Chvatal, V., "A combinatorial theorem in plane geometry," Journal of Combinatorial Theory, Series B, vol. 18, 1975, pp. 39-41.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....facilitates both. The 2 D coverage problem was posed by Victor Klee in 1973 and is better known as The Art Gallery Problem [15] For a polygon with n vertices, b observers are sufficient and sometimes necessary to cover the interior of the polygon. The first proof was given by Chv atal [4]. Later, Fisk gave a simpler proof by using a triangulation of the polygon and showing that as a triangulated polygon is 3 colorable, selecting the least used color will generate the bound [9] The placement of observers can be done in linear time [12] Alternative formulations of the 2 D coverage ....

Chv'atal, V. A combinatorial theorem in plane geometry. Journal of Comb. Theory Ser B 18 (1975), 39--41.

....polyhedral surfaces, or planar subdivisions. 1 Introduction In 1973, Victor Klee posed the problem of determining the minimum number of guards or light sources sufficient to cover or illuminate the interior of an n sided art gallery modeled by a simple polygon. Using a combinatorial argument [5], Chvatal showed that bn=3c guards are sufficient and sometimes necessary. Subsequently, Fisk [9] gave a concise and elegant proof of the same result, using the fact that the vertices of a triangulated polygon can be three colored. Since then, there has been an explosion of results in this area, ....

V. Chvatal. A combinatorial theorem in plane geometry. J. Comb. Theory Ser. B, 18, pp. 39--41, 1975.

....1.3 Art gallery problems The original art gallery problem was first proposed by V. Klee, who described the problem as the following: how many guards are necessary, and how many guards are su#cient, to guard the paintings and works of art in an art gallery with n walls Later, Chvatal [3] showed that n guards are always su#cient and occasionally necessary to guard a simple polygon with n edges. Since then, there have been numerous variations of the art gallery problem, including, but not limited to, vertex guard problem, edge guard problem, fortress and prison yard problems, ....

V. Chvatal. A combinatorial theorem in plane geometry. Journal of Combinatorial Theory, Series B 18:39--41, 1975.

....Purisima, Iztapalapa, Mexico DF, Mexico Jorge Urrutia, Department of Computer Science, University of Ottawa, Ottawa, Ontario, Canada 1. Introduction Illumination problems of several kinds have been studied intensely in recent years [6] One of the first results in this area is that of Chvatal [2] which states that any simple polygon with n vertices can be guarded (in the context of this paper illuminated) using at most [ lamps. It is also known that any family of n disjoint compact convex sets can be illuminated using at most 4n 7 lamps [5] Numerous variations of these problems have ....

....with n vertices can be guarded (in the context of this paper illuminated) using at most [ lamps. It is also known that any family of n disjoint compact convex sets can be illuminated using at most 4n 7 lamps [5] Numerous variations of these problems have been studied in the literature; see [1,2,3,4,5,6]. Normally it has been assumed that the light sources used emit light in all directions. In this paper we study a line illumination problem in which our light sources have a restricted angle of illumination, just the way a floodlight works. Thus for the rest of this paper, a floodlight is a ....

V. Chvatal, A combinatorial theorem in plane geometry, J. Comb. Theory Ser. B 18 (1975), 39-41.

....the problem of determining the minimum number of guards sufficient to cover the interior of an n sided art gallery (polygon) in 1973. Chv atal showed that bn=3c guards are sufficient and sometimes necessary to cover the interior of an n sided art gallery using a lengthy combinatorial argument [4]. Subsequently Fisk [8] gave a concise and elegant proof using the fact that the vertices of a triangulated polygon may be three colored. Since then, there has been an explosion in this area where many different variations of this original problem have been studied. The reader is referred to the ....

V. Chvatal. A combinatorial theorem in plane geometry. J. Comb. Theory Ser. B, 18:39--41, 1975.

....and FCAR 93ER0291. Address: School of Computer Science, McGill University, 3480 University, Montr eal, Qu ebec, H3A 2A7. E mail: godfried cs.mcgill.ca. Group C3, MS M986, Los Alamos National Lab, Los Alamos, NM 87545 USA. E mail: bhz c3serve.c3. lanl.gov a lengthy combinatorial argument [Chv75]. Subsequently Fisk [Fis78] gave a concise and elegant proof using the fact that the vertices of a triangulated polygon may be three colored. Avis and Toussaint [AT81] used Fisk s proof to design an O(n log n) algorithm for placing the guards. Recently, Kooshesh and Moret [KM92] showed that the ....

