A note on domino treewidthy

In [DO95], Ding and Oporowski proved that for every k and d, every graph G = (V;E) with treewidth at most k and with maximum degree at most d has a tree decomposition of width at most max(600k2d3; 5400d3), such that every vertex v 2 V belongs to at most two of the sets associated to the nodes in the tree decomposition. Such a tree decomposition was called a domino tree decomposition by Bodlaender and Engelfriet in [BE97], where they independently gave a similar result, but with a more complicated proof and with a much higher constant, which was exponential, both in k and in d. In this note, a new and easy to understand proof for the result is given. Additionally, the constant factor arising from the proof given here is smaller: it is shown that graphs with treewidth at most k and maximum degree at most d have domino treewidth at most (9k + 7)d(d+ 1) 1. The proof uses amongst others a technique from [BGHK95] (inspired by a technique from [RS95]), and some other ideas. The proof is given in Section 3. In Section 4, it is shown that a lower bound of (kd) holds: there are graphs with domino treewidth at least 1 12kd 1, treewidth at most k, and maximum degree at most d, for many values k and d. Some final remarks are made in Section 5, and it is shown that the domino treewidth of a tree is at most its maximum degree. yThis research was partially supported by ESPRIT Long Term Research Project 20244 (project ALCOM IT: Algorithms and Complexity in Information Technology).