Abstract
We prove that ⌊ n 2 ⌋ vertex guards are always sufficient and sometimes necessary to guard the surface of an n-vertex polyhedral terrain. We also show that ⌊ (4n − 4) 13 ⌋ edge guards are sometimes necessary to guard the surface of an n-vertex polyhedral terrain. The upper bound on the number of edge guards is ⌊ n 3 ⌋ (Everett and Rivera-Campo, 1994). Since both upper bounds are based on the four color theorem, no practical polynomial time algorithm achieving these bounds seems to exist, but we present a linear time algorithm for placing ⌊ 3n 5 ⌋ vertex guards for covering a polyhedral terrain and a linear time algorithm for placing ⌊ 2n 5 ⌋ edge guards.
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