Relative derived equivalences and relative Igusa-Todorov dimensions

Let $A$ be an Artin algebra and $F$ a non-zero subfunctor of ${\text{Ext}}_A^1(-,-)$. In this paper, we characterize the relative $\phi$-dimension of $A$ by the bi-functor ${\text{Ext}}_F^1(-,-)$. Furthermore, we show that the finiteness of relative $\phi$-dimension of an Artin algebra is invariant under relative derived equivalence. More precisely, for an Artin algebra $A$, assume that $F$ has enough projectives and injectives, such that there exists $G\in \text{modcat}A$ such that $\text{add}G=\mathcal{P}(F)$, where $\mathcal{P}(F)$ is the category of all $F$-projecitve $A$-modules. If $\text{cpx}T$ is a relative tilting complex over $A$ with term length $t(\text{cpx}T)$ such that $B=\text{End}(\text{cpx}T)$, then we have ${\text{phd}}_F(A)-t(T^{\bullet })\le \text{phd}(B)\le {\text{phd}}_F(A)+t(T^{\bullet })+2$.