Capturing Logarithmic Space and Polynomial Time on Chordal Claw-Free Graphs

TL;DR

This paper demonstrates that chordal claw-free graphs can be canonized and tested for isomorphism in logarithmic space using an extended logic, and that fixed-point logic with counting captures polynomial time on this class.

Contribution

It introduces a logspace-definable canonization for chordal claw-free graphs and shows fixed-point logic with counting captures polynomial time on this class.

Findings

Logarithmic-space canonization algorithm exists for chordal claw-free graphs.

Logarithmic-space isomorphism testing is possible for this graph class.

Fixed-point logic with counting captures polynomial time on chordal claw-free graphs.

Abstract

We show that the class of chordal claw-free graphs admits LREC$_=$-definable canonization. LREC$_=$ is a logic that extends first-order logic with counting by an operator that allows it to formalize a limited form of recursion. This operator can be evaluated in logarithmic space. It follows that there exists a logarithmic-space canonization algorithm, and therefore a logarithmic-space isomorphism test, for the class of chordal claw-free graphs. As a further consequence, LREC$_=$ captures logarithmic space on this graph class. Since LREC$_=$ is contained in fixed-point logic with counting, we also obtain that fixed-point logic with counting captures polynomial time on the class of chordal claw-free graphs.