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Aspects of connectivity with matroid constraints in graphs

Quentin Fortier 1
1 G-SCOP_ROSP [2016-2019] - Recherche Opérationnelle pour les Systèmes de Production [2016-2019]
G-SCOP - Laboratoire des sciences pour la conception, l'optimisation et la production
Abstract : The notion of connectivity is fundamental in graph theory. We study thoroughly a recent development in this field, with the addition of matroid constraints.Firstly, we exhibit two reduction operations on connected graphs with matroid constraints. Using these operations, we generalize the Menger's theorem on connectivity and Edmond's theorem on packing of arborescences.However, this extension of Edmond's theorem does not ensure that the arborescences are spanning. It has been conjectured that one can always find such spanning arborescences. We prove this conjecture in some cases, including matroids of rank two and transversal matroids. We disprove this conjecture in the general case by providing a counter-example with more than 300 vertices, on a parallel extension of the Fano matroid.Finally, we explore other generalizations of connectivity with matroid constraints: in mixed graphs, hypergraphs and with reachability conditions.
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Quentin Fortier. Aspects of connectivity with matroid constraints in graphs. Modeling and Simulation. Université Grenoble Alpes, 2017. English. ⟨NNT : 2017GREAM059⟩. ⟨tel-01838231⟩

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