# Generalization of locally bounded treewidth graphs

Is the following graph class known in the literature?

The class of graphs is parameterized by positive integers $d$ and $t$ and contains each graph $G=(V,E)$ such that for each vertex $v\in V$, the subgraph of $G$ induced on all vertices at distance at most $d$ from $v$ in $G$ has treewidth at most $t$.

It generalizes the concept of locally bounded treewidth, and it seems useful when searching for local structures in graphs.

The concept of local treewidth introduced in M. Frick, M.Grohe, Deciding first-order properties of locally tree-decomposable structures is more general than the definition of locally bounded treewidth in the wikipedia article to which you refer. For each graph $G$, the local treewidth of $G$ is the function $ltw^G$ which maps a radius $r$ to the maximum treewidth of $N^G_r(v)$ among all vertices $v$ of $G$, where $N^G_r(v)$ is the subgraph of $G$ induced by vertices at distance at most $r$ to $v$. A class has bounded local treewidth, if there exists a function $f$ such that $ltw^G(r) \leq f(r)$ for each $r$ and each graph $G$ belonging to the class.