V. Chvatal. A combinatorial theorem in plane geometry. J. Comb. Theory Ser. B, 18:39--41, 1975.

....Z S S S C C C XXXXXXX C C C C C C C C . 6 C C CO . s . R L q 2 T p 1 p 2 q 1 g Fig. 3. Crossing visibility segments Definition 2. 2 (i) A walk on P is a pair (l, r) of continuous functions such that: l :[0,1] # L r : 0,1] # R l (0) r(0) s, l(1) r(1) g, l(t) is visible from r(t) for all t # [0, 1] Any line segment l(t) r(t) is called a walk line segment of the walk. The point r(t)isthewalk partner of l(t) and vice versa. ii) A walk on P is called straight if both l and r are ....

....S S C C C XXXXXXX C C C C C C C C . 6 C C CO . s . R L q 2 T p 1 p 2 q 1 g Fig. 3. Crossing visibility segments Definition 2. 2 (i) A walk on P is a pair (l, r) of continuous functions such that: l : 0,1] # L r :[0,1] # R l (0) r(0) s, l(1) r(1) g, l(t) is visible from r(t) for all t # [0, 1] Any line segment l(t) r(t) is called a walk line segment of the walk. The point r(t)isthewalk partner of l(t) and vice versa. ii) A walk on P is called straight if both l and r are non decreasing with ....

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V. Chvatal, "A Combinatorial Theorem in Plane Geometry", Journal of Combinatorial Theory B 13(6) (1975) 39--41.

....technique, we can improve and simplify a number of algorithms that deal with convex decompositions of simple polygons into k gons, as well as offer a structural characterization of various attributes of such polygons. Such characterizations include a very simple proof of Chvatal s Watchman Theorem [3] and of its rectilinear version due to Kahn et al. 5] We describe optimal algorithms, based on the degree sequence of the boundary vertices of the given simple polygon and its convex decomposition, to construct the dual tree of the decomposition in linear time. The same algorithm, with minor ....

....Corollary 1. No two vertices of C i , 1 i k, are adjacent. In particular, the vertex set of any simple polygon partitioned into convex k gons is two colorable for k even, three colorable for k odd. 2 This corollary allows us to give a particularly simple proof of a theorem of Chvatal [3]. Our proof is similar to that of Fisk [4] in that we both use the 3 colorability of a triangulated simple polygon. Our folding technique not only gives us a very clear proof of 3colorability, but also shows how to reduce the problem of watching over the interior of a triangulated simple polygon ....

V. Chvatal, "A combinatorial theorem in plane geometry," J. Combin. Theory Series B 18 (1975), 39--41.

....is extremely flexible and can easily be modified to give a set of radically different decompositions. An Efficient Algorithm for Decomposing a Polygon into Star Shaped PolygonsDecember 9, 1999 2 Our algorithm follows closely Fisk s constructive proof [7] of the following theorem due to Chvatal [8]: for every polygon with n vertices there exists a decomposition into at most [n 3] disjoint star shaped polygons. 2.0 The Algorithm Before stating the algorithm we introduce a few terms. A triangulation of a simple polygon P is a planar graph formed by adding as many non intersecting edges as ....

V. Chvatal, "A combinatorial theorem in plane geometry," J. Combin. Theory. B 18, 39-41 (197S).

.... shaped gallery of n walls (polygon of n edges) what is the minimum number of guards (lights) required to guard (illuminate) its interior [Ho] Vasek Chvatal soon established what has become known as Chvatal s Art Gallery Theorem : n 3 lights are always sufficient and sometimes necessary [Ch]. In 1981 Avis and Toussaint exhibited an efficient algorithm for actually finding the locations where these lights should be installed [AT] We should add that it is tempting to place a light on every third vertex of the polygon. The reader can easily design a gallery where this placement ....

V. Chvatal, "A combinatorial theorem in plane geometry," Journal of Combinatorial Theory, Series B, vol. 18, 1975, pp. 39-41.

....We can model the warehouse as a simple polygon of n vertices in the plane. A well known theorem that concerns this type of multiple facility location problem is Chvatal s Art Gallery Theorem [O R87] This theorem states that n 3 cameras are always sufficient and sometimes necessary to do the job [Ch75]. Avis and Toussaint [AT81] presented an O(n log n) time algorithm for finding the location of these cameras. The book by O Rourke [O R87] is devoted entirely to these types of facility location problems for simple polygons as customers. For the case of simple objects such as circles and squared ....

