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Algorithmes génériques en temps constant pour la résolution de problèmes combinatoires dans la classe des rotagraphes et fasciagraphes. Application aux codes identifiants, dominants-localisateurs et dominants-total-localisateurs

Marwane Bouznif 1
1 G-SCOP_OC [?-2015] - Optimisation Combinatoire [?-2015]
G-SCOP - Laboratoire des sciences pour la conception, l'optimisation et la production
Abstract : A fasciagraph of length n and of fiber F, is constituted of n consecutive copies of a graph F, each copy being linked to the next one according to a same scheme. Rotagraphs are defines similarily, but along a circular structure. In this thesis, we caracterize a set of combinatorial problems that can be efficiently solved when applied on the class of rotagraphs and fasciagraphs. In this context, we define closed and stable (d,q,w)-properties, and we present, for such properties, an algorithm to compute an optimal solution, in constant time, for the set of fasciagraphs or rotagraphs of fixed fiber. We show that several problems, largely studied in graph theory, are caracterized by closed or stable (d,q,w)-properties. In a second part of the thesis, we adapt the generic algorithm to three problems caracterized by stable (d,q,w)-properties : the problem of minimum indentifying code, and two other, close to this one, the problem of minimum locating-dominating set et the one of minimum locating-total-dominating set. We present an implementation of our algorithm which has let us respond to open questions in a certain sub-class of rotagraphs : the circular strips of bounded height. We deduce from there other results on infinite strips of bounded height. Finaly we explore the problem of minimum identifying code in another class of graphs with repetitive structure : the fractal graphs.
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Marwane Bouznif. Algorithmes génériques en temps constant pour la résolution de problèmes combinatoires dans la classe des rotagraphes et fasciagraphes. Application aux codes identifiants, dominants-localisateurs et dominants-total-localisateurs. Mathématiques générales [math.GM]. Université de Grenoble, 2012. Français. ⟨NNT : 2012GRENM036⟩. ⟨tel-00744488⟩

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