Hint for Exercise 7.27, part 1.

Instead of using either Hamiltonian Path or X3C, use 3SAT.

Use a construction similar to that in Figure 7.4 (on p. 237).
Differences are:  
* use a single vertex for each clause 
* add a "tail" of 2 edges to the topmost vertex ("root")
* the lengths (weights) of edges are:
  - from root to literal vertices is 2
  - between literals is 1
  - coming out of clause vertices is 3

Set diameter bound to 4 and total weight bound to 3(n+m),
where n is the number of variables and m is the number of clauses.