Relative global Gorenstein dimensions

Let $\mathcal{A}$ be an abelian category. In this paper, we investigate the global $(\mathcal{X},\mathcal{Y})$-Gorenstein projective dimension $\mathrm{g}\mathrm{l}\mathrm{.}\mathrm{G}\mathrm{P}\mathrm{D}(\mathcal{X},\mathcal{Y})(\mathcal{A})$, associated to a GP-admissible pair $(\mathcal{X},\mathcal{Y})$. We give homological conditions over $(\mathcal{X},\mathcal{Y})$ that characterize it. Moreover, given a GI-admisible pair $(\mathcal{Z},\mathcal{W})$, we study conditions under which $\mathrm{g}\mathrm{l}\mathrm{.}\mathrm{G}\mathrm{I}\mathrm{D}(\mathcal{Z},\mathcal{W})(\mathcal{A})$ and $\mathrm{g}\mathrm{l}\mathrm{.}\mathrm{G}\mathrm{P}\mathrm{D}(\mathcal{X},\mathcal{Y})(\mathcal{A})$ are the same.