  Consider the following classes of pairs consisting of a group and a finite
collection of subgroups:
  $\mathcal{C}= \left\{ (G,\mathcal H) \mid \text{$\mathcal{H}$ is
hyperbolically embedded in $G$} \right\}$ and
  $ \mathcal{D}= \left\{ (G,\mathcal H) \mid \text{the relative Dehn function
of $(G,\mathcal H)$ is well-defined} \right\}.$
  Let $G$ be a group that splits as a finite graph of groups such that each
vertex group $G_v$ is assigned a finite collection of subgroups
$\mathcal{H}_v$, and each edge group $G_e$ is conjugate to a subgroup of some
$H\in \mathcal{H}_v$ if $e$ is adjacent to $v$. Then there is a finite
collection of subgroups $\mathcal{H}$ of $G$ such that:
  $\bullet$ If each $(G_v, \mathcal{H}_v)$ is in $\mathcal C$, then
$(G,\mathcal{H})$ is in $\mathcal C$.
  $\bullet$ If each $(G_v, \mathcal{H}_v)$ is in $\mathcal D$, then
$(G,\mathcal{H})$ is in $\mathcal D$.
  $\bullet$ For any vertex $v$ and for any $g\in G_v$, the element $g$ is
conjugate to an element in some $Q\in\mathcal{H}_v$ if and only if $g$ is
conjugate to an element in some $H\in\mathcal{H}$.
  That edge groups are not assumed to be finitely generated and that they do
not necessarily belong to a peripheral collection of subgroups of an adjacent
vertex are the main differences between this work and previous results in the
literature. The method of proof provides lower and upper bounds of the relative
Dehn functions in terms of the relative Dehn functions of the vertex groups.
These bounds generalize and improve analogous results in the literature.
