard algorithmic problems dealing with graphs, circuits, formulas and
constraints admit polynomial-time upper bounds if the underlying graph has
small treewidth. The same problems often encourage reducing the maximal degree
of vertices to simplify theoretical arguments or address practical concerns.
Such degree reduction can be performed through a sequence of splittings of
vertices, resulting in an _expansion_ of the original graph. We observe that
the treewidth of a graph may increase dramatically if the splittings are not
performed carefully. In this context we address the following natural question:
is it possible to reduce the maximum degree to a constant without substantially
increasing the treewidth?
  Our work answers the above question affirmatively. We prove that any simple
undirected graph G=(V, E) admits an expansion G'=(V', E') with the maximum
degree <= 3 and treewidth(G') <= treewidth(G)+1. Furthermore, such an expansion
will have no more than 2|E|+|V| vertices and 3|E| edges; it can be computed
efficiently from a tree-decomposition of G. We also construct a family of
examples for which the increase by 1 in treewidth cannot be avoided.
