Succinct representation of graphs

This work presents a different approach to the continuing work done in the development of efficient algorithms for the solution of graph problems. While until now the main effort has been directed towards designing more efficient algorithms, we try to get a better solution by changing the graph representation. We suggest a new model for graph representation, which we named Graph Construction Representation. The new model proved to have two significant advantages over the conventional graph representations: (1) For many graphs, this model will use less space, or in other words will have a "succinct" representation. (2) We developed several algorithms which accept the new model as input. The time complexity for those algorithms when applied to graphs which are represented "succinctly" is improved in comparison with the best known algorithm. In this thesis the Graph Construction Representation is investigated. For many graph families it is shown how they are represented using this model. Bounds on the size of the representation are given. Several algorithms for checking graph properties are described, among them "Connectivity", "Has a Triangle?" and "Maximum Independent Set". We identify problems which are still open concerning the representation of graphs which are related to a graph already succinctly represented. We have only partial answers for these questions. We also present another graph representation, which was named the Small Circuit Representation, for which we prove that checking many graph properties is hard.