The A4-structure of a graph

We define the A4-structure of a graph G to be the 4-uniform hypergraph on the vertex set of G whose edges are the vertex subsets inducing 2K2, C4, or P4. We show that perfection of a graph is determined by its A4-structure. We relate the A4structure to the canonical decomposition of a graph as defined by Tyshkevich [Discrete Mathematics 220 (2000) 201-238]; for example, a graph is indecomposable if and only if its A4-structure is connected. We also characterize the graphs having the same A4-structure as a split graph.