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### Linking disjoint segments into a simple polygon is hard [article]

Rain Jiang, Kai Jiang, Minghui Jiang
2021 arXiv   pre-print
Deciding whether a family of disjoint line segments in the plane can be linked into a simple polygon (or a simple polygonal chain) by adding segments between their endpoints is NP-hard.  ...  For each visibility edge of this polygon between two endpoints in V (S), we add a visibility edge between the corresponding endpoints in V (S ′ ).  ...  Then, following the other visibility edges of the simple polygon through S ′ , which are outside and between the gadgets, S can be linked into a simple polygon too.  ...

### Visibilty: Finding the Staircase Kernel in Orthogonal Polygons

Stefan A. Pape, Tzvetalin S. Vassilev
2012 American Journal of Computational and Applied Mathematics
Our result is optimal in the case of orthogonal polygon with holes, since the kernel (as proven) can consist of quadratic number of disjoint regions.  ...  We generalize the notion of visibility in the following way: We say that two points a and b in an orthogonal polygon P are visible to each other via staircase paths if and only if there exist an orthogonal  ...  Stefan Pape wants to acknowledge the help of Dr. Vassilev during the work on this paper.  ...

### Open Guard Edges and Edge Guards in Simple Polygons [chapter]

Csaba D. Tóth, Godfried T. Toussaint, Andrew Winslow
2012 Lecture Notes in Computer Science
An open edge of a simple polygon is the set of points in the relative interior of an edge.  ...  A guard edge of a polygon is an edge that sees every point inside the polygon.  ...  Similarly, if q = d, then c is not visible from the closed edge ab. We conclude that path(b, c) and path(d, a) are disjoint, and Q is a simple polygon.  ...

### Linking disjoint axis-parallel segments into a simple polygon is hard too [article]

Rain Jiang, Kai Jiang, Minghui Jiang
2021 arXiv   pre-print
Deciding whether a family of disjoint axis-parallel line segments in the plane can be linked into a simple polygon (or a simple polygonal chain) by adding segments between their endpoints is NP-hard.  ...  Suppose that S can be linked into a simple polygon. For each visibility edge of this polygon between two endpoints in V (S), we add a visibility edge between the corresponding endpoints in V (S ′ ).  ...  Then, following the other visibility edges of the simple polygon through S ′ , which are outside and between the gadgets, S can be linked into a simple polygon too.  ...

### Constructing pairwise disjoint paths with few links [chapter]

Himanshu Gupta, Rephael Wenger
1997 Lecture Notes in Computer Science
size of the polygon.  ...  Let P be a simple polygon and let {(u 1 , u 1 ), (u 2 , u 2 ), . . . , (um, u m )} be a set of m pairs of distinct vertices of P where for every distinct i, j ≤ m, there exist pairwise disjoint (nonintersecting  ...  VISIBILITY The construction of pairwise disjoint paths uses many of the ideas and notation developed in the study of polygon visibility. (See [Goodman and O'Rourke 1997, Chapter 25] .)  ...

### Constructing pairwise disjoint paths with few links

Himanshu Gupta, Rephael Wenger
2007 ACM Transactions on Algorithms
size of the polygon.  ...  Let P be a simple polygon and let {(u 1 , u 1 ), (u 2 , u 2 ), . . . , (um, u m )} be a set of m pairs of distinct vertices of P where for every distinct i, j ≤ m, there exist pairwise disjoint (nonintersecting  ...  VISIBILITY The construction of pairwise disjoint paths uses many of the ideas and notation developed in the study of polygon visibility. (See [Goodman and O'Rourke 1997, Chapter 25] .)  ...

### Page 6972 of Mathematical Reviews Vol. , Issue 95k [page]

1995 Mathematical Reviews
If the input consists of several disjoint polygons, however, it is often desirable to merge them in prepro- cessing so as to reduce a single polygon that retains the geometric characteristics of its individual  ...  We propose an O(n*) time algorithm for finding the maximum independent set of the convex visibility graph for a restricted class of simple polygons.”  ...

### 3D Visibility Representations by Regular Polygons [chapter]

Jan Štola
2010 Lecture Notes in Computer Science
We study 3D visibility representations of complete graphs where vertices are represented by equal regular polygons lying in planes parallel to the xy-plane.  ...  Edges correspond to the z-parallel visibility among these polygons.  ...  We assume that the z-coordinate of P i is i but we use it to identify polygons that can block visibility between other polygons only.  ...

