Constant-degree graph expansions that preserve the treewidth

TL;DR

This paper demonstrates that any simple graph can be expanded to have a maximum degree of 3 while increasing its treewidth by at most 1, enabling degree reduction without significant complexity increase.

Contribution

It proves the existence of constant-degree graph expansions that preserve treewidth within a small additive factor, with an efficient construction method.

Findings

Maximum degree <= 3 achieved with minimal treewidth increase

Expansion size is linear in the number of edges and vertices

Construction can be efficiently derived from a tree-decomposition

Abstract

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.