  Let $Q$ be a quiver and $R$ an associative ring. A representation by
$R$-modules of $Q$ is called strongly fp-injective if it admits a pure acyclic
injective resolution in the category of representations. It is shown that such
representations possess many nice properties. We characterize strongly
fp-injective representations under some mild assumptions, which is closely
related to strongly fp-injective $R$-modules. Subsequently, we use such
representations to define relative Gorenstein injective representations, called
Gorenstein strongly fp-injective representations, and give an explicit
characterization of the Gorenstein strongly fp-injective representations of
right rooted quivers. As an application, a model structure in the category of
representations is given.
