Joel,

In GridTools::find_cells_adjacent_to_vertex(dof_handler,center_id) you need 
to give the number of the vertex_number as center_id, not the number of the 
degree of freedom.
Then, you want to use 

Quadrature<dim>  
quadrature_formula(dof_handler.get_fe().get_unit_support_points());

to create a Quadrature object that gives you the support points on each 
cell.
Finally, you can output the global dof number and their support point on 
each cell by: 

for (unsigned int j=0; j<dofs_per_cell; ++j)
  std::cout << local_dof_indices[j] << ": "<< fe_values.quadrature_point(j) 
<< std::endl;

In this ouput, you can then search for the dofs with support_point nearest 
to the vertex you wanted to consider.

Best,
Daniel

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