Example: HP ≤ p HC Hamiltonian Path –Input: Undirected Graph G = (V,E) –Y/N Question: Does G contain a Hamiltonian Path? Hamiltonian Cycle –Input: Undirected Graph G = (V,E) –Y/N Question: Does G contain a Hamiltonian Cycle?

Specification of R(x) Consider any undirected graph G = (V,E) as input x R(x) will be a graph G’ = (V’, E’) where –V’ = V union {v} where v is not in V –E’ = E union {(v,w) | w in V} Argument that R(x) has polynomial size –We add exactly 1 node and |V| edges.

x is yes  R(x) is yes Suppose graph G has a Hamiltonian Path Let this path be v 1, v 2, …, v n We now argue that v 1, v 2, …, v n, v is a Hamiltonian Cycle in G’ –First, all nodes in V’ are included exactly once above or else v 1, v 2, …, v n would not be a HP in G –Since G’ has all the edges that G has, (v i,v i+1 ) is an edge in E’ for 1 ≤ i ≤ n-1 –Finally, since E’ contains edge (v,w) for all w in V, it must be the case that E’ contains edges (v n, v) and (v,v 1 ).

R(x) is yes  x is yes Suppose graph G’ has a Hamiltonian Cycle Let this cycle be v 1, v 2, …, v n, v We now argue that v 1, v 2, …, v n is a Hamiltonian Path in G –First, all nodes in V are included exactly once above or else v 1, v 2, …, v n, v would not be a HC in G’ –Since the only extra edges in E’ compared to E are edges involving node v, it must be the case that E contains edge (v i,v i+1 ) for 1 ≤ i ≤ n-1

Turing Reducibility Steve made a suggestion for an alternate reduction. Given graph G, output  (n 2 ) graphs G v,w = (V,E v,w ) where –E v,w = E union {(v,w)} where v,w are nodes in V This is not a polynomial-time reduction because we are outputting  (n 2 ) graphs. However, this idea can be used to show that if HC can be solved in polynomial time, then HP can be solved in polynomial time. –Run each graph G v,w through our procedure that solves HC. –If HC says yes for any one of these graphs, return yes. –Otherwise return no. This more general reduction is often called a Turing reduction. We allow ourselves to use the procedure that solves HC (or  2 ) a polynomial number of times rather than just once.