# Empilements et recouvrements en théorie des graphes

Abstract : In this thesis, we study two problems largely studied in graph theory over the last three decades : error correcting codes and domination.
First, we study two generalizations of error correcting codes: perfect codes on mixed alphabets and weighted perfect codes of radius one. These problems have been largely studied in the Hamming metric.
We study them in the Lee metric and prove both existence and inexistence results.
We also show that domination and codes satisfy strong duality on the square grid for balls without center.
Then, we study domination in products of graphs. Since Vizing conjectured in 1968 that domination is supermultiplicative on the cartesian product, relations between variants of the domination number of some product of graphs and of its factors drew much attention. After giving new bounds on the total domination number of the direct product of graphs, we determine the power domination number of products of paths. Then, we prove a Vizing like conjecture for upper total domination in cartesian product.
Next, we study domination on a structural point of view. Carrying on a study from Favaron and Henning, we give upper bounds on the paired-domination number of star-free graphs for any number of branches and of $P_5$-free graphs.
We also give infinite families of graphs for which these bounds are sharp. We finally compare upper total and upper paired-domination, two variations on domination which attracted quite some attention recently, and we give precise bounds for trees.
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Cited literature [64 references]

https://tel.archives-ouvertes.fr/tel-00181722
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Submitted on : Wednesday, October 24, 2007 - 12:24:37 PM
Last modification on : Friday, March 25, 2022 - 11:09:50 AM
Long-term archiving on: : Monday, April 12, 2010 - 12:18:14 AM

### Identifiers

• HAL Id : tel-00181722, version 1

### Citation

Paul Dorbec. Empilements et recouvrements en théorie des graphes. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2007. Français. ⟨tel-00181722⟩

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