Structural and Algorithmic Aspects of Graphs

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My research interest lies in computational combinatorics, which deals with mostly finite objects. The type of objects that I spent most of time studying are graphs. Many problems of practical interest, such as scheduling of tasks, communication network design, traffic control, etc., can be modelled by graphs. Efficient computational solutions of problems can often be derived from an understanding of the structural properties of graphs. The theory of graphs has also proven to be useful in many other sciences. Computer science, for example, is very closely related to graph theory. Developing efficient algorithms to solve problems on graphs is of major interest to computer scientists. Of course, not all problems on graphs have been shown to admit efficient algorithms. It is generally believed that certain problems do not admit efficient solutions, but to prove this is the case is a problem by itself. The P vs NP problem, established by Clay Mathematics Institute as one of the Millennium Prize Problems, reflects exactly this situation.My long-term goal is an on-going study of combinatorial problems from both the structural and computational points of view. The P vs NP problem is considered to be very difficult at the moment. I anticipate a theory will be developed along the way in finding a solution to this difficult problem. I would like to make contributions toward establishing such a theory. My objectives are therefore to continue producing mathematical results concerning the structure of combinatorial objects and using the structural properties to derive dichotomy type of theorems, which hopefully will form building blocks for the anticipated theory.Graph searching is fundamental in exploring graphs and detecting their structures. Deciding which vertices of a graph can be end-vertices of a specific graph search is a problem that has been actively studied in the recent years. Several results which characterize end-vertices of certain searches have been obtained for some classes of graphs. However, the end-vertex problem is still open for many classes of graphs.Graph structures are sometimes inherent in their orientation properties; certain graphs can only admit certain orientations. Deciding whether a partially oriented graph can be completed to an oriented graph that satisfies a prescribed property is a fundamental problem which generalizes several existing problems. Graph colouring is a central topic in graph theory and has applications in real life. Colouring problems are special partition problems which are hard in general. Discovering graph classes for which these problems admit efficient algorithmic solutions is desirable and has potential applications in real life. As short-term objectives, I will continue studying these three types of problems, i.e., end-vertex problems of various graph search algorithms, orientation completion problems, and generalized graph colouring problems.