153 research outputs found
Non-Stretchable Pseudo-Visibility Graphs
We exhibit a family of graphs which can be realized as pseudo-visibility graphs of pseudo-polygons, but not of straight-line polygons. The example is based on the characterization of vertex-edge pseudo-visibility graphs of O\u27Rourke and Streinu [Proc. ACM Symp. Comput. Geometry, Nice, France, 1997, pp. 119-128] and extends a recent result of the author [Proc. ACM Symp. Comput. Geometry, Miami Beach, 1999, pp. 274-280] on non-stretchable vertex-edge visibility graphs. We construct a pseudo-visibility graph for which there exists a unique compatible vertex-edge visibility graph, which is then shown to be non-stretchable. The construction is then extended to an infinite family. © 2004 Elsevier B.V
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
Stretchability of Star-Like Pseudo-Visibility Graphs
We present advances on the open problem of characterizing vertex-edge visibility graphs (ve-graphs), reduced by results of O\u27Rourke and Streinu to a stretchability question for pseudo-polygons. We introduce star-like pseudo-polygons as a special subclass containing all the known instances of non-stretchable pseudo-polygons. We give a complete combinatorial characterization and a linear-time decision procedure for star-like pseudo-polygon stretchability and star-like ve-graph recognition. To the best of our knowledge, this is the first problem in computational geometry for which a combinatorial characterization was found by first isolating the oriented matroid substructure and then separately solving the stretchability question. It is also the first class (as opposed to isolated examples) of oriented matroids for which an efficient stretchability decision procedure based on combinatorial criteria is given. The difficulty of the general stretchability problem implied by Mnev\u27s Universality Theorem makes this a result of independent interest in the theory of oriented matroids
Stretchability of Star-Like Pseudo-Visibility Graphs
We present advances on the open problem of characterizing vertex-edge visibility graphs (ve-graphs), reduced by results of O\u27Rourke and Streinu to a stretchability question for pseudo-polygons. We introduce star-like pseudo-polygons as a special subclass containing all the known instances of non-stretchable pseudo-polygons. We give a complete combinatorial characterization and a linear-time decision procedure for star-like pseudo-polygon stretchability and star-like ve-graph recognition. To the best of our knowledge, this is the first problem in computational geometry for which a combinatorial characterization was found by first isolating the oriented matroid substructure and then separately solving the stretchability question. It is also the first class (as opposed to isolated examples) of oriented matroids for which an efficient stretchability decision procedure based on combinatorial criteria is given. The difficulty of the general stretchability problem implied by Mnev\u27s Universality Theorem makes this a result of independent interest in the theory of oriented matroids
Terrain Visibility Graphs: Persistence Is Not Enough
In this paper, we consider the Visibility Graph Recognition and
Reconstruction problems in the context of terrains. Here, we are given a graph
with labeled vertices such that the labeling
corresponds with a Hamiltonian path . also may contain other edges. We
are interested in determining if there is a terrain with vertices such that is the visibility graph of and the
boundary of corresponds with . is said to be persistent if and only
if it satisfies the so-called X-property and Bar-property. It is known that
every "pseudo-terrain" has a persistent visibility graph and that every
persistent graph is the visibility graph for some pseudo-terrain. The
connection is not as clear for (geometric) terrains. It is known that the
visibility graph of any terrain is persistent, but it has been unclear
whether every persistent graph has a terrain such that is the
visibility graph of . There actually have been several papers that claim
this to be the case (although no formal proof has ever been published), and
recent works made steps towards building a terrain reconstruction algorithm for
any persistent graph. In this paper, we show that there exists a persistent
graph that is not the visibility graph for any terrain . This means
persistence is not enough by itself to characterize the visibility graphs of
terrains, and implies that pseudo-terrains are not stretchable.Comment: To appear in SoCG 202
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