minimum size=4pt,inner sep=0pt\]; at (0,3) ; at (3,3) ; at (0,0) ; at (3,0) ; at (0,-1.2) [1]{}; at (-1.3,4) [2]{}; at (-1.8,0.2) [3]{}; plot \[smooth cycle\] coordinates [(-0.2,3.5) (3.5,4) (3.5, 2.5) (0.5,2.5) (0.5, -0.5) (-0.5, -0.5)]{}; plot \[smooth cycle\] coordinates [(2.5,3.5) (2.5,0.7) (-1,0.7) (-1,0) (3.2, -0.5) (3.5, 3.5)]{}; plot \[smooth cycle\] coordinates [(-1,3.5) (3,-1) (4,4)]{};

=\[draw, fill=white, shape=circle, minimum size=4pt,inner sep=2pt\]; (6,-12)–(6,12); (0,-12)–(0,12); (-6,-12)–(-6,12);

at (0,-13) [$x$]{}; at (6,-13) [$y$]{};

(10, -4)–(10, -7); (10, 4)–(10, 7); (-10, -4)–(-10, -7); (-10, 4)–(-10, 7);

at (-14,4) [$r_f$]{}; at (-14,-4) [$b_f$]{};

(-12,4)–(12,4);

(-12,7)–(12,7); (-12,8)–(12,8); (-12,9)–(12,9); (-12,10)–(12,10);

(-12,-4)–(12,-4);

(-12,-7)–(12,-7); (-12,-8)–(12,-8); (-12,-9)–(12,-9); (-12,-10)–(12,-10);

\(w) at (-4.5,-4) [$w$]{}; (u) at (1.5,-4) [$u$]{}; (v) at (4.5,-4) [$v$]{};

(w1) at (-6, -2) ; (w2) at (-6, -6) ; (u1) at (0, -2) ; (u2) at (0, -6) ; (v1) at (6, -2) ; (v2) at (6, -6) ;

(u1)–(u)–(u2); (v1)–(v)–(v2); (w1)–(w)–(w2);

NP-hardness reduction for every dimension {#sec:hard3d}
-----------------------------------------

In Section \[sec:hard2d\], we have proved that the mentioned problem is -hard in 2D by a reduction from 2-coloring of 3-uniform hypergraphs. Now, let us give the proof for the case where the wireless sensors are deployed in 3D.

\[lem:hard3d\] Hyperplanar substructures recognition is -hard in 3D.

This reduction 