Empty intersection of longest path in connected graph
You need not find a vertex that does not appear in all longest paths (in fact you can't). This is neither sufficient nor necessary. It is sufficient to prove that for each vertex $v$, there exists a longest path that does not contain $v$.

Easy to see the length of the longest path is 10. For convenience, let's encode the vertices as follows.

     1
     2
    3 4
   5   6
  7 8 9 10
11        12


Then for vertices 1,2,7,11,10,12, for example, 1, there is a longest path 11-7-8-4-6-5-3-9-10-12 that does not contain 1.

For vertices 3,4,5,6,8,9, for example, 3, there is a longest path 1-2-4-6-5-7-8-9-10-12 that does not contain 3.