Intersection properties of graphs (Q1175421)

Let \(P\) denote any graph-theoretic property that implies connectedness. Then a graph \(F\) is in \(\bigcap P\) if and only if there is some graph \(G\) satisfying \(P\) and some family of connected subgraphs of \(G\) such that \(F\) is isomorphic to the intersection graph of the family. A general method is described how suitably expressed information about \(P\) can be mechanically transformed into information about \(\bigcap P\). For instance, the necessity of the asteroidal triple characterization of \textit{C. G. Lekkerkerker} and \textit{J. C. Boland} [Fundam. Math. 51, 45--64 (1962; Zbl 0105.17501)] for interval graphs follows from a simple description of paths. The method is particularily useful when \(P\) describes some class of trees closed under contraction.