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 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

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^*

Postoperative pain management in children: Guidance from the pain committee of the European Society for Paediatric Anaesthesiology (ESPA Pain Management Ladder Initiative)

The main remit of the European Society for Paediatric Anaesthesiology (ESPA) Pain Committee is to improve the quality of pain management in children. The ESPA Pain Management Ladder is a clinical practice advisory based upon expert consensus to help to ensure a basic standard of perioperative pain management for all children. Further steps are suggested to improve pain management once a basic standard has been achieved. The guidance is grouped by the type of surgical procedure and layered to suggest basic, intermediate, and advanced pain management methods. The committee members are aware that there are marked differences in financial and personal resources in different institutions and countries and also considerable variations in the availability of analgesic drugs across Europe. We recommend that the guidance should be used as a framework to guide best practice

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 $n$-node network of diameter $D$ and assuming a period $B \geq 2D+2$, our algorithm only requires $O(\log B)$ bits per node to achieve full polynomiality as it stabilizes in at most $2D-2$ rounds and $O(\min(n^2B, n^3))$ 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

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

Being Efficient in Time, Space, and Workload: a Self-Stabilizing Unison and Its Consequences

We present a self-stabilizing algorithm for the unison problem which is efficient in time, workload, and space in a weak model. Precisely, our algorithm is defined in the atomic-state model and works in anonymous asynchronous connected 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 an n-node network of diameter D and assuming the knowledge B ≥ 2D+2, our algorithm only requires Θ(log(B)) bits per node and is fully polynomial as it stabilizes in at most 2D+2 rounds and O(min(n²B, n³)) moves. In particular, it is the first self-stabilizing unison for arbitrary asynchronous anonymous networks achieving an asymptotically optimal stabilization time in rounds using a bounded memory at each node. Furthermore, we show that our solution can be used to efficiently simulate synchronous self-stabilizing algorithms in asynchronous environments. For example, this simulation allows us to design a new state-of-the-art algorithm solving both the leader election and the BFS (Breadth-First Search) spanning tree construction in any identified connected network which, to the best of our knowledge, beats all existing solutions in the literature

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