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

Created from the Publication Database of the Vienna University of Technology.