Courcelle's Theorem Without Logic

TL;DR

This paper generalizes Courcelle's Theorem by replacing the logical definability condition with a combinatorial hypothesis based on connection matrices, broadening its applicability to graph properties.

Contribution

It introduces a combinatorial framework that extends Courcelle's Theorem without relying on Monadic Second Order Logic, inspired by Lovász's work with connection matrices.

Findings

Provides a generalized version of Courcelle's Theorem

Replaces logical definability with combinatorial conditions

Broadens the scope of efficiently checkable graph properties

Abstract

Courcelle's Theorem states that on graphs $G$ of tree-width at most $k$ with a given tree-decomposition of size $t(G)$, graph properties $\mathcal{P}$ definable in Monadic Second Order Logic can be checked in linear time in the size of $t(G)$. Inspired by L. Lov\'asz' work using connection matrices instead of logic, we give a generalized version of Courcelle's theorem which replaces the definability hypothesis by a purely combinatorial hypothesis using a generalization of connection matrices.