Complete structural description of 5-holed graphs

Determine a complete structural description of 5-holed graphs, whose holes all have length five.

Background

A graph is 5-holed if every hole—an induced cycle of length at least four—has length five. The paper places these graphs within the broader study of monoholed graphs and notes that several subclasses have already been investigated. A complete structural description would clarify the organization of this graph class and could support further algorithmic results.

References

However, a complete description of $5$-holed graphs remains unknown.

— Strong isometric path complexity of graphs: Asymptotic minors, restricted holes, and graph operations  (2501.10828 - Chakraborty et al., 18 Jan 2025) in Section 1, subsection “Monoholed graphs and even-hole free graphs”