3,281 research outputs found
Vertex-Edge Pseudo-Visibility Graphs: Characterization and Recognition
We extend the notion of polygon visibility graphs to pseudo-polygons defined on generalized configurations of points. We consider both vertex-to-vertex, as well as vertex-to-edge visibility in pseudo-polygons. We study the characterization and recognition problems for vertex-edge pseudo-visibility graphs. Given a bipartite graph G satisfying three simple properties, which can all be checked in polynomial time, we show that we can define a generalized configuration of points and a pseudo-polygon on it, so that its vertex-edge pseudo-visibility graph is G. This provides a full characterization of vertex-edge pseudo-visibility graphs and a polynomial-time algorithm for the decision problem. It also implies that the decision problem for vertex visibility graphs of pseudo-polygons is in NP (as opposed to the same problem with straight-edge visibility, which is only known to be in PSPACE)
Recognizing Visibility Graphs of Polygons with Holes and Internal-External Visibility Graphs of Polygons
Visibility graph of a polygon corresponds to its internal diagonals and
boundary edges. For each vertex on the boundary of the polygon, we have a
vertex in this graph and if two vertices of the polygon see each other there is
an edge between their corresponding vertices in the graph. Two vertices of a
polygon see each other if and only if their connecting line segment completely
lies inside the polygon, and they are externally visible if and only if this
line segment completely lies outside the polygon. Recognizing visibility graphs
is the problem of deciding whether there is a simple polygon whose visibility
graph is isomorphic to a given input graph. This problem is well-known and
well-studied, but yet widely open in geometric graphs and computational
geometry.
Existential Theory of the Reals is the complexity class of problems that can
be reduced to the problem of deciding whether there exists a solution to a
quantifier-free formula F(X1,X2,...,Xn), involving equalities and inequalities
of real polynomials with real variables. The complete problems for this
complexity class are called Existential Theory of the Reals Complete.
In this paper we show that recognizing visibility graphs of polygons with
holes is Existential Theory of the Reals Complete. Moreover, we show that
recognizing visibility graphs of simple polygons when we have the internal and
external visibility graphs, is also Existential Theory of the Reals Complete.Comment: Sumbitted to COCOON2018 Conferenc
The Vertex-Edge Visibility Graph of a Polygon
We introduce a new polygon visibility graph, the vertex-edge visibility graph GV E, and demonstrate that it encodes more geometric information about the polygon than does the vertex visibility graph GV. © 1998 Elsevier Science B.V
A Characterization of Visibility Graphs for Pseudo-Polygons
In this paper, we give a characterization of the visibility graphs of
pseudo-polygons. We first identify some key combinatorial properties of
pseudo-polygons, and we then give a set of five necessary conditions based off
our identified properties. We then prove that these necessary conditions are
also sufficient via a reduction to a characterization of vertex-edge visibility
graphs given by O'Rourke and Streinu
Visibility Graphs, Dismantlability, and the Cops and Robbers Game
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 line segments
inside any polygon and, more generally, inside any simply connected planar
region with a reasonable boundary. Essentially, our problem is a type of
pursuit-evasion using the link metric rather than the Euclidean metric, and our
result provides an interesting class of infinite cop-win graphs.Comment: 23 page
Reconstructing Generalized Staircase Polygons with Uniform Step Length
Visibility graph reconstruction, which asks us to construct a polygon that
has a given visibility graph, is a fundamental problem with unknown complexity
(although visibility graph recognition is known to be in PSPACE). We show that
two classes of uniform step length polygons can be reconstructed efficiently by
finding and removing rectangles formed between consecutive convex boundary
vertices called tabs. In particular, we give an -time reconstruction
algorithm for orthogonally convex polygons, where and are the number of
vertices and edges in the visibility graph, respectively. We further show that
reconstructing a monotone chain of staircases (a histogram) is fixed-parameter
tractable, when parameterized on the number of tabs, and polynomially solvable
in time under reasonable alignment restrictions.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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