Logical properties of random graphs from small addable classes

TL;DR

This paper proves zero-one and convergence laws for monadic second-order logic on various classes of random graphs, including planar graphs and graphs with bounded tree-width, revealing logical properties of these graph classes.

Contribution

It establishes zero-one and convergence laws for MSO logic on several important graph classes, extending understanding of logical properties in random graph theory.

Findings

MSO obeys a zero-one law on connected planar graphs

MSO obeys a zero-one law on connected graphs with bounded tree-width

Dropping connectivity leads to convergence laws but not zero-one laws

Abstract

We establish zero-one laws and convergence laws for monadic second-order logic (MSO) (and, a fortiori, first-order logic) on a number of interesting graph classes. In particular, we show that MSO obeys a zero-one law on the class of connected planar graphs, the class of connected graphs of tree-width at most $k$ and the class of connected graphs excluding the $k$-clique as a minor. In each of these cases, dropping the connectivity requirement leads to a class where the zero-one law fails but a convergence law for MSO still holds.