Let F be the Cartesian product of N closed sets in ℂ. We prove that there exists a function g which is continuous on F and holomorphic on the interior of F such that ${\Gamma }_g\left(F\right):=(z,g(z\left)\right):z\in F$ is complete pluripolar in ${ℂ}^{N+1}$. Using this result, we show that if D is an analytic polyhedron then there exists a bounded holomorphic function g such that ${\Gamma }_g\left(D\right)$ is complete pluripolar in ${ℂ}^{N+1}$. These results are high-dimensional analogs of the previous ones due to Edlund [Complete pluripolar curves and graphs, Ann. Polon. Math. 84 (2004), 75-86]...