Characterization of the graphs with bounded above equivalence partition number in the class of $\mathcal U$-split graphs

It is proved that, for an arbitrary fixed $k\ge3$, the class $L^l(k)$ of graphs with equivalence partition number at most $k$ can be characterized by means of a finite list of forbidden induced subgraphs in an extension of the class of split graphs — the class of $\mathcal{U}$-split graphs. In the case $k=3$ the corresponding list as well as a description of the graphs in $L^l(3)$ in the class of $\mathcal{U}$-split graphs not being split are obtained.