11 research outputs found
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
On the connectivity of visibility graphs
The visibility graph of a finite set of points in the plane has the points as
vertices and an edge between two vertices if the line segment between them
contains no other points. This paper establishes bounds on the edge- and
vertex-connectivity of visibility graphs.
Unless all its vertices are collinear, a visibility graph has diameter at
most 2, and so it follows by a result of Plesn\'ik (1975) that its
edge-connectivity equals its minimum degree. We strengthen the result of
Plesn\'ik by showing that for any two vertices v and w in a graph of diameter
2, if deg(v) <= deg(w) then there exist deg(v) edge-disjoint vw-paths of length
at most 4. Furthermore, we find that in visibility graphs every minimum edge
cut is the set of edges incident to a vertex of minimum degree.
For vertex-connectivity, we prove that every visibility graph with n vertices
and at most l collinear vertices has connectivity at least (n-1)/(l-1), which
is tight. We also prove the qualitatively stronger result that the
vertex-connectivity is at least half the minimum degree. Finally, in the case
that l=4 we improve this bound to two thirds of the minimum degree.Comment: 16 pages, 8 figure
On Colourability of Polygon Visibility Graphs
We study the problem of colouring the visibility graphs of polygons. In particular, we provide
a polynomial algorithm for 4-colouring of the polygon visibility graphs, and prove that the 6-
colourability question is already NP-complete for them
Colouring 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 ? has chromatic number at most 3?4^{?-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 polynomial time
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