Application of the Luttinger theorem to the Kondo lattice YbRh ₂ Si ₂ suggests that its large 4f-derived Fermi surface (FS) in the paramagnetic (PM) regime should be similar in shape and volume to that of the divalent local-moment antiferromagnet (AFM) EuRh ₂ Si ₂ in its PM regime. Here we show by angle-resolved photoemission spectroscopy that paramagnetic EuRh ₂ Si ₂ has a large FS essentially similar to the one seen in YbRh ₂ Si ₂ down to 1 K. In EuRh ₂ Si ₂ the onset of AFM order below 24.5 K induces an extensive fragmentation of the FS due to Brillouin zone folding, intersection and resulting hybridization of the Fermi-surface sheets. Our results on EuRh ₂ Si ₂ indicate that the formation of the AFM state in YbRh ₂ Si ₂ is very likely also connected with similar changes in the FS, which have to be taken into account in the controversial analysis and discussion of anomalies observed at the quantum critical point in this system.