A Characterization of X kX Intersection Graphs

For two xed graphs X and Y the X Y intersection graph of a graph G is a graph whose vertices are induced subgraphs of G isomor phic to Y and where two vertices are adjacent if their intersection in G contains an induced subgraph isomorphic to X A conformal k graph is a simple hypergraph whose hyperedges are exactly k cliques in the section of the hypergraph Let kX denote the disjoint union of k copies of X We show that for any integer k and connected graph X with no bipartite blocks the family of X kX intersection graphs coincides with the family of line graphs of conformal k graphs On the other hand we obtain a Ramsey type result on vertex splitting and use it to prove that for any connected bipartite graph X with at least two vertices the family of X kX intersection graphs is strictly contained in the family of line graphs of conformal k graphs This work was partially supported by an Earmarked Research Grant from the Research Grants Council of Hong Kong X kX intersection graphs