Symbiosis is described as the close relationship between different biological species. The relationship may be deeply intimate with one partner living on another (ectosymbiosis) or inside (endosymbiosis). The latter includes situations where divorce is not possible without leading to the death of both symbiont and host (obligate symbiosis). The symbiotic relationship has been described as being mutualistic, parasitic, or commensal in nature depending on whether, respectively, both species benefit from the relation, only one benefits and the other is harmed, one benefits and the other either is helped or is not significantly harmed. In reality, it is often difficult to know which situation applies for any pair of symbiont-host. There are few really clear-cut cases and one could say instead that the range of possible interactions represents a continuum between the extreme situations of free relation and mariage with no possible divorce, harmful or mutually beneficial. The variety between these extremes is huge even for a given type of relationship, for instance parasitism, in which one species is supposed to be ``only'' harmed by its interaction with another, or for a given type of symbiont. This variety is mirrored by a huge variety of genomic and biochemical landscapes inside the symbiont world, and at the interface between symbionts and hosts. The purpose of this project is to combinatorially explore those landscapes at the molecular level, that is at the level of the genome and of two of the main types of biochemical networks that may be reconstructed from the sequenced genomes of symbionts and hosts. Such networks are the metabolic and protein-protein interaction (PPI) networks. The final objective is to try to relate the contours of the landscapes to the modus operandi of the symbiotic relation, thereby offering a hope of better understanding the latter, in particular its evolution. The symbiosis issue is vast and complex. The project will focus first on two questions concerning the evolution of symbionts, one at the genome level, namely the studies of rearrangements, and one at the biochemical network level and interface between genome and network. The evolution of symbionts may be largely dependent on the evolution of their hosts. In a third part of the project, we therefore address the question of the evolution of the intimate relations themselves by studying the co-cladogenesis (co-speciation) of hosts and symbionts, and more generally their co-evolution, that is the mutual evolutionary influence they exert on each other. Graph (tree) combinatorics and algorithmics underlie each of these problems, as well as issues related to random graph enumeration under certain models to improve confidence in the evolution and co-evolution scenarii inferred. Although this is a computational and mathematical project, it will be conducted in partnership with experimental biologists in the same laboratory as the methodological people involved in the work. The main symbiotic organisms used will come from the proteobacteria phylum.