There are two types of professional wrestlers: "babyfaces" ("good guys") and "heels" ("bad guys"). Between any pair of professional wrestlers, there may or may not be a rivalry. Suppose we have n professional wrestlers and we have a list of r pairs of wrestlers for which there are rivalries. Give an O(n + r) -time algorithm that determines whether it is possible to designate some of the wrestlers as babyfaces and the remainder as heels such that each rivalry is between a babyface and a heel. If it is possible to perform such a designation, your algorithm should produce it. What is the running time of your algorithm?

Steam in a heating system flows through tubes whose outer radius is 2.5 cm and whose walls are maintained at a temperature of 180 °c . circular aluminum alloy 2024-t6 fins ( k= 186 wm. k) of outer radius 3 cm and constant thickness 1 mm are attached to the tube. the space between the fins is 3 mm and thus there are 250 fins per meter of tube length. heat is transferred to the surrounding air at too-25 °c, with a heat transfer coefficient of 40 w/m2. k. manufacturer a claims that this system is rated at 3.5 kw per meter of tube length. if the fins are an integral part of the steam tube, find the heat transfer rate in kw for a 1-m length of tube. is manufacturer a's claim valid? compare the heat transfer rate from 1 meter of tube length if there are no fins.