20 research outputs found
String graphs. I. The number of critical nonstring graphs is infinite
AbstractString graphs (intersection graphs of curves in the plane) were originally studied in connection with RC-circuits. The family of string graphs is closed in the induced minor order, and so it is reasonable to study critical nonstring graphs (nonstring graphs such that all of their proper induced minors are string graphs). The question of whether there are infinitely many nonisomorphic critical nonstring graphs has been an open problem for some time. The main result of this paper settles this question. In a later paper of this series we show that recognizing string graphs is NP-hard
A Van Kampen type obstruction for string graphs
In this paper we prove that a graph is a string graph (the intersection graph
of curves in the plane) if and only if it admits a drawing in the plane with
certain properties. This also allows us to define an algebraic obstruction,
similar to the Van Kampen obstruction to embeddability, which must vanish for
every string graph. However, unlike in the case of graph planarity this
obstruction is incomplete.Comment: 10 pages, 3 figure
On String Graph Limits and the Structure of a Typical String Graph
We study limits of convergent sequences of string graphs, that is, graphs
with an intersection representation consisting of curves in the plane. We use
these results to study the limiting behavior of a sequence of random string
graphs. We also prove similar results for several related graph classes.Comment: 18 page
Intersection Graphs of Rays and Grounded Segments
We consider several classes of intersection graphs of line segments in the
plane and prove new equality and separation results between those classes. In
particular, we show that: (1) intersection graphs of grounded segments and
intersection graphs of downward rays form the same graph class, (2) not every
intersection graph of rays is an intersection graph of downward rays, and (3)
not every intersection graph of rays is an outer segment graph. The first
result answers an open problem posed by Cabello and Jej\v{c}i\v{c}. The third
result confirms a conjecture by Cabello. We thereby completely elucidate the
remaining open questions on the containment relations between these classes of
segment graphs. We further characterize the complexity of the recognition
problems for the classes of outer segment, grounded segment, and ray
intersection graphs. We prove that these recognition problems are complete for
the existential theory of the reals. This holds even if a 1-string realization
is given as additional input.Comment: 16 pages 12 Figure
Hierarchical Partial Planarity
In this paper we consider graphs whose edges are associated with a degree of
{\em importance}, which may depend on the type of connections they represent or
on how recently they appeared in the scene, in a streaming setting. The goal is
to construct layouts of these graphs in which the readability of an edge is
proportional to its importance, that is, more important edges have fewer
crossings. We formalize this problem and study the case in which there exist
three different degrees of importance. We give a polynomial-time testing
algorithm when the graph induced by the two most important sets of edges is
biconnected. We also discuss interesting relationships with other
constrained-planarity problems.Comment: Conference version appeared in WG201
Outerstring graphs are -bounded
An outerstring graph is an intersection graph of curves that lie in a common
half-plane and have one endpoint on the boundary of that half-plane. We prove
that the class of outerstring graphs is -bounded, which means that their
chromatic number is bounded by a function of their clique number. This
generalizes a series of previous results on -boundedness of outerstring
graphs with various additional restrictions on the shape of curves or the
number of times the pairs of curves can cross. The assumption that each curve
has an endpoint on the boundary of the half-plane is justified by the known
fact that triangle-free intersection graphs of straight-line segments can have
arbitrarily large chromatic number.Comment: Introduction extended by a survey of results on (outer)string graphs,
some minor correction