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