Chvatal, V., "A combinatorial theorem in plane geometry," Journal of Combinatorial Theory, Series B, vol. 18, 1975, pp. 39-41.

....results in this area the reader is referred to the paper by Radke [Ra88] It is expected that most of these proximity graphs can find a place in the document analysis problem where they can make practical contributions. 4.3 Polygon Decomposition 4.3. 1 Simple polygons In 1975 Vasek Chvatal [Ch75] proved that n 3 guards were always sufficient, and sometimes necessary, to guard (jointly see) the complete interior of a simple polygon (art gallery) consisting of n walls or vertices. This result has come to be known as Chvatal s Art Gallery Theorem and has since evolved to fill out an entire ....

Chvatal, V., "A combinatorial theorem in plane geometry," Journal of Combinatorial Theory Series B, vol. 18, 1975, pp. 39-41.

....to a quadrangulation because even though it discards all diagonals, it does not insert new diagonals between pairs of vertices. Although the Hamiltonian approach gives a marked improvement in the number of Steiner points used, we show that by using coloring arguments for triangulated polygons [13, 18], we can further reduce the number of Steiner points by a factor of three and this is optimal. Before proceeding, we make our definition of Steiner points more precise. As pointed out in the introduction, no Steiner points may be placed on the boundary of the polygon or on diagonals. Therefore, we ....

V. Chvatal. A combinatorial theorem in plane geometry. J. Combin. Theory Ser. B, 18:39--41, 1975.

....lemma gives 18n and Kachalski s trick reduces 7 it by 0.5. The same trick reduces Avis Horton s bound by 0.5. David Avis has found examples that require 9n edges and conjectures that the best upper bound is in fact 9n. 4.3 Polygon Decomposition 4.3. 1 Simple polygons In 1975 Vasek Chvatal [Ch75] proved that n 3 guards were always sufficient, and sometimes necessary, to guard (jointly see) the complete interior of a simple polygon (art gallery) consisting of n walls or vertices. This result has come to be known as Chvatal s Art Gallery Theorem and has since evolved to fill out an entire ....

Chvatal, V., "A combinatorial theorem in plane geometry," Journal of Combinatorial Theory Series B, vol. 18, 1975, pp. 39-41.

....and FCAR 93ER0291. Address: School of Computer Science, McGill University, 3480 University, Montr eal, Qu ebec, H3A 2A7. E mail: godfried cs.mcgill.ca. x Group C3, MS M986, Los Alamos National Lab, Los Alamos, NM 87545 USA. E mail: bhz c3serve.c3. lanl.gov a lengthy combinatorial argument [Chv75] Subsequently Fisk [Fis78] gave a concise and elegant proof using the fact that the vertices of a triangulated polygon may be three colored. Avis and Toussaint [AT81] used Fisk s proof to design an O(n log n) algorithm for placing the guards. Recently, Kooshesh and Moret [KM92] showed that the ....

V. Chvatal. A combinatorial theorem in plane geometry. J. Comb. Theory Ser. B, 18:39--41, 1975.

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Chvatal, V., "A combinatorial theorem in plane geometry," Journal of Combinatorial Theory, Series B, vol. 18, 1975, pp. 39-41.

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Chvatal, V., "A Combinatorial Theorem in Plane Geometry", Journal of Computorial Theory(B), vol. 18, 1975, pp 39-41.

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Chvatal, V., "A Combinatorial Theorem in Plane Geometry", Journal of Combinatorial Theory (B), vol. 18, 1975, pp 39-41.

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V. Chvatal, "A Combinatorial Theorem in Plane Geometry", Journal of Computorial Theory (B), vol. 18, pp 39-41, 1975.

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Chvatal, V. (1975), `A combinatorial theorem in plane geometry,' J. Combinatorial Theory, series B 18, 39--41.

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V. Chvatal, "A Combinatorial Theorem in Plane Geometry ", Journal of Combinatorial Theory (B) 18 (1975), pp 39-41.

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V. Chvatal, "A Combinatorial Theorem in Plane Geometry ", Journal of Combinatorial Theory (B) 18 (1975), pp 39-41.

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V. Chvatal. A combinatorial theorem in plane geometry. J. Combin. Theory Ser. B, 18:39--41, 1975.

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V. Chvatal. A combinatorial theorem in plane geometry. Journal of Combinatorial Theory (Series B), 18:39--41, 1975.

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