  We develope the theory of the $\mathcal{E}$-relative Igusa-Todorov functions
in an exact $ IT$-context $(\mathcal{C},\mathcal{E}).$ In the case when
$\mathcal{C}=$mod$\, (\Lambda)$ is the category of finitely generated left
$\Lambda$-modules, for an artin algebra $\Lambda,$ and $\mathcal{E}$ is the
class of all exact sequences in $\mathcal{C},$ we recover the usual
Igusa-Todorov functions. We use the setting of the exact structures and the
Auslander-Solberg relative homological theory to generalise the original
Igusa-Todorov's results. Furthermore, we introduce the $\mathcal{E}$-relative
Igusa-Todorov dimension and also we obtain relationships with the relative
global and relative finitistic dimensions and the Gorenstein homological
dimensions.
