Etude d'une nouvelle classe de graphes : les graphes hypotriangulés. (A class of graphs : hypochordal graphs)

In this thesis, we define a new class of graphs : the hypochordal graphs. These graphs satisfy that for any path of length two, there exists an chord or another path of length two between its two endpoints. This class can represent robust networks. Indeed, we show that in such graphs, in the case of an edge or a vertex deletion, the distance between any pair of nonadjacent vertices remains unchanged. Then, we study several properties for this class of graphs. Especially, after introducing a familly of specific partitions, we show the relations between some of these partitions and hypochordality. Moreover, thanks to these partitions, we characterise minimum hypochordal graphs that are, among connected hypochordal graphs, those that minimise the number of edges for a given number of vertices. In a second part, we study the complexity, for hypochordal graphs, of problems that are NP-hard in the general case. We first show that the classical problems of hamiltonian cycle, colouring, maximum clique and maximum stable remain NP-hard for this class of graphs. Then, we analyse graph modification problems : deciding the minimal number of edges to add or delete from a graph, in order to obtain an hypochordal graph. We study the complexity of these problems for several classes of graphs.