Abstract
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. © 2012 Wiley Periodicals, Inc. (Contract grant sponsors: Faculty Research Committee, Black Hills State University (M. D. B.); National Security Agency under Award No. H98230-10-1-0363 (D. B. W.).)
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