Graph optimization

I am working on an interesting graph optimization problem and I would
like to have a few expert opinions for helping me with a solution.

So here goes ...

I have a black box with a complex internal circuitry that is
represented in the form of a graph. I have to abstract the graph by
reducing the number of internal points and constructing cumulative
paths from each input to every possible output in case a path exists (a
series combination of edges). Each edge of the graph has a weight
associated with it. I have to add up all the weights to form the
resultant path.

My representation of the graph is in the form of a linked list
structure as I do not know the number of nodes in the graph apriori ..

A simple situation would be as follows ..

a b a+b
1----> 2 ----> 3 ====> 1 ----> 3

An edge from 1 to 2 with weight 'a' and an edge from 2 to 3 with weight
'b' should be transformed into an edge from 1 to 3 with weight 'a+b'.
Of course, this is a naive example but one can imagine different
scenarios with multiple input arcs and multiple output arcs into/from
internal nodes. I have to find a cumulative path from each input to
every possible output in the most efficient manner. I know the input
points as well as the output points.

Can any C++ guru suggest an detailed algorithm with pseudo code for
solving this problem ?

.