3 problems

Chordal graph representation conjecture. Let $H$ be a chordal graph. There are chordal graphs $G$ and $G'$ such that $H$ is isomorphic to both $C(G)$ and $C_R(G')$.

Reduced clique graph recognition conjecture. There is a polynomial-time algorithm for deciding whether a given graph is isomorphic to $C_R(G)$ for some chordal graph $G$.

Seven-hole conjecture. There is no chordal graph $G$ such that $C_R(G)$ contains an induced cycle with seven or more vertices.