17 research outputs found
N-free extensions of posets.Note on a theorem of P.A.Grillet
Let be the poset obtained by adding a dummy vertex on each
diagonal edge of the 's of a finite poset . We show that
is -free. It follows that this poset is the smallest
-free barycentric subdivision of the diagram of , poset whose existence
was proved by P.A. Grillet. This is also the poset obtained by the algorithm
starting with and consisting at step of adding a dummy vertex on
a diagonal edge of some in , proving that the result of this
algorithm does not depend upon the particular choice of the diagonal edge
choosen at each step. These results are linked to drawing of posets.Comment: 7 pages, 4 picture
Infinite cographs and chain complete N-free posets
We give a necessary and sufficient condition for a -free graph to be a
cograph. This allows us to obtain a simple proof of the fact that finite
-free graphs are finite cographs. We also prove that chain complete posets
whose comparability graph is a cograph are series-parallel.Comment: 7 page
Unavoidable induced subgraphs in large graphs with no homogeneous sets
A homogeneous set of an -vertex graph is a set of vertices () such that every vertex not in is either complete or
anticomplete to . A graph is called prime if it has no homogeneous set. A
chain of length is a sequence of vertices such that for every vertex
in the sequence except the first one, its immediate predecessor is its unique
neighbor or its unique non-neighbor among all of its predecessors. We prove
that for all , there exists such that every prime graph with at least
vertices contains one of the following graphs or their complements as an
induced subgraph: (1) the graph obtained from by subdividing every
edge once, (2) the line graph of , (3) the line graph of the graph in
(1), (4) the half-graph of height , (5) a prime graph induced by a chain of
length , (6) two particular graphs obtained from the half-graph of height
by making one side a clique and adding one vertex.Comment: 13 pages, 3 figure
Irreducible pairings and indecomposable tournaments
We only consider finite structures. With every totally ordered set and a
subset of , we associate the underlying tournament obtained from the transitive tournament
by reversing , i.e.,
by reversing the arcs such that . The subset is a
pairing (of ) if , a quasi-pairing (of ) if
; it is irreducible if no nontrivial interval of is
a union of connected components of the graph . In this paper, we
consider pairings and quasi-pairings in relation to tournaments. We establish
close relationships between irreducibility of pairings (or quasi-pairings) and
indecomposability of their underlying tournaments under modular decomposition.
For example, given a pairing of a totally ordered set of size at least
, the pairing is irreducible if and only if the tournament is indecomposable. This is a consequence of a more
general result characterizing indecomposable tournaments obtained from
transitive tournaments by reversing pairings. We obtain analogous results in
the case of quasi-pairings.Comment: 17 page
On minimal prime graphs and posets
We show that there are four infinite prime graphs such that every infinite
prime graph with no infinite clique embeds one of these graphs. We derive a
similar result for infinite prime posets with no infinite chain or no infinite
antichain
A survey on algorithmic aspects of modular decomposition
The modular decomposition is a technique that applies but is not restricted
to graphs. The notion of module naturally appears in the proofs of many graph
theoretical theorems. Computing the modular decomposition tree is an important
preprocessing step to solve a large number of combinatorial optimization
problems. Since the first polynomial time algorithm in the early 70's, the
algorithmic of the modular decomposition has known an important development.
This paper survey the ideas and techniques that arose from this line of
research
Graphs whose indecomposability graph is 2-covered
Given a graph , a subset of is an interval of provided
that for any and , if and only
if . For example, , and are
intervals of , called trivial intervals. A graph whose intervals are trivial
is indecomposable; otherwise, it is decomposable. According to Ille, the
indecomposability graph of an undirected indecomposable graph is the graph
whose vertices are those of and edges are the unordered
pairs of distinct vertices such that the induced subgraph is indecomposable. We characterize the indecomposable
graphs whose admits a vertex cover of size 2.Comment: 31 pages, 5 figure
More on discrete convexity
In several recent papers some concepts of convex analysis were extended to
discrete sets. This paper is one more step in this direction. It is well known
that a local minimum of a convex function is always its global minimum. We
study some discrete objects that share this property and provide several
examples of convex families related to graphs and to two-person games in normal
form
Fully polynomial FPT algorithms for some classes of bounded clique-width graphs
Parameterized complexity theory has enabled a refined classification of the
difficulty of NP-hard optimization problems on graphs with respect to key
structural properties, and so to a better understanding of their true
difficulties. More recently, hardness results for problems in P were achieved
using reasonable complexity theoretic assumptions such as: Strong Exponential
Time Hypothesis (SETH), 3SUM and All-Pairs Shortest-Paths (APSP). According to
these assumptions, many graph theoretic problems do not admit truly
subquadratic algorithms, nor even truly subcubic algorithms (Williams and
Williams, FOCS 2010 and Abboud, Grandoni, Williams, SODA 2015). A central
technique used to tackle the difficulty of the above mentioned problems is
fixed-parameter algorithms for polynomial-time problems with polynomial
dependency in the fixed parameter (P-FPT). This technique was introduced by
Abboud, Williams and Wang in SODA 2016 and continued by Husfeldt (IPEC 2016)
and Fomin et al. (SODA 2017), using the treewidth as a parameter. Applying this
technique to clique-width, another important graph parameter, remained to be
done. In this paper we study several graph theoretic problems for which
hardness results exist such as cycle problems (triangle detection, triangle
counting, girth, diameter), distance problems (diameter, eccentricities, Gromov
hyperbolicity, betweenness centrality) and maximum matching. We provide
hardness results and fully polynomial FPT algorithms, using clique-width and
some of its upper-bounds as parameters (split-width, modular-width and
-sparseness). We believe that our most important result is an -time algorithm for computing a maximum matching where
is either the modular-width or the -sparseness. The latter generalizes
many algorithms that have been introduced so far for specific subclasses such
as cographs, -lite graphs, -extendible graphs and -tidy
graphs. Our algorithms are based on preprocessing methods using modular
decomposition, split decomposition and primeval decomposition. Thus they can
also be generalized to some graph classes with unbounded clique-width