[+] Approximation schemes in minor-free graphs I: local treewidth and apex-minor-free graphs, extension to H-minor-free graphs, deletion decomposition
| In this
lecture we look at Baker's technique from another perspective which
leads to the notion of deletion decomposition: partitioning a graph
into k parts such that removing any part gives a graph of low
treewidth (in k). This idea of a simplifying decomposition is one of
the main themes of this course.
We will then see how to generalize this idea from planar graphs to apex-minor-free graphs. To this end, we introduce the notion of local treewidth, one of the major concepts in algorithmic graph structure theory. For the generalization to H-minor-free graphs we will have to look at the Robertson-Seymour decomposition of H-minor-free graphs into clique-sums of almost embeddable graphs.
All in all, we obtain a clean and simple deletion decomposition theorem for all H-minor-free graphs that can be used to obtain many approximation algorithms, PTASes, and parameterized algorithms.
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