46 research outputs found
Tree-width of hypergraphs and surface duality
In Graph Minors III, Robertson and Seymour write: "It seems that the
tree-width of a planar graph and the tree-width of its geometric dual are
approximately equal - indeed, we have convinced ourselves that they differ by
at most one". They never gave a proof of this. In this paper, we prove a
generalisation of this statement to embedding of hypergraphs on general
surfaces, and we prove that our bound is tight
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
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
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 (2â3)^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
Making local algorithms efficiently self-stabilizing in arbitrary asynchronous environments
This paper deals with the trade-off between time, workload, and versatility
in self-stabilization, a general and lightweight fault-tolerant concept in
distributed computing.In this context, we propose a transformer that provides
an asynchronous silent self-stabilizing version Trans(AlgI) of any terminating
synchronous algorithm AlgI. The transformed algorithm Trans(AlgI) works under
the distributed unfair daemon and is efficient both in moves and rounds.Our
transformer allows to easily obtain fully-polynomial silent self-stabilizing
solutions that are also asymptotically optimal in rounds.We illustrate the
efficiency and versatility of our transformer with several efficient (i.e.,
fully-polynomial) silent self-stabilizing instances solving major distributed
computing problems, namely vertex coloring, Breadth-First Search (BFS) spanning
tree construction, k-clustering, and leader election
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