3,677 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
Comparing Width Parameters on Graph Classes
We study how the relationship between non-equivalent width parameters changes
once we restrict to some special graph class. As width parameters, we consider
treewidth, clique-width, twin-width, mim-width, sim-width and tree-independence
number, whereas as graph classes we consider -subgraph-free graphs,
line graphs and their common superclass, for , of -free
graphs.
We first provide a complete comparison when restricted to
-subgraph-free graphs, showing in particular that treewidth,
clique-width, mim-width, sim-width and tree-independence number are all
equivalent. This extends a result of Gurski and Wanke (2000) stating that
treewidth and clique-width are equivalent for the class of
-subgraph-free graphs.
Next, we provide a complete comparison when restricted to line graphs,
showing in particular that, on any class of line graphs, clique-width,
mim-width, sim-width and tree-independence number are all equivalent, and
bounded if and only if the class of root graphs has bounded treewidth. This
extends a result of Gurski and Wanke (2007) stating that a class of graphs
has bounded treewidth if and only if the class of line graphs of
graphs in has bounded clique-width.
We then provide an almost-complete comparison for -free graphs,
leaving one missing case. Our main result is that -free graphs of
bounded mim-width have bounded tree-independence number. This result has
structural and algorithmic consequences. In particular, it proves a special
case of a conjecture of Dallard, Milani\v{c} and \v{S}torgel.
Finally, we consider the question of whether boundedness of a certain width
parameter is preserved under graph powers. We show that the question has a
positive answer for sim-width precisely in the case of odd powers.Comment: 31 pages, 4 figures, abstract shortened due to arXiv requirement
Parity Games of Bounded Tree-Depth
The exact complexity of solving parity games is a major open problem. Several
authors have searched for efficient algorithms over specific classes of graphs.
In particular, Obdr\v{z}\'{a}lek showed that for graphs of bounded tree-width
or clique-width, the problem is in , which was later improved by
Ganardi, who showed that it is even in (with an additional
assumption for clique-width case). Here we extend this line of research by
showing that for graphs of bounded tree-depth the problem of solving parity
games is in logspace uniform . We achieve this by first
considering a parameter that we obtain from a modification of clique-width,
which we call shallow clique-width. We subsequently provide a suitable
reduction.Comment: This is the full version of the paper that has been accepted at CSL
2023 and is going to be published in Leibniz International Proceedings in
Informatics (LIPIcs
Vertex coloring with forbidden subgraphs
Given a set of graphs, a graph is -free if does not contain any graph in as induced subgraph. A is an induced cycle of length at least . A - is a graph obtained by adding a vertex adjacent to three consecutive vertices in a . Hole-twins are closely related to the characterization of the line graphs in terms of forbidden subgraphs.
By using {\it clique-width} and {\it perfect graphs} theory, we show that (,,-)-free graphs and (,-,-)-free graphs are either perfect or have bounded clique-width. And thus the coloring of them can be done in polynomial time
Counting Euler Tours in Undirected Bounded Treewidth Graphs
We show that counting Euler tours in undirected bounded tree-width graphs is
tractable even in parallel - by proving a upper bound. This is in
stark contrast to #P-completeness of the same problem in general graphs.
Our main technical contribution is to show how (an instance of) dynamic
programming on bounded \emph{clique-width} graphs can be performed efficiently
in parallel. Thus we show that the sequential result of Espelage, Gurski and
Wanke for efficiently computing Hamiltonian paths in bounded clique-width
graphs can be adapted in the parallel setting to count the number of
Hamiltonian paths which in turn is a tool for counting the number of Euler
tours in bounded tree-width graphs. Our technique also yields parallel
algorithms for counting longest paths and bipartite perfect matchings in
bounded-clique width graphs.
While establishing that counting Euler tours in bounded tree-width graphs can
be computed by non-uniform monotone arithmetic circuits of polynomial degree
(which characterize ) is relatively easy, establishing a uniform
bound needs a careful use of polynomial interpolation.Comment: 17 pages; There was an error in the proof of the GapL upper bound
claimed in the previous version which has been subsequently remove
Compact Labelings For Efficient First-Order Model-Checking
We consider graph properties that can be checked from labels, i.e., bit
sequences, of logarithmic length attached to vertices. We prove that there
exists such a labeling for checking a first-order formula with free set
variables in the graphs of every class that is \emph{nicely locally
cwd-decomposable}. This notion generalizes that of a \emph{nicely locally
tree-decomposable} class. The graphs of such classes can be covered by graphs
of bounded \emph{clique-width} with limited overlaps. We also consider such
labelings for \emph{bounded} first-order formulas on graph classes of
\emph{bounded expansion}. Some of these results are extended to counting
queries
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
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