### Page 3898 of Mathematical Reviews Vol. , Issue 97F [page]

1997 Mathematical Reviews
Our algorithm also works for a collection of disjoint simple polygons.  ...  “In the case of staircase polygons (special convex fans), we define a class of graphs called persistent graphs and show that the visibility graph of a staircase polygon is persistent.  ...

### Finding a shortest diagonal of a simple polygon in linear time

John Hershberger, Subhash Suri
1997 Computational geometry
of vertices satisfying an additional visibility constraint.  ...  A diagonal of a planar, simple polygon P is an open line segment that connects two nonadjacent vertices and lies in the relative interior of P.  ...  Theorem 5.5 also implies a linear-time algorithm for computing a closest visible vertex pair of the polygon: a closest visible pair either forms a shortest diagonal or it is an edge of the polygon.  ...

### Vc-dimension of exterior visibility

V. Isler, S. Kannan, K. Daniilidis, P. Valtr
2004 IEEE Transactions on Pattern Analysis and Machine Intelligence
In this paper, we study the Vapnik-Chervonenkis (VC)-dimension of set systems arising in 2D polygonal and 3D polyhedral configurations where a subset consists of all points visible from one camera.  ...  Here, we consider the case of exterior visibility, where the cameras lie on a constrained area outside the polygon and have to observe the entire boundary.  ...  Polygons that are entirely visible from the viewing circle are called externally visible polygons [29] .  ...

### Circumscribing Polygons and Polygonizations for Disjoint Line Segments [article]

Hugo A. Akitaya, Matias Korman, Oliver Korten, Mikhail Rudoy, Diane L. Souvaine, Csaba D. Tóth
2021 arXiv   pre-print
We prove that every arrangement of n disjoint line segments in the plane has a subset of size Ω(√(n)) that admits a circumscribing polygon, which is the first improvement on this bound in 20 years.  ...  A circumscribing polygon is a polygonization for G if every edge in E is an edge of P.  ...  We construct a circumscribing polygon for a subset Sb2 ⊆ Sb1 incrementally, while maintaining a polygon in the segment endpoint visibility graph of Sb1 .  ...

### Colouring polygon visibility graphs and their generalizations [article]

James Davies, Tomasz Krawczyk, Rose McCarty, Bartosz Walczak
2021 arXiv   pre-print
Curve pseudo-visibility graphs generalize polygon and pseudo-polygon visibility graphs and form a hereditary class of graphs.  ...  The proof is carried through in the setting of ordered graphs; we identify two conditions satisfied by every curve pseudo-visibility graph (considered as an ordered graph) and prove that they are sufficient  ...  Acknowledgement We thank Bodhayan Roy for sharing the problem of whether polygon visibility graphs are χ-bounded.  ...

### Coverage with k-Transmitters in the Presence of Obstacles [chapter]

Brad Ballinger, Nadia Benbernou, Prosenjit Bose, Mirela Damian, Erik D. Demaine, Vida Dujmović, Robin Flatland, Ferran Hurtado, John Iacono, Anna Lubiw, Pat Morin, Vera Sacristán (+2 others)
2010 Lecture Notes in Computer Science
We develop lower and upper bounds for the number of k-transmitters that are necessary and sufficient to cover a given collection of line segments, polygonal chains and polygons.  ...  We thank Joseph O'Rourke for the pinwheel example from Fig. 6 and for initiating this line of work.  ...  A small enough gap guarantees that the visibility regions of the points in S are all disjoint, as illustrated in Figure 5 .  ...

### Growing a Tree from Its Branches

P. Bose, G. Toussaint
1995 Journal of Algorithms
Given a set Ä of Ò disjoint line segments in the plane, we show that it is always possible to form a spanning tree of the endpoints of the segments, such that each line segment is an edge of the tree and  ...  Finally, we show that the complexity of finding the minimum weight spanning tree is optimal ¢´Ò ÐÓ Òµ when we assume that the endpoints of the line segments are in general position.  ...  The input to the algorithm is a simple polygon with collinearities that is weakly-visible from an edge ÙÚ. Let the counter-clockwise order of the vertices of the polygon be Ú Ô ¼ Ô ½ Ô Ò ¾ Ù.  ...
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