A note on domino treewidth

In [3], Bodlaender and Engelfriet proved that for every k, and d, there exists a constant ck;d, such that every graph with treewidth at most k and maximum degree at most d has domino treewidth at most ck;d. This note gives a new, simpler proof of this fact, with a much better bound for ck;d, namely (9k + 7)d(d+ 1) 1. It is also shown that a lower bound of (kd) holds: there are graphs with domino treewidth at least 1 12 1, treewidth at most k, and maximum degree at most d, for many values k and d. The domino treewidth of a tree is at most its maximum degree.