  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.

Logic in Computer Science;Discrete Mathematics