Combinatorial and algorithmic aspects of identifying codes in graphs. (Aspects combinatoires et algorithmiques des codes identifiants dans les graphes)

v Combinatorial and algorithmic aspects of identifying codes in graphs Abstract: An identifying code is a set of vertices of a graph such that, on the one hand, each vertex out of the code has a neighbour in the code (the domination property), and, on the other hand, all vertices have a distinct neighbourhood within the code (the separation property). In this thesis, we investigate combinatorial and algorithmic aspects of identifying codes. For the combinatorial part, we rst study extremal questions by giving a complete characterization of all nite undirected graphs having their order minus one as the minimum size of an identifying code. We also characterize nite directed graphs, in nite undirected graphs and in nite oriented graphs having their whole vertex set as the unique identifying code. These results answer open questions that were previously studied in the literature. We then study the relationship between the minimum size of an identifying code and the maximum degree of a graph. In particular, we give several upper bounds for this parameter as a function of the order and the maximum degree. These bounds are obtained using two techniques. The rst one consists in the construction of independent sets satisfying certain properties, and the second one is the combination of two tools from the probabilistic method: the Lovász local lemma and a Cherno bound. We also provide constructions of graph families related to this type of upper bounds, and we conjecture that they are optimal up to an additive constant. We also present new lower and upper bounds for the minimum cardinality of an identifying code in speci c graph classes. We study graphs of girth at least 5 and of given minimum degree by showing that the combination of these two parameters has a strong in uence on the minimum size of an identifying code. We apply these results to random regular graphs. Then, we give lower bounds on the size of a minimum identifying code of interval and unit interval graphs. Finally, we prove several lower and upper bounds for this parameter when considering line graphs. The latter question is tackled using the new notion of an edge-identifying code. For the algorithmic part, it is known that the decision problem associated with the notion of an identifying code is NP-complete, even for restricted graph classes. We extend the known results to other classes such as split graphs, co-bipartite graphs, line graphs or interval graphs. To this end, we propose polynomial-time reductions from several classical hard algorithmic problems. These results show that in many graph classes, the identifying code problem is computationally more di cult than related problems (such as the dominating set problem). Furthermore, we extend the knowledge of the approximability of the optimization problem associated to identifying codes. We extend the known result of NP-hardness of approximating this problem within a sub-logarithmic factor (as a function of the instance graph) to bipartite, split and co-bipartite graphs, respectively. We also extend the known result of its APX-hardness for graphs of given maximum degree to a subclass of split graphs, bipartite graphs of maximum degree 4 and line graphs. Finally, we show the existence of a PTAS algorithm for unit interval graphs. An identifying code is a set of vertices of a graph such that, on the one hand, each vertex out of the code has a neighbour in the code (the domination property), and, on the other hand, all vertices have a distinct neighbourhood within the code (the separation property). In this thesis, we investigate combinatorial and algorithmic aspects of identifying codes. For the combinatorial part, we rst study extremal questions by giving a complete characterization of all nite undirected graphs having their order minus one as the minimum size of an identifying code. We also characterize nite directed graphs, in nite undirected graphs and in nite oriented graphs having their whole vertex set as the unique identifying code. These results answer open questions that were previously studied in the literature. We then study the relationship between the minimum size of an identifying code and the maximum degree of a graph. In particular, we give several upper bounds for this parameter as a function of the order and the maximum degree. These bounds are obtained using two techniques. The rst one consists in the construction of independent sets satisfying certain properties, and the second one is the combination of two tools from the probabilistic method: the Lovász local lemma and a Cherno bound. We also provide constructions of graph families related to this type of upper bounds, and we conjecture that they are optimal up to an additive constant. We also present new lower and upper bounds for the minimum cardinality of an identifying code in speci c graph classes. We study graphs of girth at least 5 and of given minimum degree by showing that the combination of these two parameters has a strong in uence on the minimum size of an identifying code. We apply these results to random regular graphs. Then, we give lower bounds on the size of a minimum identifying code of interval and unit interval graphs. Finally, we prove several lower and upper bounds for this parameter when considering line graphs. The latter question is tackled using the new notion of an edge-identifying code. For the algorithmic part, it is known that the decision problem associated with the notion of an identifying code is NP-complete, even for restricted graph classes. We extend the known results to other classes such as split graphs, co-bipartite graphs, line graphs or interval graphs. To this end, we propose polynomial-time reductions from several classical hard algorithmic problems. These results show that in many graph classes, the identifying code problem is computationally more di cult than related problems (such as the dominating set problem). Furthermore, we extend the knowledge of the approximability of the optimization problem associated to identifying codes. We extend the known result of NP-hardness of approximating this problem within a sub-logarithmic factor (as a function of the instance graph) to bipartite, split and co-bipartite graphs, respectively. We also extend the known result of its APX-hardness for graphs of given maximum degree to a subclass of split graphs, bipartite graphs of maximum degree 4 and line graphs. Finally, we show the existence of a PTAS algorithm for unit interval graphs.