Let $G$ be a finite simple graph and let $C(G)$ be the flag complex associated to $G$ (the set of vertices of $C(G)$ is the vertex set of $G$ and the set of all cliques of $G$ are its simplexes). 


Are there characterizations of contractibility of $C(G)$ ONLY in terms of the graph theoretical properties of $G$?



A flag complex is contractible iff the underlying graph is....?
It is known that every induced subcomplex of the flag complex of a graph is contractible iff the graph is chordal (no induced cycles of length 4 or more). I doubt a necessary and sufficient condition that is purely graph theoretic for contractibility of just the flag complex is possible because the barycentric subdivision of any simplicial complex is a flag complex. 


