The purpose of this note is to give a simple proof for a necessary and sufficient condition for visibility paths in simple polygons. A visibility path is a curve such that every point inside a simple polygon is visible from at least one point on the path. This result is essential for finding the shortest watchman route inside a simple polygon specially when the route is restricted to curved paths.
We devise the following dynamic algorithms for both maintaining as well as querying for the visibility and weak visibility polygons amid vertex insertions and/or deletions to the simple polygon. * A fully-dynamic algorithm for maintaining the visibility polygon of a fixed point located interior to the simple polygon amid vertex insertions and deletions to the simple polygon. The time complexity to update the visibility polygon of a point $q$ due to the insertion (resp. deletion) of vertex $v$ to (resp. from) the current simple polygon is expressed in terms of the number of combinatorial change…
Visibility graph of a simple polygon is a graph with the same vertex set in which there is an edge between a pair of vertices if and only if the segment through them lies completely inside the polygon. Each pair of adjacent vertices on the boundary of the polygon are assumed to be visible. Therefore, the visibility graph of each polygon always contains its boundary edges. This implies that we have always a Hamiltonian cycle in a visibility graph which determines the order of vertices on the boundary of the corresponding polygon. In this paper, we propose a polynomial time algorithm for determi…
We investigate a practical variant of the well-known polygonal visibility path (watchman) problem. For a polygon $P$, a minimum link visibility path is a polygonal visibility path in $P$ that has the minimum number of links. The problem of finding a minimum link visibility path is NP-hard for simple polygons. If the link-length (number of links) of a minimum link visibility path (tour) is $Opt$ for a simple polygon $P$ with $n$ vertices, we provide an algorithm with $O(kn^2)$ runtime that produces polygonal visibility paths (or tours) of link-length at most $(γ+a_l/(k-1))Opt$ (or $(γ+a_l/k)Opt…
We devise an algorithm for maintaining the visibility polygon of any query point in a dynamic polygonal domain, i.e., as the polygonal domain is modified with vertex insertions and deletions to its obstacles, we update the data structures that store the visibility polygon of the query point. After preprocessing the initial input polygonal domain to build a few data structures, our algorithm takes O(k(\lg{|VP_{\cal P'}(q)|})+(\lg{n'})^{2}+h) (resp. O(k(\lg n')^2+(\lg|VP_{\cal P'}(q)|)+h)) worst-case time to update data structures that store visibility polygon VP_{\cal P'}(q) of a query point q …
The VC-dimension plays an important role for the algorithmic problem of guarding art galleries efficiently. We prove that inside a simple polygon at most $5$ points can be shattered by $L_1$-visibility polygons and give an example where 5 points are shattered. The VC-dimension is exactly $5$. The proof idea for the upper bound is different from previous approaches. Keywords: Art gallery, VC-dimension, $L_1$-visibility, polygons
We introduce the \emph{visibility center} of a set of points inside a polygon -- a point $c_V$ such that the maximum geodesic distance from $c_V$ to see any point in the set is minimized. For a simple polygon of $n$ vertices and a set of $m$ points inside it, we give an $O((n+m) \log {(n+m)})$ time algorithm to find the visibility center. We find the visibility center of \emph{all} points in a simple polygon in $O(n \log n)$ time. Our algorithm reduces the visibility center problem to the problem of finding the geodesic center of a set of half-polygons inside a polygon, which is of independent…
The visibility graph of a simple polygon represents visibility relations between its vertices. Knowing the correct order of the vertices around the boundary of a polygon and its visibility graph, it is an open problem to locate the vertices in a plane in such a way that it will be consistent with this visibility graph. This problem has been solved for special cases when we know that the target polygon is a {\it tower} or a {\it spiral}. Knowing that a given visibility graph belongs to a simple polygon with at most three concave chains on its boundary, a {\it pseuodo-triangle}, we propose a lin…
We are interested in the problem of guarding simple orthogonal polygons with the minimum number of $ r $-guards. The interior point $ p $ belongs an orthogonal polygon $ P $ is visible from $ r $-guard $ g $, if the minimum area rectangle contained $ p $ and $ q $ lies within $ P $. A set of point guards in polygon $ P $ is named guard set (as denoted $ G $) if the union of visibility areas of these point guards be equal to polygon $ P $ i.e. every point in $ P $ be visible from at least one point guards in $ G $. For an orthogonal polygon, if dual graph of vertical decomposition is a path, it…
