localization. Thus, the unit disk graph model is used to model the network. It should be noted that realizing UDG is also an -hard problem, by reduction from [Unit disk graphs recognition]{} [@udg; @udgRecognition; @unitdiskapprox]. Since range-based localization uses UDG model, one can easily see that it is also -hard [@complexityOfWSN].

In [@cagiricithesis], it was observed, and experimentally verified that knowing which nodes are in the same group increases the accuracy, and decreases the computational time, when the number of groups is much smaller than the number of sensors. This is because conventional distance-based localization methods, such as trilateration [@trilat] and quadrilateration [@cagiricithesis], use distances from points (in our case sensors) in general position. However, in practice, objects lie on planar surfaces to form coplanar groups. Which means a large amount of computational power is spent on finding correct nodes to localize another node.

In an effort to perform a more efficient localization, we can exploit the information on the hyperplanar substructures, if provided. That is, a function $h : S \to \{1, 2, \dots, k\}$, called *hyperplanar grouping function*. Two sensor nodes $s_i$ and $s_j$ are said to be on the same hyperplanar group if, and only if, $h(i) 