Given a complex semisimple Lie algebra \(\mathfrak{g}\) and the subalgebra \(\mathfrak{g}_0 \subset \mathfrak{g}\) fixed by a diagram automorphism of \(\mathfrak{g}\), the authors investigate a branching problem on \((\mathfrak{g}, \mathfrak{g}_0)\). Let \(d\) be a saturation factor given by \((\mathfrak{g}, \mathfrak{g}_0)\), \(\rho\)(resp. \(\rho_0\)) be the half sum of positive roots in a root system of \(\mathfrak{g}\) (resp. \(\mathfrak{g}_0)\)) and \(\mu\) be a dominant weight of \(\mathfrak{g}_0\). The main result of the paper is to give a necessary and sufficient condition for the irreducible highest weight \(\mathfrak{g}_0\)-module \(V_0(d\mu)\) to occur in the restriction of the irreducible \(\mathfrak{g}\)-module \(V(d\mu)\) to \(\mathfrak{g}_0\).