FINE-GRAINED META-THEOREMS FOR VERTEX INTEGRITY

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 with q quantifiers, deciding if G satisfies can be done in time 2 O ( k 2 q + q log q ) + n O (1) ; (ii) for MSO formulas with q quantifiers, the same can be done in time 2 2O ( 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 2O(kq)and 2 2O ( 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 the best possible. More precisely, we show that it is not possible to decide FO formulas with q quantifiers in time 2 o ( k 2 q ) , and that there exists an MSO formula which cannot be decided in time 2 2o ( 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.