Let $s$ be a graph parameter (ex. diameter, domination number, etc)
A family $\mathcal{F}$ of graphs has the $s$-treewidth property if there is a function $f$ such that for any graph $G\in \mathcal{F}$, the treewidth of $G$ is at most $f(s)$.
For instance, let $s = \mathit{diameter}$, and $\mathcal{F}$ be the family of planar graphs. Then it is known that any planar graph of diameter at most $s$ has treewidth at most $O(s)$.More generally, Eppstein showed that a family of graphs has the diameter-treewidth property if and only if it excludes some apex graph as a minor.Examples of such families are graphs of constant genus, etc.
As another example, let $s = \mathit{domination{-}number}$. Fomin and Thilikos have proved an analog result to Eppstein's by showing that a family of graphs has the domination-number-treewidth propertyif and only if $\mathcal{F}$ has local-treewidth. Note that this happens if and only if $\mathcal{F}$ has the diameter-treewidth property.
Questions:
- For which graph parameters $s$ is the $s$-treewidth property known to hold on planar graphs?
- For which graph parameters $s$ is the $s$-treewidth property known to hold on graphs of bounded local-treewidth?
- Are there any other families of graphs, not comparable to graphs of bounded local-treewidth for which the $s$-treewidth property holds for some suitable parameter $s$?
I have a feeling that these questions have some relation with the theory of bidimensionality. Within this theory,there are several important parameters. For instance, the sizes of feedback vertex set, vertex cover,minimum maximal matching, face cover, dominating set, edge dominating set,R-dominating set, connected dominating set, connected edge dominating set,connected R-dominating set, etc.
- Does any parameter $s$ encountered in bidimensionality theory have the $s$-treewidth property for some suitable family of graphs?