This thesis takes place in the project ToDo 2 funded by the french National Research Agency. We deal with the resolution of two graph problems, by exact and approximation methods. For the sake of compromise between runtime and quality of the solutions, we propose a new approach by partitioning the vertices of the graph into cliques, which aims (1) to solve problems quickly with exact algortihms and (2) to ensure the quality if results with approximation algorithms. We combine our approach with filtering techniques and heuristic list. To complete this theoretical work, we implement our algorithms and compared with those existing in the literature. At the first step, we discuss the problem of independent dominating of minimum size. We solve this problem accurately and prove that there are special graphs where the problem is 2-approximable. In the second step, we solve by an exact algorithm and an approximation algorithm, the vertex cover problem and the connected vertex cover problem. Then at the end of this thesis, we extend our work to the problems in graphs including conflicts between vertices.
