13 research outputs found
Tree-width and dimension
Over the last 30 years, researchers have investigated connections between
dimension for posets and planarity for graphs. Here we extend this line of
research to the structural graph theory parameter tree-width by proving that
the dimension of a finite poset is bounded in terms of its height and the
tree-width of its cover graph.Comment: Updates on solutions of problems and on bibliograph
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
Some notes on generic rectangulations
A rectangulation is a subdivision of a rectangle into rectangles. A generic rectangulation is a rectangulation that has no crossing segments. We explain several observations and pose some questions about generic rectangulations. In particular, we show how one may "centrally invert" a generic rectangulation about any given rectangle, analogous to reflection across a circle in classical geometry. We also explore 3-dimensional orthogonal polytopes related to "marked" rectangulations and drawings of planar maps. These observations arise from viewing a generic rectangulation as topologically equivalent to a sphere
Graph Relations and Constrained Homomorphism Partial Orders
We consider constrained variants of graph homomorphisms such as embeddings,
monomorphisms, full homomorphisms, surjective homomorpshims, and locally
constrained homomorphisms. We also introduce a new variation on this theme
which derives from relations between graphs and is related to
multihomomorphisms. This gives a generalization of surjective homomorphisms and
naturally leads to notions of R-retractions, R-cores, and R-cocores of graphs.
Both R-cores and R-cocores of graphs are unique up to isomorphism and can be
computed in polynomial time.
The theory of the graph homomorphism order is well developed, and from it we
consider analogous notions defined for orders induced by constrained
homomorphisms. We identify corresponding cores, prove or disprove universality,
characterize gaps and dualities. We give a new and significantly easier proof
of the universality of the homomorphism order by showing that even the class of
oriented cycles is universal. We provide a systematic approach to simplify the
proofs of several earlier results in this area. We explore in greater detail
locally injective homomorphisms on connected graphs, characterize gaps and show
universality. We also prove that for every the homomorphism order on
the class of line graphs of graphs with maximum degree is universal
On Schnyder's Theorm
The central topic of this thesis is Schnyder's Theorem. Schnyder's Theorem provides
a characterization of planar graphs in terms of their poset dimension, as follows: a graph
G is planar if and only if the dimension of the incidence poset of G is at most three. One
of the implications of the theorem is proved by giving an explicit mapping of the vertices
to R^2 that defines a straightline embedding of the graph. The other implication is proved
by introducing the concept of normal labelling. Normal labellings of plane triangulations
can be used to obtain a realizer of the incidence poset. We present an exposition of
Schnyderâs theorem with his original proof, using normal labellings. An alternate proof
of Schnyderâs Theorem is also presented. This alternate proof does not use normal
labellings, instead we use some structural properties of a realizer of the incidence poset
to deduce the result.
Some applications and a generalization of one implication of Schnyderâs Theorem
are also presented in this work. Normal labellings of plane triangulations can be used to
obtain a barycentric embedding of a plane triangulation, and they also induce a partition
of the edge set of a plane triangulation into edge disjoint trees. These two applications
of Schnyderâs Theorem and a third one, relating realizers of the incidence poset and
canonical orderings to obtain a compact drawing of a graph, are also presented. A
generalization, to abstract simplicial complexes, of one of the implications of Schnyderâs
Theorem was proved by Ossona de Mendez. This generalization is also presented in this
work. The concept of order labelling is also introduced and we show some similarities of
the order labelling and the normal labelling. Finally, we conclude this work by showing
the source code of some implementations done in Sage