To avoid expensive round-trips between the application layer and the database layer it is crucial that data-intensive processing and calculations happen close to where the data resides -- ideally within the database engine. However, each application has its own domain and provides domain-specific languages (DSL) as a user interface to keep interactions confined within the well-known metaphors of the respective domain. Revealing the innards of the underlying data layer by forcing users to formulate problems in terms of a general database language is often not an option. To bridge that gap, we propose an approach to transform and directly compile a DSL into a general database execution plan using graph transformations. We identify the commonalities and mismatches between different models and show which parts can be cherry-picked for direct translation. Finally, we argue that graph transformations can be used in general to translate a DSL into an executable plan for a database.

This thesis focuses mainly on the Maximum Independent Set (MIS) problem. Some related graph theoretical combinatorial problems are also considered. As these problems are generally NP-hard, we study their complexity in hereditary graph classes, i.e. graph classes defined by a set F of forbidden induced subgraphs. We revise the literature about the issue, for example complexity results, applications, and techniques tackling the problem. Through considering some general approach, we exhibit several cases where the problem admits a polynomial-time solution. More specifically, we present polynomial-time algorithms for the MIS problem in: + some subclasses of $S_{2;j;k}$-free graphs (thus generalizing the classical result for $S_{1;2;k}$-free graphs); + some subclasses of $tree_{k}$-free graphs (thus generalizing the classical results for subclasses of P5-free graphs); + some subclasses of $P_{7}$-free graphs and $S_{2;2;2}$-free graphs; and various subclasses of graphs of bounded maximum degree, for example subcubic graphs. Our algorithms are based on various approaches. In particular, we characterize augmenting graphs in a subclass of $S_{2;k;k}$-free graphs and a subclass of $S_{2;2;5}$-free graphs. These characterizations are partly based on extensions of the concept of redundant set [125]. We also propose methods finding augmenting chains, an extension of the method in [99], and finding augmenting trees, an extension of the methods in [125]. We apply the augmenting vertex technique, originally used for $P_{5}$-free graphs or banner-free graphs, for some more general graph classes. We consider a general graph theoretical combinatorial problem, the so-called Maximum -Set problem. Two special cases of this problem, the so-called Maximum F-(Strongly) Independent Subgraph and Maximum F-Induced Subgraph, where F is a connected graph set, are considered. The complexity of the Maximum F-(Strongly) Independent Subgraph problem is revised and the NP-hardness of the Maximum F-Induced Subgraph problem is proved. We also extend the augmenting approach to apply it for the general Maximum Π -Set problem. We revise on classical graph transformations and give two unified views based on pseudo-boolean functions and αff-redundant vertex. We also make extensive uses of α-redundant vertices, originally mainly used for $P_{5}$-free graphs, to give polynomial solutions for some subclasses of $S_{2;2;2}$-free graphs and $tree_{k}$-free graphs. We consider some classical sequential greedy heuristic methods. We also combine classical algorithms with αff-redundant vertices to have new strategies of choosing the next vertex in greedy methods. Some aspects of the algorithms, for example forbidden induced subgraph sets and worst case results, are also considered. Finally, we restrict our attention on graphs of bounded maximum degree and subcubic graphs. Then by using some techniques, for example ff-redundant vertex, clique separator, and arguments based on distance, we general these results for some subclasses of $S_{i;j;k}$-free subcubic graphs.

This thesis focuses mainly on the Maximum Independent Set (MIS) problem. Some related graph theoretical combinatorial problems are also considered. As these problems are generally NP-hard, we study their complexity in hereditary graph classes, i.e. graph classes defined by a set F of forbidden induced subgraphs. We revise the literature about the issue, for example complexity results, applications, and techniques tackling the problem. Through considering some general approach, we exhibit several cases where the problem admits a polynomial-time solution. More specifically, we present polynomial-time algorithms for the MIS problem in: + some subclasses of $S_{2;j;k}$-free graphs (thus generalizing the classical result for $S_{1;2;k}$-free graphs); + some subclasses of $tree_{k}$-free graphs (thus generalizing the classical results for subclasses of P5-free graphs); + some subclasses of $P_{7}$-free graphs and $S_{2;2;2}$-free graphs; and various subclasses of graphs of bounded maximum degree, for example subcubic graphs. Our algorithms are based on various approaches. In particular, we characterize augmenting graphs in a subclass of $S_{2;k;k}$-free graphs and a subclass of $S_{2;2;5}$-free graphs. These characterizations are partly based on extensions of the concept of redundant set [125]. We also propose methods finding augmenting chains, an extension of the method in [99], and finding augmenting trees, an extension of the methods in [125]. We apply the augmenting vertex technique, originally used for $P_{5}$-free graphs or banner-free graphs, for some more general graph classes. We consider a general graph theoretical combinatorial problem, the so-called Maximum -Set problem. Two special cases of this problem, the so-called Maximum F-(Strongly) Independent Subgraph and Maximum F-Induced Subgraph, where F is a connected graph set, are considered. The complexity of the Maximum F-(Strongly) Independent Subgraph problem is revised and the NP-hardness of the Maximum F-Induced Subgraph problem is proved. We also extend the augmenting approach to apply it for the general Maximum Π -Set problem. We revise on classical graph transformations and give two unified views based on pseudo-boolean functions and αff-redundant vertex. We also make extensive uses of α-redundant vertices, originally mainly used for $P_{5}$-free graphs, to give polynomial solutions for some subclasses of $S_{2;2;2}$-free graphs and $tree_{k}$-free graphs. We consider some classical sequential greedy heuristic methods. We also combine classical algorithms with αff-redundant vertices to have new strategies of choosing the next vertex in greedy methods. Some aspects of the algorithms, for example forbidden induced subgraph sets and worst case results, are also considered. Finally, we restrict our attention on graphs of bounded maximum degree and subcubic graphs. Then by using some techniques, for example ff-redundant vertex, clique separator, and arguments based on distance, we general these results for some subclasses of $S_{i;j;k}$-free subcubic graphs.