Marie-Claude Heydemann

Cayley graphs and interconnection networks

Due to recent developments of parallel and distributed computing, the design and analysis of various interconnection networks has been a main topic of research for the past few years and is still stimulated by the new technologies of communication networks such as optic fibers. There are many advantages in using Cayley (di)graphs as models for interconnection networks. This work first surveys some classes of Cayley graphs which are well studied as models of interconnection networks. Results and problems related to routings in networks are then presented, with emphasis on loads of nodes and links in routings.

Cycles in the cube-connected cycles graph

In this paper we study the existence of cycles of all lengths in the cube-connected cycles graph and we establish that this graph is no far from being pancyclic in case n odd and bi-pancyclic in case n even.

Broadcasting and spanning trees in de Bruijin and Kautz networks

Abstract We prove that, for any p≤d, there exists a spanning directed p-ary tree of depth at most D⌈logpd⌉; in a de Bruijn digraph B(d,D) or in a Kautz digraph K(d,D) of degree d and diameter D. This result gives directly an upper bound of pD⌈logpd⌉ on the broadcast time of these digraphs, which improves the previously known bounds for d≥15. In the case of de Bruijn digraphs, an upper bound on the broadcast time of B(pq,D) in terms of the broadcast times of B(p,D) and B(q,D) is established. This is used to improve the upper bounds on the broadcast time of B(d,D). We obtain several results which are refinements of the following general statements: 1. (i) for any D⩾2, d⩾2, if 2 kl k , b(B(d, D))⩽( 5 4 k+3)D . 2. (ii) for any k⩾3, if 2k 1

Embedding Complete Binary Trees into Star and Pancake Graphs

Abstract. Let G and H be two simple undirected graphs. An embedding of the graph G into the graph H is an injective mapping f from the vertices of G to the vertices of H . The dilation of the embedding is the maximum distance between f(u),f(v) taken over all edges (u,v) of G . We give a construction of embeddings of dilation 1 of complete binary trees into star graphs. The height of the trees embedded with dilation 1 into the n -dimensional star graph is Ω (n log n) , which is asymptotically optimal. Constructions of embeddings of complete binary trees of dilation 2δ and 2δ +1 , δ≥ 1, into star graphs are given. The use of larger dilation allows embeddings of trees of greater height into star graphs. It is shown that all these constructions can be modified to yield embeddings of dilation 1 and 2δ , δ≥ 1 , of complete binary trees into pancake graphs.

Chemins Et Circuits Dans Les Graphes Orientes

Let D be a digraph with n vertices: we give sufficient conditions and conjectures on the number of arcs of D to insure that D has a directed cycle or a directed path of given length l , with more emphasis on the cases l = n, n - 1 or l small. We study the case where D is any digraph and the case where D is strong.

Line graphs of hypergraphs I

Abstract We define the k -line graph of a hypergraph H as the graph whose vertices are the edges of H , two vertices being joined if the edges they represent intersect in at least k elements. In this paper we show that for any integer k and any graph G there exists a partial hypergraph H of some complete h -partite hypergraph K h h x N such that G is the k -line graph of H . We also prove that, for any integer p , there exist graphs which are not the ( h - p )-line graph of some h -uniform hypergraph. As a corollary we answer a problem of C. Cook. Further we show that it is not possible to characterize the ( h - 1)-line graphs by excluding a finite number of forbidden induced subgraphs.

Embedding complete binary trees into star networks

Star networks have been proposed as a possible interconnection network for massively parallel computers. In this paper we investigate embeddings of complete binary trees into star networks. Let G and H be two networks represented by simple undirected graphs. An embedding of G into H is an injective mapping f from the vertices of G into the vertices of H. The dilation of the embedding is the maximum distance between f(u), f(v) taken over all edges (u,v) of G.

Uniform homomorphisms of de Bruijn and Kautz networks

Abstract In this paper, we study the problem of homomorphisms of a general class of line digraphs. We show that the homomorphisms can always be defined using a partial binary operation on the alphabet whose letters form labels of the vertices. We apply these results to de Bruijn and Kautz (in short B/K) digraphs to characterize their uniform homomorphisms. For d non-prime, we describe algorithms for constructing non-trivial uniform homomorphisms of d -ary B/K digraphs of diameter D onto d -ary B/K digraphs of diameter D − 1. Using the properties of the uniform homomorphisms and shortest-path spanning trees of B/K digraphs, we also describe optimal emulations of Divide&Conquer computations on B/K digraphs.

Girth in digraphs

For an integer k > 2, the best function m(n, k) is determined such that every strong digraph of order n with at least m(n, k) arcs contains a circuit of length k or less.

A new digraphs composition with applications to de Bruijn and generalized de Bruijn digraphs

Abstract In this paper, we introduce a new operation on digraphs that we apply to different cases; we give new results about de Bruijn digraphs and generalized de Bruijn digraphs. We prove that this new operation commutes with the operation of taking the line digraph. In particular, we give a simple construction of the Kautz digraph of diameter n from two de Bruijn digraphs of diameter n − 1 and n. We also study k-factors in the composed digraph with application to the counting of 1-factors of de Bruijn and Kautz digraphs.