Reducing closed TSP to open with fixed starting vertex I have implemented a branch and bound algoritm for finding a Hamiltonian cycle in my software, but I actually need to find a shortest route from fixed vertex through all verticies ending at any of them. 

As far as I understood, I can reduce closed-cycle TSP to open-cycle with fixed starting point by adding dummy vertex. But I can't find or figure out the weights of the edges of this vertex.

How can I do this?
Add two dummy vertices, $d_1,d_2$.  Add edges of weight 0 from all other vertices to $d_1$.  Add an edge of weight 0 from $d_1$ to $d_2$.  Add an edge of weight 0 from $d_2$ to your fixed starting vertex.  (If you need a complete graph, each missing edge can be replaced by an edge with weight $+\infty$ or some very large positive weight.)  Then any Hamiltonian cycle in the resulting graph will correspond to a Hamiltonian path in the graph that starts at your fixed vertex.