2,455 research outputs found
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
Parameterized Edge Hamiltonicity
We study the parameterized complexity of the classical Edge Hamiltonian Path
problem and give several fixed-parameter tractability results. First, we settle
an open question of Demaine et al. by showing that Edge Hamiltonian Path is FPT
parameterized by vertex cover, and that it also admits a cubic kernel. We then
show fixed-parameter tractability even for a generalization of the problem to
arbitrary hypergraphs, parameterized by the size of a (supplied) hitting set.
We also consider the problem parameterized by treewidth or clique-width.
Surprisingly, we show that the problem is FPT for both of these standard
parameters, in contrast to its vertex version, which is W-hard for
clique-width. Our technique, which may be of independent interest, relies on a
structural characterization of clique-width in terms of treewidth and complete
bipartite subgraphs due to Gurski and Wanke
Minimal classes of graphs of unbounded clique-width defined by finitely many forbidden induced subgraphs
We discover new hereditary classes of graphs that are minimal (with respect
to set inclusion) of unbounded clique-width. The new examples include split
permutation graphs and bichain graphs. Each of these classes is characterised
by a finite list of minimal forbidden induced subgraphs. These, therefore,
disprove a conjecture due to Daligault, Rao and Thomasse from 2010 claiming
that all such minimal classes must be defined by infinitely many forbidden
induced subgraphs.
In the same paper, Daligault, Rao and Thomasse make another conjecture that
every hereditary class of unbounded clique-width must contain a labelled
infinite antichain. We show that the two example classes we consider here
satisfy this conjecture. Indeed, they each contain a canonical labelled
infinite antichain, which leads us to propose a stronger conjecture: that every
hereditary class of graphs that is minimal of unbounded clique-width contains a
canonical labelled infinite antichain.Comment: 17 pages, 7 figure
Induced Minor Free Graphs: Isomorphism and Clique-width
Given two graphs and , we say that contains as an induced
minor if a graph isomorphic to can be obtained from by a sequence of
vertex deletions and edge contractions. We study the complexity of Graph
Isomorphism on graphs that exclude a fixed graph as an induced minor. More
precisely, we determine for every graph that Graph Isomorphism is
polynomial-time solvable on -induced-minor-free graphs or that it is
GI-complete. Additionally, we classify those graphs for which
-induced-minor-free graphs have bounded clique-width. These two results
complement similar dichotomies for graphs that exclude a fixed graph as an
induced subgraph, minor, or subgraph.Comment: 16 pages, 5 figures. An extended abstract of this paper previously
appeared in the proceedings of the 41st International Workshop on
Graph-Theoretic Concepts in Computer Science (WG 2015
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
Rank-width and Tree-width of H-minor-free Graphs
We prove that for any fixed r>=2, the tree-width of graphs not containing K_r
as a topological minor (resp. as a subgraph) is bounded by a linear (resp.
polynomial) function of their rank-width. We also present refinements of our
bounds for other graph classes such as K_r-minor free graphs and graphs of
bounded genus.Comment: 17 page
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