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Couverture de sommets sous contraintes

Abstract : This PhD thesis concerns the problem of covering finite sets in a discrete structure. This very general issue allows numerous approaches and we study some of them. The first chapter introduces the notions that are essentials to the understanding of this thesis and makes a brief state of the art on some covering problems, including the domination problem. The second chapter addresses the power dominating problem, a variation of the dominating problem with a propagation process. We study this problem on triangular grids and square grids of dimension 3. In the third chapter, we come back to the classical domination but in the context of a game, with the Maker-Breaker domination game. We study the complexity of the problem of deciding which player has a winning strategy and the minimum duration of a game if both players play perfectly. We also derive this problem for total domination and for an Avoider-Enforcer version. The fourth chapter is about the strong geodetic number: a problem with the distinctive characteristic that the covering is made by shortest paths in the graph. We study the strong geodetic number of several graph classes and its behaviour for the Cartesian product. Lastly, in the fifth chapter, we leave the realm of graphs to study the identification of points using disks. More than just covering every point of a certain set, the subset of disks covering each point must be unique to that point. We give results on particular configurations, bounds on the general case and we study the complexity of the problem when the radius of the disks is fixed
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Submitted on : Thursday, March 5, 2020 - 11:36:11 AM
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Valentin Gledel. Couverture de sommets sous contraintes. Mathématique discrète [cs.DM]. Université de Lyon, 2019. Français. ⟨NNT : 2019LYSE1130⟩. ⟨tel-02499430⟩



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