Before revision:  We show that deciding whether there exists a realization of a given graph as unit disk graphs on straight lines is NP-hard, even if the given lines are parallel to either x-axis or y-axis.
After revision:  In this paper, we first describe a polynomial-time reduction which shows that deciding whether there exists a realization of a given graph as unit disk graphs on straight lines is NP-hard, even if the given lines are parallel to either x-axis or y-axis.