  We study the singularity of the order parameter at the transition between a
critical phase and an ordered phase of bond percolation on pointed hierarchical
graphs. In pointed hierarchical graphs, the renormalization group (RG) equation
explicitly depends on the bare parameter, which causes the phase transitions
that correspond to the bifurcation of the RG fixed point. We derive the
relation between the type of this bifurcation and the type of the singularity
of the order parameter. In the case of a saddle node bifurcation, the
singularity of the order parameter is power-law or essential one depending on
the fundamental local structure of the graph. In the case of pitchfork and
transcritical bifurcations, the singularity is essential and power-law ones,
respectively. These becomes power-law and discontinuous ones, respectively, in
the absence of the first-order perturbation to the largest eigenvalue of the
combining matrix, which gives the growth rate of the cluster size. We also show
that the first-order perturbation vanishes if the relevant RG parameter is
unique and the backbone of the pointed hierarchical graph is simply connected
via nesting subunits.
