Subtree overlap graphs and the maximum independent set problem

A graph G is a subtree overlap graph if there exists a tree T and a set of subtrees fTig so that there exists a one-to-one mapping between vertices and subtrees and two subtrees overlap if and only if their respective vertices are adjacent. The class of subtree overlap graphs is proven to contain the classes of circle, spider or circle polygon, and chordal graphs. An upper bound on the size of the subtree overlap model is proven to be 3m. As well a general algorithm to nd the maximum independent set for any class of overlap graph is given, provided testing for containment and intersection in the overlap graph can be done in polynomial time, and the maximumweight independent set problem is solved for the related class of intersection graph. The complexities of the Hamiltonian Cycle, several domination problems, isomorphism and colouring are shown to be as hard for subtree overlap graphs as they are for graphs in general. Acknowledgements Many people have helped me with this thesis, not least by listening to me blather on as I tried to convince myself certain points were true and had value. Firstly, I'd like to thank my supervisor, Professor Lorna Stewart, who suggested the study of the subtree overlap graph. Secondly, I'd like to thank Phil Nadeau. Trying to explain the underlying themes of this thesis in e-mail led to the developing of the structure of this thesis as it stood in previous versions. Most importantly, having a structure made writing a great deal easier. Similarly, Mike Jones, who proofread a very early version and pointed out all the inconsistencies between de nitions. They have been, I hope, xed. Lastly, my mother, without whom this thesis would have been a lot harder to write. Thank you.