42 research outputs found
Structures combinatoires et interactions
Structures combinatoires et interaction
On the Number of Circuit-cocircuit Reversal Classes of an Oriented Matroid
The first author introduced the circuit-cocircuit reversal system of an
oriented matroid, and showed that when the underlying matroid is regular, the
cardinalities of such system and its variations are equal to special
evaluations of the Tutte polynomial (e.g., the total number of
circuit-cocircuit reversal classes equals , the number of bases of
the matroid). By relating these classes to activity classes studied by the
first author and Las Vergnas, we give an alternative proof of the above results
and a proof of the converse statements that these equalities fail whenever the
underlying matroid is not regular. Hence we extend the above results to an
equivalence of matroidal properties, thereby giving a new characterization of
regular matroids.Comment: 7 pages. v2: simplified proof, with new statements concerning other
special evaluations of the Tutte polynomia
Split decomposition and graph-labelled trees: characterizations and fully-dynamic algorithms for totally decomposable graphs
In this paper, we revisit the split decomposition of graphs and give new
combinatorial and algorithmic results for the class of totally decomposable
graphs, also known as the distance hereditary graphs, and for two non-trivial
subclasses, namely the cographs and the 3-leaf power graphs. Precisely, we give
strutural and incremental characterizations, leading to optimal fully-dynamic
recognition algorithms for vertex and edge modifications, for each of these
classes. These results rely on a new framework to represent the split
decomposition, namely the graph-labelled trees, which also captures the modular
decomposition of graphs and thereby unify these two decompositions techniques.
The point of the paper is to use bijections between these graph classes and
trees whose nodes are labelled by cliques and stars. Doing so, we are also able
to derive an intersection model for distance hereditary graphs, which answers
an open problem.Comment: extended abstract appeared in ISAAC 2007: Dynamic distance hereditary
graphs using split decompositon. In International Symposium on Algorithms and
Computation - ISAAC. Number 4835 in Lecture Notes, pages 41-51, 200
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