  A graph is Helly if its disks satisfy the Helly property, i.e., every family
of pairwise intersecting disks in G has a common intersection. It is known that
for every graph G, there exists a unique smallest Helly graph H(G) into which G
isometrically embeds; H(G) is called the injective hull of G. Motivated by
this, we investigate the structural properties of the injective hulls of
various graph classes. We say that a class of graphs $\mathcal{C}$ is closed
under Hellification if $G \in \mathcal{C}$ implies $H(G) \in \mathcal{C}$. We
identify several graph classes that are closed under Hellification. We show
that permutation graphs are not closed under Hellification, but chordal graphs,
square-chordal graphs, and distance-hereditary graphs are. Graphs that have an
efficiently computable injective hull are of particular interest. A linear-time
algorithm to construct the injective hull of any distance-hereditary graph is
provided and we show that the injective hull of several graphs from some other
well-known classes of graphs are impossible to compute in subexponential time.
In particular, there are split graphs, cocomparability graphs, bipartite graphs
G such that H(G) contains $\Omega(a^{n})$ vertices, where $n=|V(G)|$ and $a>1$.
