A characterization and hereditary properties for partition graphs

Abstract

A general partition graph is an intersection graph G on a set S so that for every maximal independent set M of vertices in G, the subsets assigned to the vertices in M partition S. It is shown that such graphs are characterized by there existing a clique cover of G so that every maximal independent set has a vertex from each clique in the cover. A process is described showing how to construct a partition graph with certain prescribed graph theoretic invariants starting with an arbitrary graph with similar invariants. Hereditary properties for various graph products are examined. It is also shown that partition graphs are preserved by removal of any vertex whose closed neighborhood properly contains the closed neighborhood of some other vertex.