Olivier Baudon

Decomposable trees: a polynomial algorithm for tripodes

In this article, we deal with graphs modelling interconnection networks of parallel systems (parallel computers, networks of workstations, etc.). We want to share the nodes of such a network between many users, each one needing a given number of nodes. Thus, a graph G with N vertices is said to be decomposable if for each set {n1,..., nk} whose sum is equal to N, there exists a partition V1,...,Vk of V(G) such that for each i, 1 ≤i≤k, |Vi| = ni and the subgraph induced by Vi is connected. We show that determining whether a given tripode (three disjoint chains connected by one extremity to a same new vertex) is decomposable can be done by a polynomial algorithm.

An Oriented Version of the 1-2-3 Conjecture

Abstract The well-known 1-2-3 Conjecture addressed by Karoński, Luczak and Thomason asks whether the edges of every undirected graph G with no isolated edge can be assigned weights from {1, 2, 3} so that the sum of incident weights at each vertex yields a proper vertex-colouring of G. In this work, we consider a similar problem for oriented graphs. We show that the arcs of every oriented graph −G⃗ can be assigned weights from {1, 2, 3} so that every two adjacent vertices of −G⃗ receive distinct sums of outgoing weights. This result is tight in the sense that some oriented graphs do not admit such an assignment using the weights from {1, 2} only. We finally prove that deciding whether two weights are sufficient for a given oriented graph is an NP-complete problem. These results also hold for product or list versions of this problem.

Routing permutations and 2---1 routing requests in the hypercube

Let Hn be the directed symmetric n-dimensional hypercube. Using the computer, we show that for anypermutation of the vertices of H4, there exists a system of pairwise arc-disjoint directed paths from each vertex to its target in the permutation. This veri3es Szymanski’s conjecture (Proceedings of the International Conference on Parallel Processing, 1989, pp. I-103–I-110) for n = 4. We also consider the so-called 2–1 routing requests in Hn, where anyvertex can be used twice as a source but onlyonce as a target; we construct for any n?3 a 2–1 request that cannot be routed in Hn byarc-disjoint paths: in other words, for n?3, Hn is not (2–1)-rearrangeable. ? 2001 Elsevier Science B.V. All rights reserved.

On the structure of arbitrarily partitionable graphs with given connectivity

A graph G=(V,E) is arbitrarily partitionable if for any sequence @t of positive integers adding up to |V|, there is a sequence of vertex-disjoint subsets of V whose orders are given by @t, and which induce connected subgraphs. Such a graph models, e.g., a computer network which may be arbitrarily partitioned into connected subnetworks. In this paper we study the structure of such graphs and prove that unlike in some related problems, arbitrarily partitionable graphs may have arbitrarily many components after removing a cutset of a given size >=2. The sizes of these components grow exponentially, though.

Structural Properties of Recursively Partitionable Graphs with Connectivity 2

Abstract A connected graph G is said to be arbitrarily partitionable (AP for short) if for every partition (n1, . . . , np) of |V (G)| there exists a partition (V1, . . . , Vp) of V (G) such that each Vi induces a connected subgraph of G on ni vertices. Some stronger versions of this property were introduced, namely the ones of being online arbitrarily partitionable and recursively arbitrarily partitionable (OL-AP and R-AP for short, respectively), in which the subgraphs induced by a partition of G must not only be connected but also fulfil additional conditions. In this paper, we point out some structural properties of OL-AP and R-AP graphs with connectivity 2. In particular, we show that deleting a cut pair of these graphs results in a graph with a bounded number of components, some of whom have a small number of vertices. We obtain these results by studying a simple class of 2-connected graphs called balloons.

Routing Permutations in the Hypercube

We study an n-dimensional directed symmetric hypercube Hn, in which every pair of adjacent vertices is connected by two arcs of opposite directions. Using the computer, we show that for H4 and for any permutation on its vertices, there exists a system of pairwise arc-disjoint directed paths from each vertex to its target in the permutation. This gives the answer to Szymanskis conjecture [Szy89] for dimension 4. In addition to this study, we consider in Hn the so-called 2-1 routing requests, that is routing requests where any vertex of Hn can be used twice as a source, but only once as a target. We give two such routing requests which cannot be routed in H3. Moreover, we show that for any dimension n ≥ 3, it is possible to find a 2-1 routing request gn such that gn cannot be routed in Hn : in other words, for any n ≥ 3, Hn is not (2-1)-rearrangeable.

2-1 routing requests in the hypercube

Abstract Let H n be the directed symmetric n-dimensional hypercube. We consider in H n the so-called 2-1 routing requests, where any vertex of H n can be used twice as a source, but only once as a target. In order to disprove the Szymanskis conjecture for a given dimension n, it is necessary to obtain 2-1 routing requests in H n − 1 that cannot be routed. In [O. Baudon, G. Fertin, and I. Havel, Routing permutations in the hypercube, Discrete Applied Mathematics 113 (1) (September 2001) 43–58], we showed that in H 3 there exists exactly two routing requests which cannot be routed, nonequivalent by automorphism. Moreover, we showed that for one of them, called g 3 , it is possible to extend it for any dimension n ≥ 3 in a 2-1 routing request g n that cannot be routed in H n . Considering distances between sources and their respective target in g 4 , we have, using a computer, studying all the 2-1 routing requests of H 4 having similar properties on distances, in order to find more not routable 2-1 routing request in dimension 4 and higher. We have by this way obtained several (a dozen) non routable 2-1 rout- ing requests in H 4 . Some of them may be extended to higher dimensions, but not all.