$4.$ An edge $\backslash ($ e $\backslash )$ can be $(\backslash ($ x $\backslash ))$ always full, $($y$)$ sometimes full, $($z$)$ never full; it can be $(\backslash ($ x$^\{\backslash$prime$\}\backslash ))$ always crossing, $(\backslash ($ y$^\{\backslash$prime$\}\backslash ))$ sometimes crossing, $(\backslash (\backslash$left$.$z$^\{\backslash$prime$\}\backslash$right$)\backslash )$ never crossing. So there are nine possible combinations: $(\backslash (\backslash$left$.$x x$^\{\backslash$prime$\}\backslash$right$)\backslash )$ always full and always crossing, $(\backslash ($ x$^\{\backslash$prime$\}$ y$^\{\backslash$prime$\}\backslash ))$ always full and sometimes crossing, and so on$.$ Or are there? Maybe some possibilities are impossible. Let's draw a table:

Take a piece of paper, complete the $\backslash (3\backslash$times $3\backslash )$ table below by either finding an example illustrating that the combination is possible or by concluding that it is impossible. To do so$,$ you will need the Max Flow Min Cut Theorem plus a bit of your own thinking. Once you have completed the table, select which of the following statements are true: There is a flow network and an edge $\backslash ($ e $\backslash )$ that is$...$

always full and always crossing.

always full and sometimes crossing.

always full and never crossing.

sometimes full and always crossing.

sometimes full and sometimes crossing.

sometimes full and never crossing.

never full and always crossing.

never full and sometimes crossing.

never full and never crossing.