A note on domino treewidth

In [DO95], Ding and Oporowski proved that for every ✁ , and ✂ , there exists a constant ✄✆☎✞✝✟ , such that every graph with treewidth at most ✁ and maximum degree at most ✂ has domino treewidth at most ✄✆☎✞✝✟ . This note gives a new simple proof of this fact, with a better bound for ✄ ☎✞✝✟ , namely ✠☛✡☞✁✍✌✏✎☞✑✒✂✓✠✔✂✕✌✗✖✘✑✚✙✛✖ . It is also shown that a lower bound of ✜✕✠☛✁✢✂✢✑ holds: there are graphs with domino treewidth at least ✣ ✣✥✤ ✁✦✂✧✙★✖ , treewidth at most ✁ , and maximum degree at most ✂ , for many values ✁ and ✂ . The domino treewidth of a tree is at most its maximum degree.


Introduction
In [DO95], Ding and Oporowski proved that for every ✩ and ✪ , every graph ✫ ✬ ✭ ✥✮ ✰✯ ✲✱ ✧✳ with treewidth at most ✩ and with maximum degree at most ✪ has a tree decomposition of width at most ✴ ✶✵ ✦✷ ✭ ✹✸ ✢✺ ✻✺ ✻✩ ✓✼ ✽✪ ✿✾ ✻✯ ❁❀ ❃❂ ✿✺ ✢✺ ✻✪ ✻✾ ✘✳ , such that every vertex ❄ ❆❅ ❇✮ 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 ✩ and in ✪ .
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 ✩ and maximum degree at most ✪ have domino treewidth at most ✭ ✥❈ ✻✩ ❊❉ ★❋ ✦✳ •✪ ❍✭ ✥✪ ■❉ ❑❏ ▲✳ ◆▼ ❖❏ .
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 some cases, ❤ will be considered a rooted tree; a specific node of ❤ is considered to be the root.A tree decomposition ✭ ✥❸ ✈✯ ✲❤ ■✳ with ❤ a rooted tree is called a rooted tree decomposition.For a node ❝ ■❅ ✗❡ , we call the set ( ❴ the bag of ❝ .

Proof: Choose an arbitrary root
❬ ✆② ❵❅ ✈❡ ❵❛ ✲② is the lowest common ancestor of two nodes in ➈ ❙ ❣ .We claim that this set ➈ ✼ fulfils the conditions.Clearly, ➈ ❙ is adjacent to a node in ❤ ❜➐ , then there are two cases: ♣ ❝ ➒ is an ancestor of a node in ❤ ❜➐ .Then ❝ ➒ is the unique parent of the root of ❤ ■➐ .
♣ ❝ ➒ is a child of a node in ❤ ❜➐ .We claim that there can be only one node fulfilling this case (for this tree ❤ ❜➐ ): suppose , such that every vertex ❄ that is adjacent to a vertex in ➧ belongs to ➧ ➏ ( ❴ ➫➥ ➏ ( ❴ ➨ .
Choose an arbitrary root ➎ ➦❅ ④❡ , and view ❤ as a rooted tree.We will process ❤ in a bottom-up order: a node is processed after all its children are processed.While processing vertices, we maintain a set ➈ ❙ ⑦ ❆❡ , which is initially empty, and a set ❼ ①➐ ✚⑦ ❖❼ , for which initially ❼ ①➐ ➯✬ ♥❼ .The idea is that nodes are added to ➈ ❙ until finally the requested set is found, and that ❼ ①➐ gives those vertices in ❼ that still can belong to a connected component with too many vertices in ❼ in it.