A hidden guard set $ G $ is a set of point guards in polygon $ P $ that all points of the polygon are visible from some guards in $ G $ under the constraint that no two guards may see each other. In this paper, we consider the problem for finding minimum hidden guard sets in histogram polygons under orthogonal visibility. Two points $ p $ and $ q $ are orthogonally visible if the orthogonal bounding rectangle for $ p $ and $ q $ lies within $ P $. It is known that the problem is NP-hard for simple polygon with general visibility and it is true for simple orthogonal polygon. We proposed a linea…
For two points in the closure of a simple polygon $P$, we say that they see each other in $P$ if the line segment uniting them does not intersect the exterior of $P$. The visibility graph of $P$ is the graph whose vertex set is the vertex set of $P$ and two vertices are joined by an edge if they see each other in $P$. The characterization of visibility graphs has been an open problem for decades, and a significant effort has been made to characterize visibility graphs of restricted polygon classes. Among them is the convex fan: a simple polygon with a convex vertex (i.e. whose internal angle i…
An ortho-polygon visibility representation of an $n$-vertex embedded graph $G$ (OPVR of $G$) is an embedding-preserving drawing of $G$ that maps every vertex to a distinct orthogonal polygon and each edge to a vertical or horizontal visibility between its end-vertices. The vertex complexity of an OPVR of $G$ is the minimum $k$ such that every polygon has at most $k$ reflex corners. We present polynomial time algorithms that test whether $G$ has an OPVR and, if so, compute one of minimum vertex complexity. We argue that the existence and the vertex complexity of an OPVR of $G$ are related to it…
Curve pseudo-visibility graphs generalize polygon and pseudo-polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number $ω$ has chromatic number at most $3\cdot 4^{ω-1}$. 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 for the claimed bound. The proof is algorithmic: both the clique number and a colouring with the claimed number of colours can be computed in polyn…
Let $P$ be a simple polygon with $n$ vertices, and let $q \in P$ be a point in $P$. Let $k \in \{0, \dots, n - 1\}$. A point $p \in P$ is $k$-visible from $q$ if and only if the line segment $pq$ crosses the boundary of $P$ at most $k$ times. The $k$-visibility region of $q$ in $P$ is the set of all points that are $k$-visible from $q$. We study the problem of computing the $k$-visibility region in the limited workspace model, where the input resides in a random-access read-only memory of $O(n)$ words, each with $Ω(\log{n})$ bits. The algorithm can read and write $O(s)$ additional words of wor…
Let $P$ be a simple polygon with $n$ vertices, and let $A$ be a set of $m$ points or line segments inside $P$. We develop data structures that can efficiently count the number of objects from $A$ that are visible to a query point or a query segment. Our main aim is to obtain fast, $O(\mathop{\textrm{polylog}} nm$), query times, while using as little space as possible. In case the query is a single point, a simple visibility-polygon-based solution achieves $O(\log nm)$ query time using $O(nm^2)$ space. In case $A$ also contains only points, we present a smaller, $O(n + m^{2 + \varepsilon}\log n…
In this paper, we characterize planar point sets that can be partitioned into disjoint polygons of arbitrarily specified sizes. We provide an algorithm to construct such a partition, if it exists, in polynomial time. We show that this problem is equivalent to finding a specified $2$-factor in the visibility graph of the point set. The characterization for the case where all cycles have length $3$ also translates to finding a $K_3$-factor of the visibility graph of the point set. We show that the generalized problem of finding a $K_k$-factor of the visibility graph of a given point set for $k \…
We study versions of cop and robber pursuit-evasion games on the visibility graphs of polygons, and inside polygons with straight and curved sides. Each player has full information about the other player's location, players take turns, and the robber is captured when the cop arrives at the same point as the robber. In visibility graphs we show the cop can always win because visibility graphs are dismantlable, which is interesting as one of the few results relating visibility graphs to other known graph classes. We extend this to show that the cop wins games in which players move along straight…
The art gallery problem enquires about the least number of guards that are sufficient to ensure that an art gallery, represented by a polygon $P$, is fully guarded. In 1998, the problems of finding the minimum number of point guards, vertex guards, and edge guards required to guard $P$ were shown to be APX-hard by Eidenbenz, Widmayer and Stamm. In 1987, Ghosh presented approximation algorithms for vertex guards and edge guards that achieved a ratio of $\mathcal{O}(\log n)$, which was improved upto $\mathcal{O}(\log\log OPT)$ by King and Kirkpatrick in 2011. It has been conjectured that constan…
