Research in minimal separators and minimal triangulations is undergoing an expanding phase, with several new researchers joining the field and exciting results published yearly. The applicants of this project proposal have been heavily involved in resear ch in several aspects of minimal separators and triangulations, and today we are in contact with all the main developments at the forefront of this research area. As we have recently gathered a deeper insight into the structural properties of minimal tria ngulations, just at the time when the research community interest in the topic is increasing, we are in the fortunate situation of having several ideas for new results that should be investigated and subsequently published as soon as possible. This will a dvance the state-of-the-art within the project topic, and have important implications for related theoretical and practical problems. It will also ensure that Bergen and Norway can keep, and probably even advance, its good position in the research communi ty within this area. In order to be able to achieve this we are in need of Ph.D. and post doctoral positions, which constitute the basis of this project proposal.