Graphs are important tools for modeling the real problems in science. Many Np-hard problems can be expressed in the language of Graph Problems. There are different approaches such as approximation algorithm, fixed parameter algorithm, randomize algorithm and heuristic method to overcome hard problems and designing optimal algorithms but these aproches can not find the optimal answer in the shortest possible time. One of the most beautiful approaches to overcome hard problems is the use of structural properties of graphs in designing optimal algorithms. Tree-width is a useful structural parameter in the effect on structural and algorithmic graph theory.
