Capturing Logarithmic Space and Polynomial Time on Chordal Claw-Free Graphs (1802.10331v4)

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.