Unit disk graphs are the intersection graphs of unit radius disks in the
Euclidean plane.

Deciding whether there exists an embedding of a given unit disk graph, i.e.
unit disk graph recognition, is an important geometric problem, and has many
application areas.

In general, this problem is known to be $\exists\mathbb{R}$-complete.

In some applications, the objects that correspond to unit disks, have
predefined (geometrical) structures to be placed on.

Hence, many researchers attacked this problem by restricting the domain of the
disk centers.

One example to such applications is wireless sensor networks, where each disk
corresponds to a wireless sensor node, and a pair of intersecting disks
corresponds to a pair of sensors being able to communicate with one another.

It is usually assumed that the nodes have identical sensing ranges, and thus a
unit disk graph model is used to model problems concerning wireless sensor
networks.

We attack the unit disk graph realization problem on a restricted domain, by
assuming a scenario where the wireless sensor nodes are deployed on the
corridors of a building.

Based on this scenario, we impose a geometric constraint such that the unit
disks must be centered onto given straight lines.

In this paper, we first describe a polynomial-time reduction which shows that
deciding whether a graph can be realized as unit disks onto given straight
lines is NP-hard, when the given lines are parallel to either $x$-axis or
$y$-axis.

Using the reduction we described, we also show that this problem is NP-complete
when the given lines are only parallel to $x$-axis (and one another).

We obtain those results using the idea of logic engine introduced by Fekete,
Houle, and Whitesides in 1997.