\in V(T)$ such that $u,v \in B_x$, and for every vertex $u \in V(G)$ the set of nodes $\{ x\in V(T) : u \in B_x \}$ induce a subtree of $T$. The vertices of $T$ are referred to as *nodes* and the sets $B_x$ are called bags.

A *rooted tree decomposition* $(T,{\mathcal}{X},r)$ is a tree decomposition with a distinguished node $r$ selected as the *root*. A *nice tree decomposition* $(T,{\mathcal}{X},r)$ (see [@Klo94]) is a rooted tree decomposition where $T$ is binary and for each node $x$ with two children $y,z$ it holds $B_x =B_y =B_z$ and for each node $x$ with one child $y$ it holds $B_x =B_y \cup \{u\}$ or $B_x =B_y \setminus \{ u\}$ for some $u \in V(G)$. The *width* of a tree decomposition is the size of a largest bag minus one. The *treewidth* of a graph, denoted by ${{\mathbf{tw}}}(G)$, is the smallest width of a tree decomposition of $G$. A *treewidth-modulator* of a graph $G$ is a set $X \subseteq V(G)$ such that ${{\mathbf{tw}}}(G-X) {\leqslant}t$, for some fixed constant $t$.

Given a bag $B$ (resp. a node $x$) of a rooted tree decomposition $T$, we denote by $G_B$ (resp. $G_x$), the subgraph induced by the 