5,449 research outputs found
On Visibility and Blockers
This expository paper discusses some conjectures related to visibility and
blockers for sets of points in the plane
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
Blocking Coloured Point Sets
This paper studies problems related to visibility among points in the plane.
A point \emph{blocks} two points and if is in the interior of
the line segment . A set of points is \emph{-blocked} if each
point in is assigned one of colours, such that distinct points are assigned the same colour if and only if some other point in blocks
and . The focus of this paper is the conjecture that each -blocked
set has bounded size (as a function of ). Results in the literature imply
that every 2-blocked set has at most 3 points, and every 3-blocked set has at
most 6 points. We prove that every 4-blocked set has at most 12 points, and
that this bound is tight. In fact, we characterise all sets
such that some 4-blocked set has exactly points in
the -th colour class. Amongst other results, for infinitely many values of
, we construct -blocked sets with points
On colouring point visibility graphs
In this paper we show that it can be decided in polynomial time whether or
not the visibility graph of a given point set is 4-colourable, and such a
4-colouring, if it exists, can also be constructed in polynomial time. We show
that the problem of deciding whether the visibility graph of a point set is
5-colourable, is NP-complete. We give an example of a point visibility graph
that has chromatic number 6 while its clique number is only 4
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