2-connected claw-free chordal graphs are cycle extendable

A cycle $C$ of length $k$ in graph $G$ is extendable if there is another cycle $C^{\mathrm{\prime }}$ in $G$ with $V(C)\subset V(C^{\mathrm{\prime }})$ and length $k+1$. A graph is cycle extendable if every non-Hamiltonian cycle is extendable. In 1990 Hendry conjectured that any Hamiltonian chordal graph (a Hamiltonian graph with no induced cycle of length greater than three) is cycle extendable, and this conjecture has been verified for Hamiltonian chordal graphs which are interval graphs, planar graphs, and split graphs. We prove that any 2-connected claw-free chordal graph is cycle extendable.