On the Recognition of General Partition Graphs

A graph G is a general partition graph if there is some set S and an assignment of non-empty subsets S x ⊆ S to the vertices of G such that two vertices x and y are adjacent if and only if S x ∩ S y ≠ O and for every maximal independent set M the set {S m | m ∈ M} is a partition of S. For every minor closed family of graphs there exists a polynomial time algorithm that checks if an element of the family is a general partition graph.