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E. Eiben, R. Ganian, S. Szeider:
"Solving Problems on Graphs of High Rank-Width";
Algorithmica (online), 80 (2017), 80; 30 pages.

A modulator in a graph is a vertex set whose deletion places the
considered graph into some specified graph class. The cardinality of a
modulator to various graph classes has long been used as a structural
parameter which can be exploited to obtain fixed-parameter algorithms
for a range of hard problems. Here we investigate what happens when a
graph contains a modulator which is large but "well-structured" (in
the sense of having bounded rank-width). Can such modulators still be
exploited to obtain efficient algorithms? And is it even possible to
find such modulators efficiently? We first show that the parameters
derived from such well-structured modulators are more powerful for
fixed-parameter algorithms than the cardinality of modulators and
rank-width itself. Then, we develop a fixed-parameter algorithm for
finding such well-structured modulators to every graph class which can
be characterized by a finite set of forbidden induced subgraphs. We
proceed by showing how well-structured modulators can be used to
obtain efficient parameterized algorithms for Minimum Vertex Cover and
Maximum Clique. Finally, we use the concept of well-structured
modulators to develop an algorithmic meta-theorem for deciding
problems expressible in monadic second order logic, and prove that
this result is tight in the sense that it cannot be generalized to
LinEMSO problems.

http://dx.doi.org/10.1007/s00453-017-0290-8

https://link.springer.com/article/10.1007%2Fs00453-017-0290-8#citeas