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Homework 1

Algorithms on Directed Graphs, Winter 2018/9 Due: 2.11.2018 by 16:00

Exercise 1.Prove the followings:

(i) every strong tournament has a Hamiltonian cycle,

(ii) Use (i) to prove that there is a polynomial time algorithm to find a Hamiltonian path in tournaments.

Exercise 2.Let F be a minimum feedback arc set of G. Prove that the digraph obtained fromGby reversing all ofF arcs is acyclic.

Exercise 3.Recall the proof of Theorem FVS on Tournaments from the lecture note. Let Z be a subtournament of T obtained by adding X to the vertices in the longest common subsequence of L and R. Prove Z is the largest acyclic subtournament ofT containing all elements of X and no element ofS−X.

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