14 research outputs found
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
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
Graphs with Plane Outside-Obstacle Representations
An \emph{obstacle representation} of a graph consists of a set of polygonal
obstacles and a distinct point for each vertex such that two points see each
other if and only if the corresponding vertices are adjacent. Obstacle
representations are a recent generalization of classical polygon--vertex
visibility graphs, for which the characterization and recognition problems are
long-standing open questions.
In this paper, we study \emph{plane outside-obstacle representations}, where
all obstacles lie in the unbounded face of the representation and no two
visibility segments cross. We give a combinatorial characterization of the
biconnected graphs that admit such a representation. Based on this
characterization, we present a simple linear-time recognition algorithm for
these graphs. As a side result, we show that the plane vertex--polygon
visibility graphs are exactly the maximal outerplanar graphs and that every
chordal outerplanar graph has an outside-obstacle representation.Comment: 12 pages, 7 figure
Visibility graphs of towers
AbstractA tower is a polygon consisting of two reflex chains sharing one common endpoint, together with one edge joining the other endpoints of the chains. A linear time algorithm is given to recognize the [vertex] visibility graphs of towers, and these graphs are characterized as bipartite permutation graphs with an added Hamiltonian cycle. Similar results have been obtained independently by Choi, Shin and Chwa (1992)
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
Recognition and Complexity of Point Visibility Graphs
A point visibility graph is a graph induced by a set of points in the plane, where every vertex corresponds to a point, and two vertices are adjacent whenever the two corresponding points are visible from each other, that is, the open segment between them does not contain any other point of the set.
We study the recognition problem for point visibility graphs: given a simple undirected graph, decide whether it is the visibility graph of some point set in the plane. We show that the problem is complete for the existential theory of the reals. Hence the problem is as hard as deciding the existence of a real solution to a system of polynomial inequalities. The proof involves simple substructures forcing collinearities in all realizations of some visibility graphs, which are applied to the algebraic universality constructions of Mnev and Richter-Gebert. This solves a longstanding open question and paves the way for the analysis of other classes of visibility graphs.
Furthermore, as a corollary of one of our construction, we show that there exist point visibility graphs that do not admit any geometric realization with points having integer coordinates
Coloring polygon visibility graphs and their generalizations
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 · 4ω−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 coloring with the
claimed number of colors can be computed in polynomial time