Capturing Polynomial Time using Modular Decomposition

TL;DR

This paper introduces the Modular Decomposition Theorem, linking graph decomposition to logical definability, and shows fixed point logic with counting captures polynomial time on permutation graphs.

Contribution

It presents a new theorem connecting modular decomposition to definability, enabling polynomial time capture results for specific graph classes.

Findings

Modular decomposition is definable in symmetric transitive closure logic with counting.

The modular decomposition tree can be computed in logarithmic space.

Fixed point logic with counting captures polynomial time on permutation graphs.

Abstract

The question of whether there is a logic that captures polynomial time is one of the main open problems in descriptive complexity theory and database theory. In 2010 Grohe showed that fixed point logic with counting captures polynomial time on all classes of graphs with excluded minors. We now consider classes of graphs with excluded induced subgraphs. For such graph classes, an effective graph decomposition, called modular decomposition, was introduced by Gallai in 1976. The graphs that are non-decomposable with respect to modular decomposition are called prime. We present a tool, the Modular Decomposition Theorem, that reduces (definable) canonization of a graph class C to (definable) canonization of the class of prime graphs of C that are colored with binary relations on a linearly ordered set. By an application of the Modular Decomposition Theorem, we show that fixed point logic with counting captures polynomial time on the class of permutation graphs. Within the proof of the Modular Decomposition Theorem, we show that the modular decomposition of a graph is definable in symmetric transitive closure logic with counting. We obtain that the modular decomposition tree is computable in logarithmic space. It follows that cograph recognition and cograph canonization is computable in logarithmic space.