Capturing Polynomial Time on Interval Graphs

TL;DR

This paper characterizes all polynomial-time computable queries on interval graphs using fixed-point logic with counting, establishing a logical framework for understanding computational complexity on this graph class.

Contribution

It proves that fixed-point logic with counting captures polynomial time on unordered interval graphs, a novel result for a class defined by forbidden induced subgraphs, using a new modular decomposition approach.

Findings

Fixed-point logic with counting captures polynomial time on interval graphs.

A canonical form of interval graphs is defined via a new modular decomposition.

FP+C does not capture polynomial time on chordal or incomparability graphs.

Abstract

We prove a characterization of all polynomial-time computable queries on the class of interval graphs by sentences of fixed-point logic with counting. More precisely, it is shown that on the class of unordered interval graphs, any query is polynomial-time computable if and only if it is definable in fixed-point logic with counting. This result is one of the first establishing the capturing of polynomial time on a graph class which is defined by forbidden induced subgraphs. For this, we define a canonical form of interval graphs using a type of modular decomposition, which is different from the method of tree decomposition that is used in most known capturing results for other graph classes, specifically those defined by forbidden minors. The method might also be of independent interest for its conceptual simplicity. Furthermore, it is shown that fixed-point logic with counting is not expressive enough to capture polynomial time on the classes of chordal graphs or incomparability graphs.