3,571 research outputs found
Induced subgraphs and tree decompositions VI. Graphs with 2-cutsets
This paper continues a series of papers investigating the following question:
which hereditary graph classes have bounded treewidth? We call a graph
-clean if it does not contain as an induced subgraph the complete graph
, the complete bipartite graph , subdivisions of a -wall, and line graphs of subdivisions of a -wall. It is known
that graphs with bounded treewidth must be -clean for some ; however, it
is not true that every -clean graph has bounded treewidth. In this paper, we
show that three types of cutsets, namely clique cutsets, 2-cutsets, and
1-joins, interact well with treewidth and with each other, so graphs that are
decomposable by these cutsets into basic classes of bounded treewidth have
bounded treewidth. We apply this result to two hereditary graph classes, the
class of (, wheel)-free graphs and the class of graphs with no cycle
with a unique chord. These classes were previously studied and decomposition
theorems were obtained for both classes. Our main results are that -clean
(, wheel)-free graphs have bounded treewidth and that -clean graphs
with no cycle with a unique chord have bounded treewidth
Product structure of graph classes with bounded treewidth
We show that many graphs with bounded treewidth can be described as subgraphs
of the strong product of a graph with smaller treewidth and a bounded-size
complete graph. To this end, define the "underlying treewidth" of a graph class
to be the minimum non-negative integer such that, for some
function , for every graph there is a graph with
such that is isomorphic to a subgraph of . We introduce disjointed coverings of graphs
and show they determine the underlying treewidth of any graph class. Using this
result, we prove that the class of planar graphs has underlying treewidth 3;
the class of -minor-free graphs has underlying treewidth (for ); and the class of -minor-free graphs has underlying
treewidth . In general, we prove that a monotone class has bounded
underlying treewidth if and only if it excludes some fixed topological minor.
We also study the underlying treewidth of graph classes defined by an excluded
subgraph or excluded induced subgraph. We show that the class of graphs with no
subgraph has bounded underlying treewidth if and only if every component of
is a subdivided star, and that the class of graphs with no induced
subgraph has bounded underlying treewidth if and only if every component of
is a star
EPG-representations with small grid-size
In an EPG-representation of a graph each vertex is represented by a path
in the rectangular grid, and is an edge in if and only if the paths
representing an share a grid-edge. Requiring paths representing edges
to be x-monotone or, even stronger, both x- and y-monotone gives rise to three
natural variants of EPG-representations, one where edges have no monotonicity
requirements and two with the aforementioned monotonicity requirements. The
focus of this paper is understanding how small a grid can be achieved for such
EPG-representations with respect to various graph parameters.
We show that there are -edge graphs that require a grid of area
in any variant of EPG-representations. Similarly there are
pathwidth- graphs that require height and area in
any variant of EPG-representations. We prove a matching upper bound of
area for all pathwidth- graphs in the strongest model, the one where edges
are required to be both x- and y-monotone. Thus in this strongest model, the
result implies, for example, , and area bounds
for bounded pathwidth graphs, bounded treewidth graphs and all classes of
graphs that exclude a fixed minor, respectively. For the model with no
restrictions on the monotonicity of the edges, stronger results can be achieved
for some graph classes, for example an area bound for bounded treewidth
graphs and bound for graphs of bounded genus.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Cliquewidth and dimension
We prove that every poset with bounded cliquewidth and with sufficiently
large dimension contains the standard example of dimension as a subposet.
This applies in particular to posets whose cover graphs have bounded treewidth,
as the cliquewidth of a poset is bounded in terms of the treewidth of the cover
graph. For the latter posets, we prove a stronger statement: every such poset
with sufficiently large dimension contains the Kelly example of dimension
as a subposet. Using this result, we obtain a full characterization of the
minor-closed graph classes such that posets with cover graphs in
have bounded dimension: they are exactly the classes excluding
the cover graph of some Kelly example. Finally, we consider a variant of poset
dimension called Boolean dimension, and we prove that posets with bounded
cliquewidth have bounded Boolean dimension.
The proofs rely on Colcombet's deterministic version of Simon's factorization
theorem, which is a fundamental tool in formal language and automata theory,
and which we believe deserves a wider recognition in structural and algorithmic
graph theory
On the expressive power of permanents and perfect matchings of matrices of bounded pathwidth/cliquewidth
Some 25 years ago Valiant introduced an algebraic model of computation in
order to study the complexity of evaluating families of polynomials. The theory
was introduced along with the complexity classes VP and VNP which are analogues
of the classical classes P and NP. Families of polynomials that are difficult
to evaluate (that is, VNP-complete) includes the permanent and hamiltonian
polynomials. In a previous paper the authors together with P. Koiran studied
the expressive power of permanent and hamiltonian polynomials of matrices of
bounded treewidth, as well as the expressive power of perfect matchings of
planar graphs. It was established that the permanent and hamiltonian
polynomials of matrices of bounded treewidth are equivalent to arithmetic
formulas. Also, the sum of weights of perfect matchings of planar graphs was
shown to be equivalent to (weakly) skew circuits. In this paper we continue the
research in the direction described above, and study the expressive power of
permanents, hamiltonians and perfect matchings of matrices that have bounded
pathwidth or bounded cliquewidth. In particular, we prove that permanents,
hamiltonians and perfect matchings of matrices that have bounded pathwidth
express exactly arithmetic formulas. This is an improvement of our previous
result for matrices of bounded treewidth. Also, for matrices of bounded
weighted cliquewidth we show membership in VP for these polynomials.Comment: 21 page
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