The domino treewidth theorem
In this section, we prove the main result of this section.The technique is inspired by a technique from [BGHK95], which was again inspired by a technique from [RS95].
Procedure MAKEDEC (graph ➮ ➽✬ ❷✭ ✥✮ ➱ ✯ ✲✱ ➱ ✳ , vertex set ❼ ) has the following steps: 1. Obtain a set ➬ ❆⑦ ❑✮ •➱ , such that every connected component of ❼ .This is the root of the new tree decomposition.Make ➎ adjacent to the roots of each of the tree decompositions, obtained in the previous step.The result is the output of the procedure.
Assume that the set ➬ found in Step 1 is at most of the size, guaranteed to exist by Corollary 2.3 on the preceding page, i.e., we have: Claim 3.1.1Let ➮ ✐✬ ➇✭ ✥✮ ❍➱ )✯ ❁✱ ✍➱ ➍✳ be a connected graph, and ❼ ❽⑦ ★✮ ❍➱ , ❼ Ï➼ ✬ tÐ .When MAKEDEC(➮ ♠✯ ✆❼ ) is called, the procedure outputs a rooted domino tree decomposition of ➮ , such that vertices in ❼ only belong to the root bag of the domino tree decomposition.
Proof: First, observe that the first parameter of a recursive call to MAKEDEC always is a connected graph, and the second parameter of every recursive call to MAKEDEC is always a non-empty set: every connected component of ➮ ➩➄ ✮ ➱ ▼ ➩➬ ✗▼ ➘❼ ➆➅ must contain vertices adjacent to ➬ ➏ ❼ .Thus, the recursive calls done to MAKEDEC involve graphs with fewer vertices, hence the procedure terminates. Let , then Ñ and ❄ belong both to the root bag ( ❰ .Otherwise, Ò and Ó belong to the same connected component ➮ ✶❴ of ➮ ➩➄ ✮ ❍➱ ❖▼ ❻➬ ✈▼ ❻❼ ➆➅ , and by induction, there will be a bag containing both Ò and Ó .In both cases, there is a bag in the resulting decomposition that contains both Ò and Ó .
If ✫ is not connected, then make separate domino tree decompositions for each connected component, and connect these to a tree in an arbitrary way.➣ The new idea in the proof can be found in step 2 of the procedure MAKEDEC: by adding the neighbours of the vertices in set ➬ ➏ ❼ to the root bag of the tree decomposition to make, we do not have to use these vertices at lower levels of the tree decomposition anymore.Apart from this idea, the structure of the algorithm is similar to algorithms found in [RS95,BGHK95].
Proof: Use the procedure, given in the proof above.Excluding the time spent in recursive calls of MAKEDEC, one call of MAKEDEC uses Ö ✶✭ ✹× ❐✳ time.There are Ö ✶✭ ✹× ❐✳ such calls (e.g., every vertex belongs to at most two bags, hence a tree decomposition with Ö ✶✭ ✹× ❐✳ nodes is obtained, and the number of recursive calls of MAKEDEC equals the number of nodes of the resulting tree decomposition), so the total time is bounded by Ö ✶✭ ☛× ➹✼ ▲✳ .➣

A lower bound
In this section, we show that a general bound like obtained in the previous section must always be of order P ◗✭ ✥✩ ✓✪ |✳ .

Final remarks
It is possible to give a modified version of the procedure of Corollary 3.2 on page 146, that yields domino tree decompositions of somewhat larger width (but still of Ö ✶✭ ✥✩ ✓✪ ✿✼ ❃✳ , but that uses Ö ✶✭ ☛× ◗ñ ëò ✻ó ✰× ❐✳ time instead of Ö ✶✭ ☛× ➹✼ ▲✳ time.However, the proof in [BE97] can be turned into an algorithm that uses linear time.It is not known how much time a procedure based upon the proof by Ding and Oporowski [DO95] would take.
The proof given in this paper seems unable to yield linear time algorithms -the approach typically leads to algorithmic results of ◗✭ ☛× ◗ñ ëò ✻ó (× ❐✳ time.It is open whether domino tree decompositions of Ö ✶✭ ✒✩ ✓✪ ✻✼ ❃✳ width can be obtained with a linear time algorithm.
Another interesting open problem is whether a bound of Ö ✶✭ ✒✩ ✼ ✪ ✿✳ can essentially be improved.It would be interesting to see if better bounds, e.g., a bound of Ö ✶✭ ✥✩ ✓✪ |✳ can be proved, and whether better lower bounds are possible.
In some special cases, better bounds can be obtained.For instance, for trees we have the following easy result.
Theorem 5.1 The domino treewidth of a tree is at most its maximum degree.
Proof: Let ❤ be a tree with maximum degree ✪ .Choose an arbitrary root ➎ , and view ❤ as a rooted tree.Let ❤ ❜➐ ➯✬ ⑩✭ ✥✮ ❯➐ ✥✯ ✲✱ )➐ ➫✳ be the tree, obtained by removing all leaves from ❤ .Consider the following tree decomposition of ❤ : ✭ ↔❬ ☞( ❵ô ◗❛ ✆❄ õ❅ ♠✮ ❯➐ ✹❣ ✻✯ ✲❤ ❜➐ ë✳ , where each set ( ❵ô consists of ❄ and all children of ❄ in ❤ .One easily verifies that this is a domino tree decomposition of ❤ with width at most ✪ .➣ So for trees (and similarly for forests), the domino treewidth is linear in its degree.(Note also that the domino treewidth of a graph with maximum degree ✪ ✈Þ ➇❏ is at least ï ↔✭ ✥✪ ◗❉ ➆❏ ▲✳ ✲➡ ✦➋ ❃ð ❜▼ ★❏ : at most two bags can contain a vertex of degree ✪ and all its neighbours.)It seems interesting to see if it is also possible to obtain similar bounds for other restricted classes of graphs of bounded treewidth, e.g., series parallel graphs, Halin graphs, or arbitrary graphs of treewidth two.