This thesis focuses on two concepts which are widely studied in the field of computational geometry. Namely, visibility and unit disk graphs. In the field of visibility, we have studied the conflict-free chromatic guarding of polygons, for which we have described a polynomial-time algorithm that uses $O(n \log^2 n)$ colors to guard a polygon in a conflict-free setting, and proper coloring of polygon visibility graphs, for which we have described an algorithm that returns a proper 4-coloring for a simple polygon. Besides, we have shown that the 5-colorability problem is NP-complete on visibilit…
We consider extending visibility polygon $(VP)$ of a given point $q$ $(VP(q))$, inside a simple polygon $¶$ by converting some edges of $¶$ to mirrors. We will show that several variations of the problem of finding mirror-edges to add at least $k$ units of area to $VP(q)$ are NP-complete, or NP-hard. Which $k$ is a given value. We deal with both single and multiple reflecting mirrors, and also specular or diffuse types of reflections. In specular reflection, a single incoming direction is reflected into a single outgoing direction. In this paper diffuse reflection is regarded as reflecting lig…
We address the following problem: Given a simple polygon $P$ with $n$ vertices and two points $s$ and $t$ inside it, find a minimum link path between them such that a given target point $q$ is visible from at least one point on the path. The method is based on partitioning a portion of $P$ into a number of faces of equal link distance from a source point. This partitioning is essentially a shortest path map (SPM). In this paper, we present an optimal algorithm with $O(n)$ time bound, which is the same as the time complexity of the standard minimum link paths problem.
Given a simple polygon $ \mathcal {P} $ of $ n $ vertices in the Plane. We study the problem of computing the visibility area from a given viewpoint $ q $ inside $ \mathcal {P} $ where only sub-linear variables are allowed for working space. Without any memory-constrained, this problem was previously solved in $ O(n) $-time and $ O(n) $-variables space. In a newer research, the visibility area of a point be computed in $ O(n) $-time, using $ O(\sqrt{n}) $ variables for working space. In this paper, we present an optimal-time algorithm, using $ O(c/\log n) $ variables space for computing visibi…
An ortho-polygon visibility representation $Γ$ of a $1$-plane graph $G$ (OPVR of $G$) is an embedding preserving drawing that maps each vertex of $G$ to a distinct orthogonal polygon and each edge of $G$ to a vertical or horizontal visibility between its end-vertices. The representation $Γ$ has vertex complexity $k$ if every polygon of $Γ$ has at most $k$ reflex corners. It is known that $3$-connected $1$-plane graphs admit an OPVR with vertex complexity at most twelve, while vertex complexity at least two may be required in some cases. In this paper, we reduce this gap by showing that vertex …
Let $s$ be a point in a polygonal domain $\mathcal{P}$ of $h-1$ holes and $n$ vertices. We consider a quickest visibility query problem. Given a query point $q$ in $\mathcal{P}$, the goal is to find a shortest path in $\mathcal{P}$ to move from $s$ to see $q$ as quickly as possible. Previously, Arkin et al. (SoCG 2015) built a data structure of size $O(n^22^{α(n)}\log n)$ that can answer each query in $O(K\log^2 n)$ time, where $α(n)$ is the inverse Ackermann function and $K$ is the size of the visibility polygon of $q$ in $\mathcal{P}$ (and $K$ can be $Θ(n)$ in the worst case). In this paper,…
Many problems in Discrete and Computational Geometry deal with simple polygons or polygonal regions. Many algorithms and data-structures perform considerably faster, if the underlying polygonal region has low local complexity. One obstacle to make this intuition rigorous, is the lack of a formal definition of local complexity. Here, we give two possible definitions and show how they are related in a combinatorial sense. We say that a polygon $P$ has point visibility width $w=pvw$, if there is no point $q\in P$ that sees more than $w$ reflex vertices. We say that a polygon $P$ has chord visibil…
We study the problem of colouring the vertices of a polygon, such that every viewer in it can see a unique colour. The goal is to minimise the number of colours used. This is also known as the conflict-free chromatic guarding problem with vertex guards, and is motivated, e.g., by the problem of radio frequency assignment to sensors placed at the polygon vertices. We first study the scenario in which viewers can be all points of the polygon (such as a mobile robot which moves in the interior of the polygon). We efficiently solve the related problem of minimising the number of guards and approxi…
