. Proof, . Oddpath-hornunsat-?-ps, N. Pace, and . Pace, Given a directed graph G and a starting node u, we will create a sat-graph SAT-G, such that (G, u) ? GEOGRAPHY if and only if (SAT-G, u) ? ODDPATH-HORNUNSAT. For this purpose, we will do the following actions: (1) First, we take the directed graph G = (V, E) and for each vertex v ? V, we create a new state v and add an edge v ? v . We will call the vertex v as the clone vertex of v, this way, we create a new graph G = (V , E )

?. Next, G. V-in-the-graph, and E. , we create a new Boolean variable x v which will be linked to vertex v. We say that x v is represented by the vertex v

. Proof, We prove ODDPATH-HORNUNSAT is NP and PSPACE-complete in Theorems 3

S. A. Cook, The complexity of theorem-proving procedures, Proceedings of the third annual ACM symposium on Theory of computing , STOC '71, pp.151-158, 1971.
DOI : 10.1145/800157.805047

L. Fortnow, The status of the P versus NP problem, Communications of the ACM, vol.52, issue.9, pp.78-86, 2009.
DOI : 10.1145/1562164.1562186

A. M. Turing, On Computable Numbers, with an Application to the Entscheidungsproblem, Proceedings of the London Mathematical Society, vol.42, pp.230-265, 1936.

C. H. Papadimitriou, Computational complexity, 1994.

T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms, 2001.

M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, 1979.