Logical limit laws for minor-closed classes of graphs

Abstract

Let 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 connected graphs in G on n vertices, and the convergence law in MSO holds if we draw uniformly at random from all graphs in 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 the same length ≈ 5.39 · 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.

Logical laws for short existential monadic second-order sentences about graphs

“…that express vertices of a graph, quantifiers ∀, ∃ and parentheses (, ). Monadic second order, or MSO, sentences (see [2,4,8,9]) are built of the above symbols of the first order language, as well as the variables X, Y, X 1 , . .…”

“…An important tool that allows exploiting combinatorial techniques for proving results in logic is a class of games called Ehrenfeucht games [2,4,5,6,7,8,10]. We consider the following modification of this game (which is also called the 1,2-Fagin game, see [4]).…”

Short Monadic Second Order Sentences about Sparse Random Graphs

“…In the case of monadic second order logic, players can choose subsets as well. Similarly, at the ν-th round (1 ≤ ν ≤ k) of the game EHR MSO (A, B, k) [6,9,20] Spoiler chooses one of the A, B. Say, he chooses A.…”

“…that express vertices of a graph, quantifiers ∀, ∃ and parentheses (, ). Monadic second order, or MSO, formulae (see [9], [16]) are built of the above symbols of the first order language, as well as the variables X, Y, X 1 , . .…”

The first order convergence law fails for random perfect graphs

“…Several restricted classes of graphs have been studied with respect to the limiting behaviour of FO properties, and usually one proves either a zero-one law (that is, every FO property has limiting probability ∈ {0, 1}) or a convergence law (that is, every FO property has a limiting probability). For instance, a zero-one law has been proved for trees [6] and for graphs not containing a clique of fixed size [4], while a convergence law has been proved for d-regular graphs for fixed d [5], and for forests and planar graphs [2].…”