 
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 i1 , v i } # E or the arc a i = (v i1 , 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
