  We consider the extended discrete KP hierarchy and show that similarity
reduction of its subhierarchies lead to purely discrete equations with
dependence on some number of parameters together with equations governing
deformations with respect to these parameters. It is written down discrete
equations which naturally generalize the first discrete Painlev\'e equation
$\mathrm{dP}_{\rm I}$ in a sense that autonomous version of these equations
admit the limit to the first Painlev\'e equation. It is shown that each of
these equations describes B\"acklund transformations of Veselov-Shabat periodic
dressing lattices with odd period known also as Noumi-Yamada systems of type
$A_{2(n-1)}^{(1)}$.
