Parameterized Complexity of Elimination Distance to First-Order Logic Properties
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The elimination distance to some target graph property P is a general graph modification parameter introduced by Bulian and Dawar. We initiate the study of elimination distances to graph properties expressible in first-order logic. We delimit the problem's fixed-parameter tractability by identifying sufficient and necessary conditions on the structure of prefixes of first-order logic formulas. Our main result is the following meta-theorem: for every graph property P expressible by a first order-logic formula ϕ∈Σ_3, that is, of the form ϕ=∃x_1∃x_2⋯∃x_r ∀y_1∀y_2⋯∀y_s ∃z_1∃z_2⋯∃z_t ψ, where ψis a quantifier-free first-order formula, checking whether the elimination distance of a graph to P does not exceed k, is fixed-parameter tractable parameterized by k. Properties of graphs expressible by formulas from Σ_3 include being of bounded degree, excluding a forbidden subgraph, or containing a bounded dominating set. We complement this theorem by showing that such a general statement does not hold for formulas with even slightly more expressive prefix structure: there are formulas ϕ∈Π_3, for which computing elimination distance is W[2]-hard.
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