A Note on Orientations of Mixed Graphs # Esther M. Arkin + Refael Hassin # Summary: A Note on Orientations of Mixed Graphs # Esther M. Arkin + Refael Hassin # October 27, 2004 Abstract We consider orientation problems on mixed graphs in which the goal is to obtain a directed graph satisfying certain connectivity requirements. Keywords: Mixed graphs, orientations, NP­complete. 1 Introduction Let G = (V, E, A) be a mixed graph with a set of vertices V , a set of (undirected) edges E and a set of (directed) arcs A. For vertices s and t, an s - t path is a sequence s = v 0 , a 1 , v 1 , a 2 , v 2 , ..., a k , v k = t such that for i = 1, ..., k v i # V , a i is either an edge a i = {v i-1 , v i } # E or the arc a i = (v i-1 , v i ) # A. By orienting an edge e = {v i , v j } # E we mean replacing e by exactly one of the two arcs (v i , v j ) or (v j , v i ). An orientation of G is an orientation of all the edges in E. In this paper we refer by `disjoint paths' to `edge/arc internally disjoint paths'. This paper considers several orientation problems on mixed graphs. The objective is to obtain a directed graph satisfying certain connectivity requirements. We begin, in Section 2, with pair connectivity problems, in which a list of pairs of vertices is given, and we require the resulting directed graph to have a directed path between each pair of them. This problem is polynomially solvable for undirected graphs [4], however, we prove that it is NP­complete for mixed graphs. In the case of two pairs of vertices we give a polynomial time algorithm based on a set of necessary and su#cient conditions. In Section 3 we consider higher Collections: Mathematics