Let \(\langle A; p,f\rangle\) be an algebraic system containing a binary predicate symbol \(p\) and a unary function symbol \(f\). An algebraic system \(\langle A; p,f\rangle\) is said to be an endograph if it satisfies the quasi-identity \((\forall x)(\forall y)(p(x,y)\to p(f(x),f(y)))\). The author proves that a set of minimal quasivarieties of endographs which has no independent basis of quasi-identities has the cardinality of the continuum.