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Definitions: For a graph $G$, a $k$-tree completion of $G$ is a $k$-tree obtained by adding edges to $G$ (if $G$ has a $k$-tree completion, $G$ is said to be a partial $k$-tree). The least integer $k$ such that $G$ has a $k$-tree completion is called the treewidth of $G$. The maximum degree of a graph $G$ is denoted by $\Delta(G)$.

Question: Suppose that $G$ has a $k$-tree completion.
Is there an upper bound $U$ in terms of $k$ and $\Delta(G)$ such that
there exist a $k$-tree completion $G'$ of $G$ with $\Delta(G')\leq U\,?$

(note: To be explicit, $U$ is a function of $k$ and $\Delta(G)$ )

Context: I have a graph parameter $x(G)$ such that $x(G)\leq \Delta(G)+p$ for every chordal graph $G$ where $p$ is a constant. Using this result, I am trying to give an upper bound for $x(G)$ in general graphs in terms of treewidth of $G$ and $\Delta(G)$.

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  • $\begingroup$ So do you care that the completion is a k-tree completion and not a k’-tree completion for a k’ which is a function of the treewidth k and max degree of G? $\endgroup$
    – daniello
    Oct 9 '20 at 2:53
  • $\begingroup$ @daniello Not really. It is perfectly fine if it is a k'-tree completion where k' is a function of treewidth k of G and max degree of G. $\endgroup$ Oct 9 '20 at 3:53
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It is known that a graph of treewidth $k$ and maximum degree $\Delta$ has tree partition width at most $O(k\Delta)$. See Wood, arXiv:math/0602507.

From a tree partition of width $O(k\Delta)$ a triangulation of width $O(k\Delta)$ and maximum degree $O(k\Delta^2)$ follows pretty directly (start with the tree partition, make every bag into a clique, make every vertex $v$ in $U$ adjacent to all vertices in all bags appearing together with neighbors of $v$).

So you get a (minimal) triangulation with the bounds above.

Note: I would not be surprised if you can get a better degree bound.. the additional $\Delta$ term feels unnecessary.

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  • $\begingroup$ Note: You ask for a k’-tree completion, this answer gives a triangulation. From your comments my guess is that a triangulation of width k’ is good enough for you. If not, you can probably still get a k’-tree completion with some additional massaging on top of this. $\endgroup$
    – daniello
    Oct 9 '20 at 6:47
  • $\begingroup$ Yeah, a trangulation (a.k.a. chordal completion) is good enough for the purpose. Thanks. $\endgroup$ Oct 9 '20 at 6:53
  • $\begingroup$ It's rather unfortunate that the bound involves $\Delta^2$. For the parameter $x(G)$ I am dealing with, it is known that $x(G)\leq \Delta(G)^2+1$ for every graph. :) $\endgroup$ Oct 9 '20 at 6:57

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