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 Sx ⊆ S to the vertices of G such that two vertices x and y are adjacent if and only if Sx ∩ Sy ≠ Ø and for every maximal independent set M the set {Sm | 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. The triangle condition says that for every maximal independent set M and for every edge (x,y) with x,y ∉ M there is a vertex m ∈ M such that {x,y,m} induces a triangle in G. It is known that the triangle condition is necessary for a graph to be a general partition graph (but in general not sufficient). We show that for AT-free graphs this condition is also sufficient and this leads to an efficient algorithm that demonstrates whether or not an AT-free graph is a general partition graph. We show that the triangle condition can be checked in polynomial time for planar graphs and circle graphs. It is unknown if the triangle condition is also a sufficient condition for planar graphs to be a general partition graph. For circle graphs we show that the triangle condition is not sufficient.