We establish connections between parameterized/kernelization complexity of graph modification problems and expressibility in logic. For a first-order logic formula φ, we consider the problem of deciding whether an input graph can be modified by removing/adding at most k vertices/edges such that the resulting modification has the property expressible by φ. We provide sufficient and necessary conditions on the structure of the prefix of φ specifying when the corresponding graph modification problem is fixed-parameter tractable (parameterized by k) and when it admits a polynomial kernel.
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We are grateful to Pål Drange for his very helpful remarks.
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The two first authors have been supported by the Research Council of Norway via the projects “CLASSIS” and “MULTIVAL”. The third author has been supported by projects DEMOGRAPH (ANR-16-CE40-0028) and ESIGMA (ANR-17-CE23-0010). All authors have been supported by the Research Council of Norway and the French Ministry of Europe and Foreign Affairs, via the Franco-Norwegian project PHC AURORA 2019.
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Fomin, F.V., Golovach, P.A. & Thilikos, D.M. On the Parameterized Complexity of Graph Modification to First-Order Logic Properties. Theory Comput Syst 64, 251–271 (2020). https://doi.org/10.1007/s00224-019-09938-8
- First-order logic
- Graph modification
- Parameterized complexity
- Descriptive complexity