  This article gives a summary of the author's Ph.D. dissertation
(arXiv:1609.06297). In addition to an overview of notions and results, it also
provides sketches of various proofs and simplified presentations of certain
abstract results of the dissertation, that concern tree representations of
structures. Further, some extensions of the dissertation results are presented.
These include the connections of the model-theoretic notions introduced in the
thesis with fixed parameter tractability and notions in the structure theory of
sparse graph classes. The constructive aspects of the proofs of the
model-theoretic results of the dissertation are used to obtain (algorithmic)
meta-kernels for various dense graphs such as graphs of bounded clique-width
and subclasses of these like $m$-partite cographs and graph classes of bounded
shrub-depth. Finally, the article presents updated definitions and results
concerning the notion of logical fractals which is a generalization of the
Equivalent Bounded Substructure Property from the dissertation. In particular,
our results show that (natural finitary adaptations of) both the upward and
downward versions of the L\"owenheim-Skolem theorem from classical model theory
can be recovered in a variety of algorithmically interesting settings, and
further in most cases, in effective form and even for logics beyond first order
logic.
