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

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 (K₁,K₂)-intersection graph of a graph G is precisely the line graph of G.

Let (, 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 and , 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 (or ). 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).