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

1 Stable 2-Pairs and (X; Y)-Intersection Graphs 2 Abstract Given two xed 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 that is iso-morphic to Y , and 2. two vertices are adjacent ii the intersection of their corresponding sub-graphs 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 deened 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 for such stable 2-pairs, we obtain the desired characterizaton of a 2-pair (X; Y) for which the family of (X; Y)-intersection graphs coincides with L(B) (or L(B)). An interesting aspect of the characterization is that it is based on a graph determined by the structure of (X; Y).