A cycle $C$ in a graph is extendable if there exists a cycle $C'$ such that $V(C) \subseteq V(C')$ and $|V(C')|$ = $|V(C)|$ + 1. A graph is cycle extendable if every non-Hamiltonian cycle in the graph is extendable. An open question is whether or not every Hamiltonian chordal graph is cycle extendable. We show that Hamiltonian spider intersection graphs, a subclass of Hamiltonian chordal graphs, are cycle extendable. Our result generalizes known results on cycle extendability in interval graphs and split graphs.

A cycle C in a graph is extendable if there exists a cycle ${C^{\prime}}$ such that $V(C) \subseteq V({C^{\prime}})$ and $\mid V({C^{\prime}}) \mid = \mid V(C) \mid + 1$. A graph is cycle extendable if every non‐Hamiltonian cycle in the graph is extendable. An unresolved question is whether or not every Hamiltonian chordal graph is cycle extendable. We show that Hamiltonian graphs in classes such as interval, split, and in some subclasses of strongly chordal graphs, are cycle extendable. We also address efficiently finding a Hamilton cycle in some cases. A unifying theme to our approach is the use of appropriate vertex elimination orders.