On the representation of triangulation graphs in trees (Q794674)

A graph G is said to be tree-representable if there exist a tree T and a family \(T_ i\) (\(i\in I)\) of subtrees of T such that G is isomorphic to the intersection graph of this family. It has been shown independently by several authors that if G is a finite graph then G is tree-representable if and only if G is chordal, i.e., no cycle of length at least 4 is an induced subgraph of G. It is not true that every infinite chordal graph is tree-representable. The author's main result is a characterization of all tree-representable graphs, given by means of simplicial decompositions.