I found this question here
The question is 
I wrote the LP problem as this:
Let $x_{ij}$ be the maximum no.of flights between city i and city j.
Let $a_0$ be the artificial link and $x_0$ be the amount of flights from Juneau to Dallas.
$s_0$ is the sourse and $s_i$ is the sink
Maximize $z=x_0$ subject to:
$x_{s0,1}<=3\\ x_{1,2}<=3\\ x_{1,3}<=2\\x_{2,si}<=2\\x_{3,si}<=1$
$x_0=x_{s0,1}\\x_{3,si}+x_{2,si}=x_{0}\\x_{s0,1}=x_{1,3}+x_{1,2}\\x_{1,2}=x_{2,si}\\x_{1,3}=x_{3,si}\\x_{i,j}>=0$
I am not sure if I have included the constraint Connecting flights must stop in Seattle and then stop in Los Angeles and Denver?
Have I written the LP problem correctly?
If I am not mistaken, this is the maximal flow analysis. The maximum number of flights from Juneau to Seattle determines the maximum flow of 3 and these three flights can be flown, one through Los Angeles and two through Denver. This problem could be an illustration to explain edge capacity and thus maximum flow in a network of directed graph. Maximum would not occur if all 3 starting from Juneau be directed through the nodes 1 and 2 because the edge capacity node 2,Si is only 2. One more observation, had there be an edge between Los Angeles and Denver, then the problem is a little deeper.
As such you do not need a LP problem ( which is, by the way, overspecified).