Preface The problem of enumerating combinatorial objects with certain criteria is a fundamental and important problem in mathematics and theoretical computer science. In the literature, there are mainly three directions in the development of the enumeration of combinatorial objects: 1) to develop delicately mathematical methods for counting the number of all objects under criteria, 2) to develop algorithmic methods for exhaustively generating objects in a particular combinatorial class without

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... epetition, and 3) to develop algorithms for randomly generating an object from a specific class under priori probability. While the first direction has been well studied in early years, the latter two have attracted a lot of attention with the advance of computer in recent years. This thesis is associated with the second direction: exhaustive generation, which systematically generates all objects of a particular class rather than print out all objects into a paper or a computer file. In early years, several researchers have studied the exhaustive generation of objects in small combinatorial classes. In recent years, more and more questions are asked to generate larger lists of combinatorial objects. With the aid of a computer, it would be possible not only to count but also to list all objects in larger classes without duplications. In many cases, the number of objects under study increases exponentially as the problem size increases. It is high demanding to design efficient algorithms in terms of time and space complexities. Note that the time complexities of such algorithms are measured by the total amount of changes in the data structures, not the time required to print out all objects. This thesis considers a special but very meaningful exhaustive generation problem which asks to systematically generate all colored and rooted outerplanar graphs with at most given number (≥ 1) of vertices without repetition. This problem is motivated by the fact that about 94.3% of molecules in the NCI database can be represented as outerplanar graphs [73] , where nodes represent atoms and edges represent the bonds between two atoms. The exhaustive list of outerplanar graphs has many applications in various areas such as chemistry, medicine, biology and computer science. However, to the best of our knowledge, few papers have been published for the exhaustive generation of outerplanar graphs in any classes. The reasons are twofold: on the one hand, few researchers notice the extensive applications of the exhaustive list of outerplanar graphs, and on the other hand, it is difficult to design efficient i algorithms to generate all outerplanar graphs without repetition because of the symmetry of graphical structures. In this thesis, we design an efficient algorithm that can systematically generate all required outerplanar graphs in constant time per each in the worse case with only ( ) space. The proposed algorithm does not require any duplication test when a new graph is generated. The key for designing this efficient algorithm is to choose a "canonical" representation for each colored and rooted outerplanar graph such that the canonical representation of any given outerplanar graph can be obtained from the canonical representation of another outerplanar graph with a constant-size change. This shares the spirit of most of the efficient generation algorithms of rooted trees such as [123] . To be more specific, the basic idea of our algorithm is presented as follows. We first introduce a canonical embedding as the representation for each outerplanar graph to avoid duplications. In doing so, the original problem of the thesis reduces to the problem of generating all canonical embeddings of colored and rooted outerplanar with at most vertices. Then, for each canonical outerplanar embedding, we define a unique canonical outerplanar embedding as its parent-embedding so that each pair of parent-embedding and child-embedding has constant-size differences. Based on this relationship, all canonical outerplanar embeddings are arranged into a tree structure, called a family tree ℱ, where each node in ℱ corresponds to a canonical embedding, and each edge in ℱ corresponds to the parent-child relationship between two canonical embeddings. Finally, we generate all canonical outerplanar embeddings by traversing the family tree ℱ with the depth-first search. In this way, we can systematically generate all colored and rooted outerplanar graphs without repetition, and moreover, the computation time for the changes between two successive graphs is constant in the worst case and the total space for the entire generation is ( ). The idea of our algorithm may be applied to the exhaustive generation problems of new families of tractable planar graphs or other classes of decomposable combinatorial structures. These algorithms are very useful and can find wide applications. For example, the exhaustive list of combinatorial objects can be used to search solutions under given constraints. The author hopes that the work in this thesis would be helpful to solve both practical and theoretical problems and stimulate future studies.