24 research outputs found
The Effect of Planarization on Width
We study the effects of planarization (the construction of a planar diagram
from a non-planar graph by replacing each crossing by a new vertex) on
graph width parameters. We show that for treewidth, pathwidth, branchwidth,
clique-width, and tree-depth there exists a family of -vertex graphs with
bounded parameter value, all of whose planarizations have parameter value
. However, for bandwidth, cutwidth, and carving width, every graph
with bounded parameter value has a planarization of linear size whose parameter
value remains bounded. The same is true for the treewidth, pathwidth, and
branchwidth of graphs of bounded degree.Comment: 15 pages, 6 figures. To appear at the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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
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
On the relationship between NLC-width and linear NLC-width
AbstractIn this paper, we consider NLC-width, NLCT-width, and linear NLC-width bounded graphs. We show that the set of all complete binary trees has unbounded linear NLC-width and that the set of all co-graphs has unbounded NLCT-width. Since trees have NLCT-width 3 and co-graphs have NLC-width 1, it follows that the family of linear NLC-width bounded graph classes is a proper subfamily of the family of NLCT-width bounded graph classes and that the family of NLCT-width bounded graph classes is a proper subfamily of the family of NLC-width bounded graph classes
Cliquewidth and knowledge compilation
In this paper we study the role of cliquewidth in succinct representation of Boolean functions. Our main statement is the following: Let Z be a Boolean circuit having cliquewidth k. Then there is another circuit Z * computing the same function as Z having treewidth at most 18k + 2 and which has at most 4|Z| gates where |Z| is the number of gates of Z. In this sense, cliquewidth is not more ‘powerful’ than treewidth for the purpose of representation of Boolean functions. We believe this is quite a surprising fact because it contrasts the situation with graphs where an upper bound on the treewidth implies an upper bound on the cliquewidth but not vice versa.
We demonstrate the usefulness of the new theorem for knowledge compilation. In particular, we show that a circuit Z of cliquewidth k can be compiled into a Decomposable Negation Normal Form (dnnf) of size O(918k k 2|Z|) and the same runtime. To the best of our knowledge, this is the first result on efficient knowledge compilation parameterized by cliquewidth of a Boolean circuit
-sails and sparse hereditary classes of unbounded tree-width
It has long been known that the following basic objects are obstructions to
bounded tree-width: for arbitrarily large , the complete graph ,
the complete bipartite graph , a subdivision of the -wall and the line graph of a subdivision of the -wall. We now add a further \emph{boundary object} to this list, a
subdivision of a \emph{-sail}.
These results have been obtained by studying sparse hereditary
\emph{path-star} graph classes, each of which consists of the finite induced
subgraphs of a single infinite graph whose edges can be decomposed into a path
(or forest of paths) with a forest of stars, characterised by an infinite word
over a possibly infinite alphabet. We show that a path-star class whose
infinite graph has an unbounded number of stars, each of which connects an
unbounded number of times to the path, has unbounded tree-width. In addition,
we show that such a class is not a subclass of circle graphs, a hereditary
class whose unavoidable induced subgraphs with large treewidth were identified
by Hickingbotham, Illingworth, Mohar and Wood
\cite{hickingbotham:treewidth_circlegraphs:}.
We identify a collection of \emph{nested} words with a recursive structure
that exhibit interesting characteristics when used to define a path-star graph
class. These graph classes do not contain any of the four basic obstructions
but instead contain graphs that have large tree-width if and only if they
contain arbitrarily large subdivisions of a -sail. Furthermore, like classes
of bounded degree or classes excluding a fixed minor, these sparse graph
classes do not contain a minimal class of unbounded tree-width