I am interested in the following variation of decomposing a graph into a tree. It is related to both the standard tree-decomposition of a graph and tree-cut decompositions.
Given a graph $G$, we want to find a tree $T$ and a partition $\{X_v \subseteq V(G): v \in V(T)\}$ of the vertices of $G$ indexed by the vertices of $T$, such that for all edges $xy \in E(G)$, either $x$ and $y$ are contained in the same set $X_v$, or there exists an edge $uv \in E(T)$ such that $x \in X_u$, $y \in X_v$. The width of the decomposition is the maximum size of $X_v$.
I remember having seen this type of decomposition analyzed in a paper, but I can no longer find a reference for the work. The specific question I'm interested in is the following.
Is it true that bounded treewidth and bounded degree suffice to force a bounded width decomposition of the above type?
Note that if the question holds, then it also implies that there exists a tree-decomposition of the graph such that every vertex is in a bounded number of bags, answering a question of Monkeymaths. To see this, assume $G$ has bounded degree and bounded treewidth, and assume that there exists a tree $T$ and partition $\{X_v: v \in V(T)\}$ as above. We may assume that $G$ is connected. It follows that the tree $T$ has bounded degree as well since for a $v \in V(T)$, every neighbor of $u$ of $v$ in $T$ must have a vertex $x \in X_u$ which is adjacent some vertex $y \in X_v$ by connectivity. To get the desired tree-decomposition of $G$, we subdivide each edge of $T$ to get a tree $T'$ and define $\{X'_v: v\in V(T')\}$ as follows. For $v \in V(T) \cap V(T')$, let $X'_v = X_v$. For edges $e = uv$, let $v_e$ be the vertex of $T'$ corresponding to the subdivided edge $e$. Define $X'_{v_e} = X_u \cup X_v$. Then $\{X'_v: v\in V(T')\}$ form the bags of a tree decomposition of bounded width. As the tree $T$ has bounded degree, every vertex of $G$ is in a bounded number of bags of $\{X'_v: v\in V(T')\}$, as desired.
I do not know whether if $G$ has bounded degree, bounded treewidth, and a bounded width tree-decomposition such that every vertex is in a bounded number of bags it follows that $G$ has the type of tree-partition decomposition of bounded width we defined above.
