39 research outputs found
A single exponential bound for the redundant vertex Theorem on surfaces
Let s1, t1,. . . sk, tk be vertices in a graph G embedded on a surface \sigma
of genus g. A vertex v of G is "redundant" if there exist k vertex disjoint
paths linking si and ti (1 \lequal i \lequal k) in G if and only if such paths
also exist in G - v. Robertson and Seymour proved in Graph Minors VII that if v
is "far" from the vertices si and tj and v is surrounded in a planar part of
\sigma by l(g, k) disjoint cycles, then v is redundant. Unfortunately, their
proof of the existence of l(g, k) is not constructive. In this paper, we give
an explicit single exponential bound in g and k
Tree-width of hypergraphs and surface duality
In Graph Minor III, Robertson and Seymour conjecture that the tree-width of a
planar graph and that of its dual differ by at most one. We prove that given a
hypergraph H on a surface of Euler genus k, the tree-width of H^* is at most
the maximum of tw(H) + 1 + k and the maximum size of a hyperedge of H^*
Treewidth of planar graphs: connections with duality
International audienceRobertson and Seymour conjectured that the treewidth of a planar graph and the treewidth of its geometric dual differ by at most one. Lapoire solved the conjecture in the affirmative, using algebraic techniques. We give here a much shorter proof of this result
Computing branchwidth via efficient triangulations and blocks
International audienceMinimal triangulations and potential maximal cliques are the main ingredients for a number of polynomial time algorithms on different graph classes computing the treewidth of a graph. Potential maximal cliques are also the main engine of the fastest so far, exact (exponential) treewidth algorithm. Based on the recent results of Mazoit, we define the structures that can be regarded as minimal triangulations and potential maximal cliques for branchwidth: efficient triangulations and blocks. We show how blocks can be used to construct an algorithm computing the branchwidth of a graph on n vertices in time (2√3)^n · n^O(1)
Trade-off between Time, Space, and Workload: the case of the Self-stabilizing Unison
We present a self-stabilizing algorithm for the (asynchronous) unison problem
which achieves an efficient trade-off between time, workload, and space in a
weak model. Precisely, our algorithm is defined in the atomic-state model and
works in anonymous networks in which even local ports are unlabeled. It makes
no assumption on the daemon and thus stabilizes under the weakest one: the
distributed unfair daemon.
In a -node network of diameter and assuming a period ,
our algorithm only requires bits per node to achieve full
polynomiality as it stabilizes in at most rounds and moves. In particular and to the best of our knowledge, it is the first
self-stabilizing unison for arbitrary anonymous networks achieving an
asymptotically optimal stabilization time in rounds using a bounded memory at
each node.
Finally, we show that our solution allows to efficiently simulate synchronous
self-stabilizing algorithms in an asynchronous environment. This provides a new
state-of-the-art algorithm solving both the leader election and the spanning
tree construction problem in any identified connected network which, to the
best of our knowledge, beat all existing solutions of the literature.Comment: arXiv admin note: substantial text overlap with arXiv:2307.0663
Branchwidth of graphic matroids.
Answering a question of Geelen, Gerards, Robertson and Whittle, we prove that the branchwidth of a bridgeless graph is equal to the branch- width of its cycle matroid. Our proof is based on branch-decompositions of hypergraph
Distributed Certification for Classes of Dense Graphs
A proof-labeling scheme (PLS) for a boolean predicate on labeled graphs
is a mechanism used for certifying the legality with respect to of global
network states in a distributed manner. In a PLS, a certificate is assigned to
each processing node of the network, and the nodes are in charge of checking
that the collection of certificates forms a global proof that the system is in
a correct state, by exchanging the certificates once, between neighbors only.
The main measure of complexity is the size of the certificates. Many PLSs have
been designed for certifying specific predicates, including cycle-freeness,
minimum-weight spanning tree, planarity, etc.
In 2021, a breakthrough has been obtained, as a meta-theorem stating that a
large set of properties have compact PLSs in a large class of networks. Namely,
for every property on labeled graphs, there exists a PLS
for with -bit certificates for all graphs of bounded
tree-depth. This result has been extended to the larger class of graphs with
bounded {tree-width}, using certificates on bits.
We extend this result even further, to the larger class of graphs with
bounded clique-width, which, as opposed to the other two aforementioned
classes, includes dense graphs. We show that, for every
property on labeled graphs, there exists a PLS for with bit certificates for all graphs of bounded clique-width
An Unified FPT Algorithm for Width of Partition Functions
During the last decades, several polynomial-time algorithms have been designed that decide whether a graph has tree-width (resp., path-width, branch-width, etc.) at most k, where k is a fixed parameter. Amini et al. (Discrete Mathematics'09) use the notions of partitioning-trees and partition functions as a generalized view of classical decompositions of graphs, namely tree decomposition, path decomposition, branch decomposition, etc. In this paper, we propose a set of simple sufficient conditions on a partition function Φ, that ensures the existence of a linear-time explicit algorithm deciding if a set A has Φ-width at most k (k fixed). In particular, the algorithm we propose unifies the existing algorithms for tree-width, path-width, linear-width, branch-width, carving-width and cut-width. It also provides the first Fixed Parameter Tractable linear-time algorithm to decide if the q-branched tree-width, defined by Fomin et al. (Algorithmica'09), of a graph is at most k (k and q are fixed). Moreover, the algorithm is able to decide if the special tree-width, defined by Courcelle (FSTTCS'10), is at most k, in linear-time where k is a Fixed Parameter. Our decision algorithm can be turned into a constructive one by following the ideas of Bodlaender and Kloks (J. of Alg. 1996).Au cours de ces dernières années, plusieurs algorithmes polynomiaux ont été conçus pour décider si un graphe a largeur arborescente (resp., largeur en chemin, branch-width, etc) au plus k, où k est un paramètre fixe. Amini et al. (Discrete Mathematics'09) ont utilisé les notions d'arbres de partition et de fonctions de partition comme une vision généralisée des décompositions des graphes classiques, à savoir la décomposition arborescente, la décomposition en chemin, la décomposition en branche, etc. Dans cet article, nous proposons un ensemble de conditions sur une fonction de partition Φ, qui assure l'existence d'un algorithme explicite en temps linéaire pour décider si un ensemble A a Φ-largeur au plus k (oú k est fixé). En particulier, l'algorithme que nous proposons unifie les algorithmes existants pour la largeur arborescente, largeur en chemin, la largeur linéaire, la largeur de branche, cut-width et carving-width. Il est également le premier algorithme FPT pour décider si la largeur arborescente q-ramifié, définie par Fomin et al. (Algorithmica'09), d'un graphe est au plus k (k et q sont fixées). De plus, l'algorithme est capable de décider si la largeur arborescente spéciale, définie par Courcelle (FSTTCS'10), est plus k, où k est un paramètre fixé. Notre algorithme de décision peut être transformé en un algorithme constructif en suivant les idées de Bodlaender et Kloks (J. of Alg., 1996)