Question: Suppose you are given an undirected graph G = (V,E), V = n. We want to determine whether the graph contains a simple cycle of

Suppose you are given an undirected graph G = (V,E), V = n. We want to determine

whether the graph contains a simple cycle of length ceiling[n/2]. Let's call this the Half Tour problem.

a) Show that Half Tour is in NP, by giving a polynomial-time algorithm for verifying a

certicate which is a (claimed) simple cycle of length ceiling[n/2] in a graph G = (V,E).

b) Describe the steps of a reduction from an arbitrary Hamiltonian cycle instance G,

to a specially-constructed Half Tour instance G', to deciding whether the original graph G

has a Hamiltonian cycle.

c) Prove that G has a Hamiltonian cycle => G' has a half tour.

d) Prove that G' has a half tour => G has a Hamiltonian cycle.

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