872 research outputs found
Bounding clique-width via perfect graphs
We continue the study into the clique-width of graph classes defined by two forbidden induced graphs. We present three new classes of bounded clique-width and one of unbounded clique-width. The four new graph classes have in common that one of their two forbidden induced subgraphs is the diamond. To prove boundedness of clique-width for the first three cases we develop a technique based on bounding clique covering number in combination with reduction to subclasses of perfect graphs. We extend our proof of unboundedness for the fourth case to show that Graph Isomorphism is Graph Isomorphism-complete on the same graph class
Bounding clique-width via perfect graphs
Given two graphs H1 and H2, a graph G is (H1,H2)-free if it contains no subgraph isomorphic to H1 or H2. We continue a recent study into the clique-width of (H1,H2)-free graphs and present three new classes of (H1,H2)-free graphs that have bounded clique-width. We also show the implications of our results for the computational complexity of the Colouring problem restricted to (H1,H2)-free graphs. The three new graph classes have in common that one of their two forbidden induced subgraphs is the diamond (the graph obtained from a clique on four vertices by deleting one edge). To prove boundedness of their clique-width we develop a technique based on bounding clique covering number in combination with reduction to subclasses of perfect graphs
Fast Parallel Fixed-Parameter Algorithms via Color Coding
Fixed-parameter algorithms have been successfully applied to solve numerous
difficult problems within acceptable time bounds on large inputs. However, most
fixed-parameter algorithms are inherently \emph{sequential} and, thus, make no
use of the parallel hardware present in modern computers. We show that parallel
fixed-parameter algorithms do not only exist for numerous parameterized
problems from the literature -- including vertex cover, packing problems,
cluster editing, cutting vertices, finding embeddings, or finding matchings --
but that there are parallel algorithms working in \emph{constant} time or at
least in time \emph{depending only on the parameter} (and not on the size of
the input) for these problems. Phrased in terms of complexity classes, we place
numerous natural parameterized problems in parameterized versions of AC. On
a more technical level, we show how the \emph{color coding} method can be
implemented in constant time and apply it to embedding problems for graphs of
bounded tree-width or tree-depth and to model checking first-order formulas in
graphs of bounded degree
Practical and Efficient Split Decomposition via Graph-Labelled Trees
Split decomposition of graphs was introduced by Cunningham (under the name
join decomposition) as a generalization of the modular decomposition. This
paper undertakes an investigation into the algorithmic properties of split
decomposition. We do so in the context of graph-labelled trees (GLTs), a new
combinatorial object designed to simplify its consideration. GLTs are used to
derive an incremental characterization of split decomposition, with a simple
combinatorial description, and to explore its properties with respect to
Lexicographic Breadth-First Search (LBFS). Applying the incremental
characterization to an LBFS ordering results in a split decomposition algorithm
that runs in time , where is the inverse Ackermann
function, whose value is smaller than 4 for any practical graph. Compared to
Dahlhaus' linear-time split decomposition algorithm [Dahlhaus'00], which does
not rely on an incremental construction, our algorithm is just as fast in all
but the asymptotic sense and full implementation details are given in this
paper. Also, our algorithm extends to circle graph recognition, whereas no such
extension is known for Dahlhaus' algorithm. The companion paper [Gioan et al.]
uses our algorithm to derive the first sub-quadratic circle graph recognition
algorithm
Advances in Learning Bayesian Networks of Bounded Treewidth
This work presents novel algorithms for learning Bayesian network structures
with bounded treewidth. Both exact and approximate methods are developed. The
exact method combines mixed-integer linear programming formulations for
structure learning and treewidth computation. The approximate method consists
in uniformly sampling -trees (maximal graphs of treewidth ), and
subsequently selecting, exactly or approximately, the best structure whose
moral graph is a subgraph of that -tree. Some properties of these methods
are discussed and proven. The approaches are empirically compared to each other
and to a state-of-the-art method for learning bounded treewidth structures on a
collection of public data sets with up to 100 variables. The experiments show
that our exact algorithm outperforms the state of the art, and that the
approximate approach is fairly accurate.Comment: 23 pages, 2 figures, 3 table
Open problems on graph coloring for special graph classes.
For a given graph G and integer k, the Coloring problem is that of testing whether G has a k-coloring, that is, whether there exists a vertex mapping c:V→{1,2,…}c:V→{1,2,…} such that c(u)≠c(v)c(u)≠c(v) for every edge uv∈Euv∈E. We survey known results on the computational complexity of Coloring for graph classes that are hereditary or for which some graph parameter is bounded. We also consider coloring variants, such as precoloring extensions and list colorings and give some open problems in the area of on-line coloring
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