$3.$ Suppose that we are given a flow network G and max flow f on G$.$ We know that all the capacities are integers in G$,$ and all flow values are also integers. Suppose that the capacity of an edge $($u$,$v$)$ is increased by $1.$

$($a$)$ How can you decide if the increase in the capacity affects the max flow value? Find an efficient algorithm to decide this.$($b$)$ If f is not max$-$flow any more, what is the new max$-$flow value?$($c$)$ How do you find the new max$-$flow function in an efficient way?