Fine-grained Meta-Theorems for Vertex Integrity
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Vertex Integrity is a graph measure which sits squarely between two more well-studied notions, namely vertex cover and tree-depth, and that has recently gained attention as a structural graph parameter. In this paper we investigate the algorithmic trade-offs involved with this parameter from the point of view of algorithmic meta-theorems for First-Order (FO) and Monadic Second Order (MSO) logic. Our positive results are the following: (i) given a graph G of vertex integrity k and an FO formula ϕ with q quantifiers, deciding if G satisfies ϕ can be done in time 2^O(k^2q+qlog q)+n^O(1); (ii) for MSO formulas with q quantifiers, the same can be done in time 2^2^O(k^2+kq)+n^O(1). Both results are obtained using kernelization arguments, which pre-process the input to sizes 2^O(k^2)q and 2^O(k^2+kq) respectively. The complexities of our meta-theorems are significantly better than the corresponding meta-theorems for tree-depth, which involve towers of exponentials. However, they are worse than the roughly 2^O(kq) and 2^2^O(k+q) complexities known for corresponding meta-theorems for vertex cover. To explain this deterioration we present two formula constructions which lead to fine-grained complexity lower bounds and establish that the dependence of our meta-theorems on k is best possible. More precisely, we show that it is not possible to decide FO formulas with q quantifiers in time 2^o(k^2q), and that there exists a constant-size MSO formula which cannot be decided in time 2^2^o(k^2), both under the ETH. Hence, the quadratic blow-up in the dependence on k is unavoidable and vertex integrity has a complexity for FO and MSO logic which is truly intermediate between vertex cover and tree-depth.
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