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 A₄‐structure to the canonical decomposition of a graph as defined by Tyshkevich [Discrete Math 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.