Kolaitis and Kopparty have shown that for any first-order formula with
parity quantifiers over the language of graphs there is a family of
multi-variate polynomials of constant-degree that agree with the
formula on all but a $2^{-\Omega(n)}$-fraction of the graphs with $n$
vertices. The proof yields a bound on the ... more >>>

We show that if $\mathcal C$ is a class of graphs which is
"nowhere dense" then first-order model-checking is
fixed-parameter tractable on $\mathcal C$. As all graph classes which exclude a fixed minor, or are of bounded local tree-width or locally exclude a minor are nowhere dense, this generalises algorithmic ... more >>>