Logical limit laws for minor-closed classes of graphs

Abstract

Let $\mathcal G$ be an addable, minor-closed class of graphs. We prove that the zero-one law holds in monadic second-order logic (MSO) for the random graph drawn uniformly at random from all {\em connected} graphs in $\mathcal G$ on $n$ vertices, and the convergence law in MSO holds if we draw uniformly at random from all graphs in $\mathcal G$ on $n$ vertices. We also prove analogues of these results for the class of graphs embeddable on a fixed surface, provided we restrict attention to first order logic (FO). Moreover, the limiting probability that a given FO sentence is satisfied is independent of the surface $S$. We also prove that the closure of the set of limiting probabilities is always the finite union of at least two disjoint intervals, and that it is the same for FO and MSO. For the classes of forests and planar graphs we are able to determine the closure of the set of limiting probabilities precisely. For planar graphs it consists of exactly 108 intervals, each of length $\approx 5\cdot 10^{-6}$. Finally, we analyse examples of non-addable classes where the behaviour is quite different. For instance, the zero-one law does not hold for the random caterpillar on $n$ vertices, even in FO.