# Question: A path cover of a directed graph G V

A path cover of a directed graph G = (V, E) is a set P of vertex-disjoint paths such that every vertex in V is included in exactly one path in P. Paths may start and end anywhere, and they may be of any length, including 0. A minimum path cover of G is a path cover containing the fewest possible paths.

a. Give an efficient algorithm to find a minimum path cover of a directed acyclic graph G = (V, E). (Hint: Assuming that V = {1, 2, . . . ,n}, construct the graph G' = (V', E'), where

b. and run a maximum-flow algorithm.)

c. Does your algorithm work for directed graphs that contain cycles? Explain.

a. Give an efficient algorithm to find a minimum path cover of a directed acyclic graph G = (V, E). (Hint: Assuming that V = {1, 2, . . . ,n}, construct the graph G' = (V', E'), where

b. and run a maximum-flow algorithm.)

c. Does your algorithm work for directed graphs that contain cycles? Explain.

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