Stable 2-pairs and (X, Y)-intersection graphs

Abstract Given two fixed graphs X and Y , the (X,Y) -intersection graph of a graph G is a graph where 1. each vertex corresponds to a distinct induced subgraph in G isomorphic to Y , and 2. two vertices are adjacent iff the intersection of their corresponding subgraphs contains an induced subgraph isomorphic to X . This notion generalizes the classical concept of line graphs since the (K 1 ,K 2 ) -intersection graph of a graph G is precisely the line graph of G . Let L ( B ) ( L ( B ∗ ) , respectively) denote the family of line graphs of bipartite graphs (bipartite multigraphs, respectively), and refer to a pair (X,Y) as a 2-pair if Y contains exactly two induced subgraphs isomorphic to X . Then L ( B ) and L ( B ∗ ) , respectively, are the smallest families amongst the families of (X,Y) -intersection graphs defined by so called hereditary 2-pairs and hereditary non-compact 2-pairs. Furthermore, they can be characterized through forbidden induced subgraphs. With this motivation, we investigate the properties of a 2-pair (X,Y) for which the family of (X,Y) -intersection graphs coincides with L ( B ) (or L ( B ∗ ) ). For this purpose, we introduce a notion of stability of a 2-pair and obtain the desired characterization for such stable 2-pairs. An interesting aspect of the characterization is that it is based on a graph determined by the structure of (X,Y) .