Matrix partition of chordal graphs

Matrix partitions are a common generalization of graph colourings, homomorphisms, and many partitions that play a role in the study of perfect graphs. Many results were obtained in the literature, focusing on the complexity of the problem, depending on the matrix. Another direction in the literature was classifying the matrix partition problems according to whether they have a finite or an infinite number of forbidden subgraphs. In this thesis we survey all these results, and we also make a contribution to the second classification (finite versus infinite number of minimal forbidden graphs), for small matrices $M$ and chordal graphs $G$. While this classification is not complete, even for 3 by 3 matrices, we do classify certain subset of 3 by 3 matrices. These results indicate the kind of methods that are useful in the classification and may possibly suggest how a general classification may look.