Proof: The proof is by induction on the vertex number and uses the well-known fact that every chordal graph has some so-called simplicial vertex---a vertex with complete neighborhood. The details are left to the reader.

Proof of the [BG81]-Theorem: Let T be a maximum spanning tree of C_M(G). We define T_x := T[{CV(T)/ xC}] for every vertex x of G. Since T has the inclusion-property by the Proposition, each T_x is connected.

If S is contractible to T, then this representation can be extended to some representation by subtrees of S in an obvious way.

Conversely, assume G is the intersection graph of subtrees (T_x/ xV(G)) of some tree T. We may assume that T is a clique tree, i.e. the graphs a* are distinct cliques of G. In other words, T is a spanning tree of C_M(G). If T were not maximum, then, by the Proposition, the inclusion property would be violated. Then C₀ C_i would not include C₀ C_t for some C_i lying between C₀ and C_t on T. Then there were some vertex x occuring in C₀ and C_t, but not in C_i, a contradiction to T_x being connected.