Guarding polyhedral terrains

Bose, Prosenjit; Shermer, Thomas C.; Toussaint, Godfried T.; Zhu, Binhai

doi:10.1016/0925-7721(95)00034-8

Cited by 53 publications

(33 citation statements)

References 9 publications

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“…On the other hand, p(G) ≥ 2 for all plane graphs G with g(G) ≥ 3. This was first proved by Bose et al [3] (see also [17]) by using the Four-Color Theorem, and afterwards by Bose et al [4] without using it. The following sketch of the second proof is similar in spirit to some of the ideas in the present paper.…”

Section: It Is Clear That For Every Plane Graph G P(g(g)) ≤ P(g) ≤ G(g)mentioning

confidence: 91%

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Polychromatic Colorings of Plane Graphs

Alon

Berke

Buchin

et al. 2009

Discrete Comput Geom

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We show that the vertices of any plane graph in which every face is incident to at least g vertices can be colored by (3g − 5)/4 colors so that every color appears in every face. This is nearly tight, as there are plane graphs where all faces are incident to at least g vertices and that admit no vertex coloring of this type with more than (3g + 1)/4 colors. We further show that the problem of determining whether a plane graph admits a vertex coloring by k colors in which all colors appear in every face is in P for k = 2 and is N P-complete for k = 3, 4. We refine this result for polychromatic 3-colorings restricted to 2-connected graphs which have face sizes from a prescribed (possibly infinite) set of integers. Thereby we find an almost complete characterization of these sets of integers (face sizes) for which the corresponding decision problem is in P, and for the others it is N P-complete.

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Section: It Is Clear That For Every Plane Graph G P(g(g)) ≤ P(g) ≤ G(g)mentioning

confidence: 91%

“…In [3] it is shown that one can guard any plane graph on n vertices with no faces of size 1 or 2 by n 2 guards. This clearly follows from the fact that p(G) ≥ 2 for any such graph.…”

Section: Theorem 3 the Decision Problem Whether A Plane Graph Is Polymentioning

confidence: 99%

Polychromatic Colorings of Plane Graphs

Alon

Berke

Buchin

et al. 2009

Discrete Comput Geom

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“…The definitions of Manhattan towers and visibility are mostly from [8] and [4], respectively. Let Z k be the plane {z = k} for k ≥ 0.…”

Section: Definitionsmentioning

confidence: 99%

“…It is known that ⌊n/2⌋ is both the lower bound [4] and the upper bound [3] of vertex guards of a polyhedral terrain. Also, the minimum vertex-guard problem is known to be NP-hard [7].…”

Section: Introductionmentioning

confidence: 99%

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Visibility Problems for Manhattan Towers

Iwamoto

Kitagaki

2016

IEICE Trans. Inf. & Syst.

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SUMMARY A Manhattan tower is a monotone orthogonal polyhedron lying in the halfspace z ≥ 0 such that (i) its intersection with the xy-plane is a simply connected orthogonal polygon, and (ii) the horizontal cross section at higher levels is nested in that for lower levels. Here, a monotone polyhedron meets each vertical line in a single segment or not at all. We study the computational complexity of finding the minimum number of guards which can observe the side and upper surfaces of a Manhattan tower. It is shown that the vertex-guarding, edge-guarding, and face-guarding problems for Manhattan towers are NP-hard.

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A note on coverings of plane graphs

Rivera-Campo

1995

Journal of Graph Theory

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Guarding polyhedral terrains

Cited by 53 publications

References 9 publications

Polychromatic Colorings of Plane Graphs

Polychromatic Colorings of Plane Graphs

Visibility Problems for Manhattan Towers

A note on coverings of plane graphs

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