Decidewhetheryouthinkthefollowingstatementistrueorfalse.Let[mathjaxinline]G[/mathjaxinline]beanarbitraryflownetwork,withasource[mathjaxinline]s[/mathjaxinline],asink[mathjaxinline]t[/mathjaxinline],andapositiveintegercapacity[mathjaxinline]c_e[/mathjaxinline]oneveryedge[mathjaxinline]e[/mathjaxinline].If[mathjaxinline]f[/mathjaxinline]isamaximum[mathjaxinline]s-t[/mathjaxinline]flowin[mathjaxinline]G[/mathjaxinline],then[mathjaxinline]f[/mathjaxinline]saturateseveryedgeoutof[mathjaxinline]s[/mathjaxinline]withflow(i.e.,foralledges[mathjaxinline]e[/mathjaxinline]outof[mathjaxinline]s[/mathjaxinline],wehave[mathjaxinline]f(e)=c_e[/mathjaxinline]).
Decidewhetheryouthinkthefollowingstatementistrueorfalse.Let$G$beanarbitraryflownetwork,withasource$s$,asink$t$,andapositiveintegercapacity$c_e$oneveryedge$e$.Let$(A,B)$beamimimum$s-t$cutwithrespecttothesecapacities${c_e:einE}$.Nowsupposeweadd1toeverycapacity,then$(A,B)$isstillaminimum$s-t$cutwithrespecttothesenewcapacities${1+c_e:einE}$.