Robertson and Seymour proved that graphs are well-quasi-ordered by the minor relation. In other words, given infinitely many graphs, one graph contains another as a minor. An application of this theorem is that every property that is closed under deleting vertices, edges, and contracting edges can be characterized by finitely many graphs, and hence can be decided in polynomial time. In this thesis we are concerned with the topological minor relation. We say that a graph G contains another graph H as a topological minor if H can be obtained from a subgraph of G by repeatedly deleting a vertex of degree two and adding an edge incident with the neighbors of the deleted vertex. Unlike the relation of minor, the topological minor relation does not well-quasi-order graphs in general. However, Robertson conjectured in the late 1980's that for every positive integer k, the topological minor relation well-quasi-orders graphs that do not contain a topological minor isomorphic to the path of length k with each edge duplicated. This thesis consists of two main results. The first one is a structure theorem for excluding a fixed graph as a topological minor, which is analogous to a cornerstone result of Robertson and Seymour, who gave such a structure for graphs that exclude a fixed minor. Results for topological minors were previously obtained by Grohe and Marx and by Dvorak, but we push one of the bounds in their theorems to the optimal value. This improvement is needed for the next theorem. The second main result is a proof of Robertson's conjecture. As a corollary, properties on certain graphs closed under deleting vertices, edges, and "suppressing" vertices of degree two can be characterized by finitely many graphs, and hence can be decided in polynomial time.

Le thème central à cette thèse est l'étude des propriétés des classes de graphes définies par sous-structures interdites et leurs applications.La première direction que nous suivons a trait aux beaux ordres. À l'aide de théorèmes de décomposition dans les classes de graphes interdisant une sous-structure, nous identifions celles qui sont bellement-ordonnées. Les ordres et sous-structures considérés sont ceux associés aux notions de contraction et mineur induit. Ensuite, toujours en considérant des classes de graphes définies par sous-structures interdites, nous obtenons des bornes sur des invariants comme le degré, la largeur arborescente, la tree-cut width et un nouvel invariant généralisant la maille.La troisième direction est l'étude des relations entre les invariants combinatoires liés aux problèmes de packing et de couverture de graphes. Dans cette direction, nous établissons de nouvelles relations entre ces invariants pour certaines classes de graphes. Nous présentons également des applications algorithmiques de ces résultats. / The central theme of this thesis is the study of the properties of the classes of graphs defined by forbidden substructures and their applications.The first direction that we follow concerns well-quasi-orders. Using decomposition theorems on graph classes forbidding one substructure, we identify those that are well-quasi-ordered. The orders and substructures that we consider are those related to the notions of contraction and induced minor.Then, still considering classes of graphs defined by forbidden substructures, we obtain bounds on invariants such as degree, treewidth, tree-cut width, and a new invariant generalizing the girth.The third direction is the study of the links between the combinatorial invariants related to problems of packing and covering of graphs. In this direction, we establish new connections between these invariants for some classes of graphs. We also present algorithmic applications of the results.