8 research outputs found
Line graphs of bounded clique-width
AbstractWe show that a set of graphs has bounded tree-width or bounded path-width if and only if the corresponding set of line graphs has bounded clique-width or bounded linear clique-width, respectively. This relationship implies some interesting algorithmic properties and re-proves already known results in a very simple way. It also shows that the minimization problem for NLC-width is NP-complete
Boundary properties of graphs
A set of graphs may acquire various desirable properties, if we apply suitable restrictions
on the set. We investigate the following two questions: How far, exactly, must one restrict
the structure of a graph to obtain a certain interesting property? What kind of tools are
helpful to classify sets of graphs into those which satisfy a property and those that do not?
Equipped with a containment relation, a graph class is a special example of a partially
ordered set. We introduce the notion of a boundary ideal as a generalisation of a notion
introduced by Alekseev in 2003, to provide a tool to indicate whether a partially ordered set
satisfies a desirable property or not. This tool can give a complete characterisation of lower
ideals defined by a finite forbidden set, into those that satisfy the given property and to
those that do not. In the case of graphs, a lower ideal with respect to the induced subgraph
relation is known as a hereditary graph class.
We study three interrelated types of properties for hereditary graph classes: the existence
of an efficient solution to an algorithmic graph problem, the boundedness of the graph
parameter known as clique-width, and well-quasi-orderability by the induced subgraph relation.
It was shown by Courcelle, Makowsky and Rotics in 2000 that, for a graph class, boundedness
of clique-width immediately implies an efficient solution to a wide range of algorithmic
problems. This serves as one of the motivations to study clique-width. As for well-quasiorderability,
we conjecture that every hereditary graph class that is well-quasi-ordered by
the induced subgraph relation also has bounded clique-width.
We discover the first boundary classes for several algorithmic graph problems, including
the Hamiltonian cycle problem. We also give polynomial-time algorithms for the dominating
induced matching problem, for some restricted graph classes.
After discussing the special importance of bipartite graphs in the study of clique-width,
we describe a general framework for constructing bipartite graphs of large clique-width. As
a consequence, we find a new minimal class of unbounded clique-width.
We prove numerous positive and negative results regarding the well-quasi-orderability of
classes of bipartite graphs. This completes a characterisation of the well-quasi-orderability of
all classes of bipartite graphs defined by one forbidden induced bipartite subgraph. We also
make considerable progress in characterising general graph classes defined by two forbidden
induced subgraphs, reducing the task to a small finite number of open cases. Finally, we
show that, in general, for hereditary graph classes defined by a forbidden set of bounded
finite size, a similar reduction is not usually possible, but the number of boundary classes
to determine well-quasi-orderability is nevertheless finite.
Our results, together with the notion of boundary ideals, are also relevant for the study
of other partially ordered sets in mathematics, such as permutations ordered by the pattern
containment relation
Boundary properties of graphs
A set of graphs may acquire various desirable properties, if we apply suitable restrictions on the set. We investigate the following two questions: How far, exactly, must one restrict the structure of a graph to obtain a certain interesting property? What kind of tools are helpful to classify sets of graphs into those which satisfy a property and those that do not? Equipped with a containment relation, a graph class is a special example of a partially ordered set. We introduce the notion of a boundary ideal as a generalisation of a notion introduced by Alekseev in 2003, to provide a tool to indicate whether a partially ordered set satisfies a desirable property or not. This tool can give a complete characterisation of lower ideals defined by a finite forbidden set, into those that satisfy the given property and to those that do not. In the case of graphs, a lower ideal with respect to the induced subgraph relation is known as a hereditary graph class. We study three interrelated types of properties for hereditary graph classes: the existence of an efficient solution to an algorithmic graph problem, the boundedness of the graph parameter known as clique-width, and well-quasi-orderability by the induced subgraph relation. It was shown by Courcelle, Makowsky and Rotics in 2000 that, for a graph class, boundedness of clique-width immediately implies an efficient solution to a wide range of algorithmic problems. This serves as one of the motivations to study clique-width. As for well-quasiorderability, we conjecture that every hereditary graph class that is well-quasi-ordered by the induced subgraph relation also has bounded clique-width. We discover the first boundary classes for several algorithmic graph problems, including the Hamiltonian cycle problem. We also give polynomial-time algorithms for the dominating induced matching problem, for some restricted graph classes. After discussing the special importance of bipartite graphs in the study of clique-width, we describe a general framework for constructing bipartite graphs of large clique-width. As a consequence, we find a new minimal class of unbounded clique-width. We prove numerous positive and negative results regarding the well-quasi-orderability of classes of bipartite graphs. This completes a characterisation of the well-quasi-orderability of all classes of bipartite graphs defined by one forbidden induced bipartite subgraph. We also make considerable progress in characterising general graph classes defined by two forbidden induced subgraphs, reducing the task to a small finite number of open cases. Finally, we show that, in general, for hereditary graph classes defined by a forbidden set of bounded finite size, a similar reduction is not usually possible, but the number of boundary classes to determine well-quasi-orderability is nevertheless finite. Our results, together with the notion of boundary ideals, are also relevant for the study of other partially ordered sets in mathematics, such as permutations ordered by the pattern containment relation.EThOS - Electronic Theses Online ServiceEngineering and Physical Sciences Research Council (EPSRC)University of Warwick. Centre for Discrete Mathematics and its Applications (DIMAP)GBUnited Kingdo